Algebra I: Linear Functions

Okay, here is a comprehensive, deeply structured lesson on Algebra I: Linear Functions, designed for high school students with a focus on depth, clarity, and real-world applications.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're planning a road trip. You know how much gas your car holds, and you know (roughly) how many miles you get per gallon. How do you figure out if you can make it to the next town without stopping for gas? Or, perhaps you're starting a small business selling custom t-shirts. You have fixed costs like equipment and rent, and variable costs like the cost of each blank t-shirt. How do you determine how many shirts you need to sell to break even, or to make a profit? These scenarios, and countless others, can be solved using the power of linear functions. Linear functions provide a simple yet powerful way to model relationships where one quantity changes at a constant rate relative to another. Think of them as the foundation for understanding more complex mathematical models used in everything from predicting stock prices to designing bridges.

### 1.2 Why This Matters

Linear functions are not just abstract mathematical concepts; they are fundamental tools for understanding and modeling the world around us. They are the building blocks upon which more advanced mathematical concepts, such as calculus and differential equations, are built. Understanding linear functions is crucial for success in higher-level mathematics courses. Beyond academics, linear functions have direct applications in various careers. Economists use them to model supply and demand, engineers use them to design structures, and data scientists use them to analyze trends. Mastering linear functions provides you with a valuable skillset applicable to a wide range of fields, enhancing your problem-solving abilities and critical thinking skills. Moreover, understanding linear relationships can help you make informed decisions in your personal life, from budgeting your finances to analyzing data presented in the news.

### 1.3 Learning Journey Preview

In this lesson, we will embark on a journey to explore the world of linear functions. We'll start by defining what a linear function is and how it differs from other types of functions. We'll then delve into the various ways to represent linear functions: graphically, algebraically (using equations), and numerically (using tables). We'll learn how to identify key features of linear functions, such as slope and y-intercept, and understand their significance. We'll then explore how to write equations of linear functions given different information, such as two points or a point and a slope. We will analyze real-world scenarios that can be modeled by linear functions, and learn how to interpret the results in context. Finally, we'll examine systems of linear equations and how to solve them graphically and algebraically. Each concept will build upon the previous one, providing you with a solid foundation in linear functions and their applications.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

Explain the definition of a linear function and differentiate it from non-linear functions using examples and graphical representations.
Identify and interpret the slope and y-intercept of a linear function in various forms (equation, graph, table, real-world context).
Write the equation of a linear function in slope-intercept form, point-slope form, and standard form, given different information (two points, a point and a slope, a graph).
Apply linear functions to model real-world scenarios, analyze the relationships between variables, and make predictions based on the model.
Solve systems of linear equations graphically and algebraically (substitution, elimination) and interpret the solution in the context of the problem.
Analyze the effect of changing the slope or y-intercept of a linear function on its graph and real-world interpretation.
Evaluate the appropriateness of a linear model for a given data set and justify your reasoning.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into linear functions, it's essential to have a solid grasp of the following concepts:

Variables and Expressions: Understanding what variables represent and how to manipulate algebraic expressions.
Order of Operations: Knowing the correct order to perform mathematical operations (PEMDAS/BODMAS).
Coordinate Plane: Familiarity with the Cartesian coordinate plane, including plotting points and identifying quadrants.
Solving Basic Equations: Ability to solve simple algebraic equations for a single variable (e.g., x + 3 = 7, 2x = 10).
Basic Arithmetic: Proficiency with addition, subtraction, multiplication, and division of real numbers (including fractions and decimals).
Number Systems: Understanding the difference between integers, rational numbers, and real numbers.

Quick Review:

Coordinate Plane: The coordinate plane is formed by two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical). Points are located using ordered pairs (x, y).
Solving Equations: The goal is to isolate the variable on one side of the equation by performing the same operations on both sides. Remember to use inverse operations (addition/subtraction, multiplication/division).

If you need a refresher on any of these topics, you can find helpful resources online (Khan Academy, YouTube tutorials) or in your previous math textbooks. Mastering these fundamentals will make learning about linear functions much easier.

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## 4. MAIN CONTENT

### 4.1 Definition of a Linear Function

Overview: A linear function is a mathematical relationship between two variables that produces a straight line when graphed. It represents a constant rate of change between the input (independent variable) and the output (dependent variable).

The Core Concept: A linear function can be defined as a function whose graph is a straight line. This means that for every equal change in the input variable (usually denoted as 'x'), there is a corresponding equal change in the output variable (usually denoted as 'y'). The relationship is characterized by a constant rate of change, which is known as the slope. Linear functions are often expressed in the form y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept (the point where the line crosses the y-axis).

The key characteristics that distinguish a linear function from a non-linear function are:

1. Constant Rate of Change (Slope): The ratio of the change in y to the change in x is constant throughout the function.
2. Straight Line Graph: When plotted on a coordinate plane, the points of a linear function form a straight line.
3. Equation Form: The function can be expressed in a standard algebraic form (e.g., y = mx + b, Ax + By = C).

Non-linear functions, on the other hand, have a variable rate of change, resulting in a curved graph. Examples of non-linear functions include quadratic functions (y = x^2), exponential functions (y = 2^x), and trigonometric functions (y = sin(x)). The difference lies in how the output (y) changes as the input (x) changes; linear functions have a steady, constant change, while non-linear functions have a changing rate of change.

Concrete Examples:

Example 1: The Cost of Text Messages
Setup: A cell phone plan charges a flat monthly fee of $10, plus $0.05 for each text message sent.
Process: Let 'x' represent the number of text messages sent in a month, and let 'y' represent the total monthly cost. The total cost can be expressed as y = 0.05x + 10. If you send 100 text messages, the cost is y = 0.05(100) + 10 = $15. If you send 200 text messages, the cost is y = 0.05(200) + 10 = $20. The cost increases by $5 for every 100 text messages, demonstrating a constant rate of change.
Result: The equation y = 0.05x + 10 represents a linear function because the cost increases at a constant rate of $0.05 per text message.
Why this matters: This illustrates how linear functions can model real-world pricing scenarios where there's a fixed cost and a variable cost that increases linearly with usage.

Example 2: Distance Traveled at a Constant Speed
Setup: A car travels at a constant speed of 60 miles per hour.
Process: Let 'x' represent the time in hours, and let 'y' represent the distance traveled in miles. The distance traveled can be expressed as y = 60x. After 1 hour, the car has traveled 60 miles. After 2 hours, the car has traveled 120 miles. The distance increases by 60 miles for every additional hour of travel.
Result: The equation y = 60x represents a linear function because the distance increases at a constant rate of 60 miles per hour.
Why this matters: This demonstrates how linear functions can model situations involving constant motion or rates, which are common in physics and engineering.

Analogies & Mental Models:

"Think of it like climbing a staircase." Each step you take (change in x) raises you by the same amount (change in y). The slope is like the steepness of the staircase. A steeper staircase (larger slope) means you rise more for each step.
How the analogy maps to the concept: The consistent height of each step corresponds to the constant rate of change in a linear function.
Where the analogy breaks down (limitations): A staircase is discrete (you can only be on a step), while a linear function is continuous (you can have values in between integer values of x).

Common Misconceptions:

❌ Students often think that any equation with 'x' and 'y' is a linear function.
✓ Actually, the key is that the relationship between 'x' and 'y' must be a constant rate of change, resulting in a straight line graph. Equations with terms like x^2, sqrt(x), or xy are non-linear.
Why this confusion happens: Students may focus on the presence of variables without understanding the underlying relationship between them.

Visual Description:

Imagine a straight line drawn on a graph. This line represents a linear function. The line can be slanted upwards or downwards, or it can be horizontal. The steepness of the line is determined by the slope. A line sloping upwards from left to right has a positive slope, while a line sloping downwards has a negative slope. A horizontal line has a slope of zero. The point where the line crosses the y-axis is the y-intercept.

Practice Check:

Which of the following equations represents a linear function?
a) y = 2x + 5
b) y = x^2 - 3
c) y = 1/x
d) y = sqrt(x)

Answer: a) y = 2x + 5. This equation is in the form y = mx + b, where m = 2 and b = 5, representing a constant rate of change.

Connection to Other Sections:

This section provides the fundamental definition of a linear function, which is essential for understanding all subsequent sections. We will build upon this definition to explore different representations of linear functions, how to calculate slope and y-intercept, and how to apply linear functions to real-world problems.

### 4.2 Slope of a Linear Function

Overview: The slope of a linear function measures its steepness and direction. It represents the constant rate of change between the dependent and independent variables.

The Core Concept: The slope of a line is a numerical value that describes both the direction and the steepness of the line. It is often referred to as "rise over run," where "rise" represents the vertical change (change in y) and "run" represents the horizontal change (change in x) between any two points on the line. Mathematically, the slope (m) is calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are any two distinct points on the line.

A positive slope indicates that the line is increasing (sloping upwards from left to right), while a negative slope indicates that the line is decreasing (sloping downwards from left to right). A slope of zero indicates a horizontal line, and an undefined slope indicates a vertical line.

The magnitude of the slope also indicates the steepness of the line. A larger absolute value of the slope indicates a steeper line, while a smaller absolute value indicates a less steep line. For example, a line with a slope of 2 is steeper than a line with a slope of 1.

Understanding the slope is crucial for interpreting the relationship between the variables in a linear function. It tells us how much the dependent variable changes for every unit change in the independent variable.

Concrete Examples:

Example 1: Calculating Slope from Two Points
Setup: Consider a line passing through the points (1, 3) and (4, 9).
Process: Using the slope formula, m = (9 - 3) / (4 - 1) = 6 / 3 = 2.
Result: The slope of the line is 2. This means that for every 1 unit increase in x, the value of y increases by 2 units.
Why this matters: This demonstrates how to calculate the slope using the slope formula, given any two points on the line.

Example 2: Interpreting Slope in a Real-World Context
Setup: A graph shows the relationship between the number of hours worked (x) and the amount earned (y) at an hourly wage. The line passes through the points (0, 0) and (5, 75).
Process: The slope is m = (75 - 0) / (5 - 0) = 75 / 5 = 15.
Result: The slope of the line is 15. This means that the hourly wage is $15 per hour.
Why this matters: This demonstrates how the slope can be interpreted as a rate of change in a real-world scenario, such as an hourly wage.

Analogies & Mental Models:

"Think of slope as the grade of a hill." A steeper hill has a larger slope, while a gentle hill has a smaller slope. A flat ground has a slope of zero.
How the analogy maps to the concept: The steepness of the hill corresponds to the rate of change in the linear function.
Where the analogy breaks down (limitations): Hills can have variable slopes, while linear functions have a constant slope.

Common Misconceptions:

❌ Students often confuse the order of subtraction in the slope formula, calculating (x2 - x1) / (y2 - y1) instead of (y2 - y1) / (x2 - x1).
✓ Actually, the change in y (rise) must be divided by the change in x (run) to correctly calculate the slope.
Why this confusion happens: Memorizing the formula without understanding the concept of rise over run can lead to this error.

Visual Description:

Imagine two points on a line. Draw a vertical line segment connecting the two points (the "rise"). Then, draw a horizontal line segment connecting the two points (the "run"). The slope is the ratio of the length of the rise to the length of the run. If the line goes up as you move from left to right, the rise is positive. If the line goes down as you move from left to right, the rise is negative.

Practice Check:

A line passes through the points (2, 5) and (6, 13). What is the slope of the line?

Answer: m = (13 - 5) / (6 - 2) = 8 / 4 = 2. The slope of the line is 2.

Connection to Other Sections:

Understanding the slope is essential for writing equations of linear functions in different forms. It also allows us to interpret the meaning of linear functions in real-world contexts and make predictions based on the model.

### 4.3 Y-Intercept of a Linear Function

Overview: The y-intercept is the point where the line crosses the y-axis. It represents the value of the dependent variable when the independent variable is zero.

The Core Concept: The y-intercept of a linear function is the point where the line intersects the y-axis. At this point, the x-coordinate is always zero. Therefore, the y-intercept is represented by the ordered pair (0, b), where 'b' is the y-value.

The y-intercept is a crucial feature of a linear function because it represents the initial value or starting point of the relationship between the variables. In the equation y = mx + b, 'b' directly represents the y-intercept.

Understanding the y-intercept allows us to interpret the meaning of the linear function in a real-world context. It often represents a fixed cost, an initial amount, or a starting value.

Concrete Examples:

Example 1: Identifying the Y-Intercept from an Equation
Setup: Consider the equation y = 3x + 2.
Process: The equation is in slope-intercept form (y = mx + b), where b = 2.
Result: The y-intercept is (0, 2).
Why this matters: This demonstrates how to directly identify the y-intercept from the slope-intercept form of a linear equation.

Example 2: Interpreting the Y-Intercept in a Real-World Context
Setup: A graph shows the amount of water in a tank (y) over time (x). The line intersects the y-axis at the point (0, 50).
Process: The y-intercept is (0, 50).
Result: This means that the tank initially contained 50 gallons of water.
Why this matters: This demonstrates how the y-intercept can be interpreted as an initial value in a real-world scenario.

Analogies & Mental Models:

"Think of the y-intercept as the starting point of a race." It's where you begin before you start moving.
How the analogy maps to the concept: The starting point corresponds to the initial value of the linear function.
Where the analogy breaks down (limitations): Races have a definite end, while linear functions can continue indefinitely.

Common Misconceptions:

❌ Students often confuse the y-intercept with the x-intercept.
✓ Actually, the y-intercept is the point where the line crosses the y-axis (x = 0), while the x-intercept is the point where the line crosses the x-axis (y = 0).
Why this confusion happens: Students may not fully understand the definitions of the x and y axes.

Visual Description:

Imagine a line drawn on a graph. The y-intercept is the point where the line crosses the vertical y-axis. It's the height of the line when it first starts (at x=0).

Practice Check:

What is the y-intercept of the line represented by the equation y = -2x + 7?

Answer: The y-intercept is (0, 7).

Connection to Other Sections:

Understanding the y-intercept, along with the slope, is essential for writing and interpreting linear functions. It allows us to fully describe the relationship between the variables and make accurate predictions.

### 4.4 Slope-Intercept Form (y = mx + b)

Overview: Slope-intercept form is a standard way to represent a linear equation, making it easy to identify the slope and y-intercept.

The Core Concept: The slope-intercept form of a linear equation is given by:

y = mx + b

where:

'y' is the dependent variable
'x' is the independent variable
'm' is the slope of the line
'b' is the y-intercept of the line (the point where the line crosses the y-axis)

This form is particularly useful because it directly reveals the slope and y-intercept, which are key features of a linear function. Given an equation in slope-intercept form, you can immediately identify the slope and y-intercept and use them to graph the line or interpret its meaning in a real-world context.

To convert an equation into slope-intercept form, you need to isolate 'y' on one side of the equation. This typically involves using algebraic manipulations such as adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Concrete Examples:

Example 1: Identifying Slope and Y-Intercept from an Equation
Setup: Consider the equation y = -4x + 6.
Process: The equation is already in slope-intercept form.
Result: The slope is -4, and the y-intercept is (0, 6).
Why this matters: This demonstrates how to directly identify the slope and y-intercept from an equation in slope-intercept form.

Example 2: Converting an Equation to Slope-Intercept Form
Setup: Consider the equation 2x + 3y = 9.
Process: To convert this to slope-intercept form, we need to isolate 'y':
Subtract 2x from both sides: 3y = -2x + 9
Divide both sides by 3: y = (-2/3)x + 3
Result: The slope-intercept form of the equation is y = (-2/3)x + 3. The slope is -2/3, and the y-intercept is (0, 3).
Why this matters: This demonstrates how to use algebraic manipulations to convert an equation to slope-intercept form.

Analogies & Mental Models:

"Think of slope-intercept form as a recipe for a line." The slope tells you how much to change the y-value for each change in the x-value, and the y-intercept tells you where to start.
How the analogy maps to the concept: The slope and y-intercept are the key ingredients for defining the line.
Where the analogy breaks down (limitations): A recipe produces a physical object, while slope-intercept form represents a mathematical relationship.

Common Misconceptions:

❌ Students often forget to isolate 'y' when converting an equation to slope-intercept form.
✓ Actually, 'y' must be completely isolated on one side of the equation to correctly identify the slope and y-intercept.
Why this confusion happens: Students may focus on manipulating the equation without understanding the goal of isolating 'y'.

Visual Description:

Imagine a line on a graph. The slope-intercept form tells you two things: where the line crosses the y-axis (the y-intercept) and how steep the line is (the slope). You can start at the y-intercept and then use the slope to find other points on the line.

Practice Check:

What is the slope and y-intercept of the line represented by the equation 4x - 2y = 8?

Answer: First, convert to slope-intercept form: -2y = -4x + 8, y = 2x - 4. The slope is 2, and the y-intercept is (0, -4).

Connection to Other Sections:

Slope-intercept form is a fundamental concept that is used throughout the study of linear functions. It is essential for graphing lines, writing equations of lines, and solving systems of linear equations.

### 4.5 Point-Slope Form (y - y1 = m(x - x1))

Overview: Point-slope form is another way to represent a linear equation, particularly useful when you know a point on the line and the slope.

The Core Concept: The point-slope form of a linear equation is given by:

y - y1 = m(x - x1)

where:

'y' and 'x' are the variables
'm' is the slope of the line
(x1, y1) is a known point on the line

This form is particularly useful when you are given a point on the line and the slope, and you need to write the equation of the line. It avoids the need to calculate the y-intercept directly.

To use the point-slope form, simply substitute the given point (x1, y1) and the slope 'm' into the equation. Then, you can simplify the equation to obtain the slope-intercept form (y = mx + b) if desired.

Concrete Examples:

Example 1: Writing an Equation Given a Point and Slope
Setup: A line passes through the point (2, 5) and has a slope of 3.
Process: Using the point-slope form, we have: y - 5 = 3(x - 2). Simplifying, we get y - 5 = 3x - 6, and then y = 3x - 1.
Result: The equation of the line in slope-intercept form is y = 3x - 1.
Why this matters: This demonstrates how to use the point-slope form to write the equation of a line when given a point and slope.

Example 2: Using Point-Slope Form in a Real-World Context
Setup: The cost of renting a car is $20 plus $0.10 per mile. You know that renting the car and driving 100 miles costs $30.
Process: We can consider the point (100, 30) as a known point on the line. The slope is $0.10 per mile. Using the point-slope form, we have: y - 30 = 0.10(x - 100).
Result: This equation can be used to calculate the cost of renting the car for any number of miles driven.
Why this matters: This demonstrates how point-slope form can be used to model real-world scenarios involving a fixed cost and a variable cost that increases linearly.

Analogies & Mental Models:

"Think of point-slope form as a GPS that needs a starting point and a direction." The point is the starting location, and the slope is the direction you need to travel.
How the analogy maps to the concept: The point and slope are the necessary information to define the line.
Where the analogy breaks down (limitations): A GPS can change direction, while a linear function has a constant slope.

Common Misconceptions:

❌ Students often forget to distribute the slope 'm' to both terms inside the parentheses (x - x1).
✓ Actually, the slope must be multiplied by both 'x' and '-x1' to correctly simplify the equation.
Why this confusion happens: Students may focus on the first term and forget to distribute the slope to the second term.

Visual Description:

Imagine a point on a line. The point-slope form tells you how to draw the line from that point. You know the slope, which tells you how much to move vertically for every unit you move horizontally. So, starting at the point, you can use the slope to find other points on the line.

Practice Check:

Write the equation of the line that passes through the point (-1, 4) and has a slope of -2.

Answer: Using point-slope form: y - 4 = -2(x - (-1)), which simplifies to y - 4 = -2(x + 1). In slope-intercept form: y = -2x + 2.

Connection to Other Sections:

Point-slope form is a valuable tool for writing equations of linear functions, especially when given a point and slope. It can be easily converted to slope-intercept form, making it a versatile tool for solving problems involving linear functions.

### 4.6 Standard Form (Ax + By = C)

Overview: Standard form is another way to represent a linear equation, often used for its symmetrical properties and ease in certain calculations.

The Core Concept: The standard form of a linear equation is given by:

Ax + By = C

where:

A, B, and C are constants (usually integers)
A and B cannot both be zero
x and y are variables

While it doesn't directly show the slope and y-intercept, standard form is useful for several reasons:

1. Symmetry: It treats x and y more symmetrically than slope-intercept form.
2. Integer Coefficients: Often, A, B, and C are integers, which can simplify calculations.
3. Finding Intercepts: It's easy to find the x and y intercepts from standard form. To find the x-intercept, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y.

To convert an equation from slope-intercept form or point-slope form to standard form, you need to rearrange the equation to have the form Ax + By = C.

Concrete Examples:

Example 1: Converting from Slope-Intercept Form to Standard Form
Setup: Consider the equation y = 2x + 3.
Process: To convert to standard form, we need to move the 'x' term to the left side:
Subtract 2x from both sides: -2x + y = 3
Result: The equation in standard form is -2x + y = 3. We could also multiply both sides by -1 to get 2x - y = -3, which is also considered standard form.
Why this matters: This demonstrates how to convert from slope-intercept form to standard form.

Example 2: Converting from Point-Slope Form to Standard Form
Setup: Consider the equation y - 1 = -3(x + 2).
Process: First, simplify: y - 1 = -3x - 6. Then, move the 'x' term to the left side and the constant term to the right side: 3x + y = -5
Result: The equation in standard form is 3x + y = -5.
Why this matters: This demonstrates how to convert from point-slope form to standard form.

Example 3: Finding Intercepts from Standard Form
Setup: Consider the equation 4x + 5y = 20.
Process: To find the x-intercept, set y = 0: 4x + 5(0) = 20, 4x = 20, x = 5. The x-intercept is (5, 0). To find the y-intercept, set x = 0: 4(0) + 5y = 20, 5y = 20, y = 4. The y-intercept is (0, 4).
Result: We found the x and y intercepts easily from standard form.
Why this matters: This demonstrates the utility of standard form for quickly finding intercepts.

Analogies & Mental Models:

"Think of standard form as organizing ingredients in a pantry." You have the 'x ingredients', the 'y ingredients', and then everything else (the constant) on the other side.
How the analogy maps to the concept: It's about organizing the terms in a specific way.
Where the analogy breaks down (limitations): Pantry organization doesn't have mathematical rules.

Common Misconceptions:

❌ Students often think that A, B, and C must be positive.
✓ Actually, only A and B cannot both be zero. They can be positive or negative.
Why this confusion happens: Often, students are presented with examples where A, B, and C are positive, but the definition doesn't require it.

Visual Description:

While standard form doesn't directly show the slope and y-intercept, you can visualize it by rearranging the equation to slope-intercept form. However, a key visual is that it's relatively easy to see where the line crosses the axes (the intercepts) directly from the equation.

Practice Check:

Convert the equation y = -1/2x + 4 to standard form.

Answer: Multiply both sides by 2: 2y = -x + 8. Then, add x to both sides: x + 2y = 8.

Connection to Other Sections:

Understanding standard form provides a more complete understanding of the different ways to represent linear equations. It also provides a useful tool for finding intercepts and solving certain types of linear problems.

### 4.7 Graphing Linear Functions

Overview: Graphing linear functions allows for a visual representation of the relationship between the variables, making it easier to understand and analyze.

The Core Concept: Graphing a linear function involves plotting points that satisfy the equation on a coordinate plane and then connecting those points with a straight line. There are several methods for graphing linear functions:

1. Using Slope-Intercept Form (y = mx + b):
Plot the y-intercept (0, b) on the y-axis.
Use the slope 'm' to find another point on the line. Remember that slope is rise over run. From the y-intercept, move 'run' units horizontally and 'rise' units vertically to find another point.
Connect the two points with a straight line.

2. Using Two Points:
Find two points that satisfy the equation. This can be done by choosing any two values for 'x' and calculating the corresponding values for 'y'.
Plot the two points on the coordinate plane.
Connect the two points with a straight line.

3. Using Intercepts:
Find the x-intercept by setting y = 0 and solving for x. Plot the point (x, 0).
Find the y-intercept by setting x = 0 and solving for y. Plot the point (0, y).
Connect the two points with a straight line.

The graph of a linear function is always a straight line. The slope of the line indicates its steepness and direction, while the y-intercept indicates where the line crosses the y-axis.

Concrete Examples:

Example 1: Graphing Using Slope-Intercept Form
Setup: Consider the equation y = 2x - 1.
Process: The y-intercept is (0, -1). The slope is 2, which can be written as 2/1. From the y-intercept, move 1 unit to the right and 2 units up to find another point (1, 1).
Result: Plot the points (0, -1) and (1, 1) and connect them with a straight line.
Why this matters: This demonstrates how to graph a linear function using the slope-intercept form.

Example 2: Graphing Using Two Points
Setup: Consider the equation y = -x + 3.
Process: Choose two values for x, such as x = 0 and x = 3. When x = 0, y = 3. When x = 3, y = 0. So, the points are (0, 3) and (3, 0).
* Result: Plot the points (0, 3) and (3, 0) and connect them with

Okay, here's a comprehensive and deeply structured lesson plan on Algebra I: Linear Functions, adhering to all the detailed requirements and aiming for exceptional clarity, depth, and engagement.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're planning a road trip. You know how much gas your car uses per mile, and you have a starting amount of gas in the tank. How far can you drive before you need to refuel? Or perhaps you're starting a small business making and selling bracelets. You have some initial startup costs for materials, and then you make a certain profit on each bracelet you sell. How many bracelets do you need to sell to break even, and start making a profit? These seemingly different scenarios share a common mathematical thread: they can be modeled using linear functions. Linear functions are not just abstract equations; they are powerful tools that help us understand and predict relationships in the world around us, from simple scenarios like calculating gas mileage to more complex situations like predicting population growth. They are the foundational building blocks for more advanced mathematical concepts you'll encounter later on.

### 1.2 Why This Matters

Understanding linear functions is crucial for several reasons. First, they are fundamental to many real-world applications. Whether you're calculating your budget, analyzing data trends, or designing a structure, linear functions provide a framework for understanding and predicting outcomes. They are also essential for success in higher-level math courses such as Algebra II, Trigonometry, and Calculus. Moreover, many careers rely heavily on the principles of linear functions. From engineers and scientists who use them to model physical phenomena, to economists and financial analysts who use them to forecast market trends, the ability to work with linear functions is a highly valuable skill. This knowledge builds directly on your prior understanding of basic arithmetic and algebraic concepts like variables and equations, and it sets the stage for exploring more complex functions and mathematical models in the future. Learning about linear functions opens doors to careers you might not have even considered.

### 1.3 Learning Journey Preview

In this lesson, we will embark on a journey to explore the world of linear functions. We will start by defining what a linear function is and learning how to represent it in different forms: equations, graphs, and tables. We'll then delve into the key characteristics of linear functions, such as slope and y-intercept, and understand how these features affect the behavior of the line. We will explore different ways to write the equations of lines, including slope-intercept form, point-slope form, and standard form. We will also learn how to graph linear functions, interpret their graphs, and solve problems involving linear functions in real-world contexts. Finally, we'll explore systems of linear equations and how to solve them graphically and algebraically. Each concept will build upon the previous one, providing you with a solid foundation in linear functions and their applications.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

Define a linear function and identify its key characteristics, including slope and y-intercept.
Represent a linear function in different forms: equation, graph, and table.
Write the equation of a line in slope-intercept form, point-slope form, and standard form, given sufficient information (e.g., two points, slope and a point).
Graph linear functions accurately using various methods, including plotting points and using slope-intercept form.
Interpret the slope and y-intercept of a linear function in the context of a real-world problem.
Solve real-world problems involving linear functions, including finding the equation of a line that models a given situation and making predictions based on the linear model.
Solve systems of linear equations graphically and algebraically (substitution and elimination methods).
Analyze the relationship between two or more linear functions and determine whether they are parallel, perpendicular, or intersecting.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into linear functions, it's essential to have a solid understanding of the following concepts:

Variables and Expressions: You should be comfortable working with variables (like x and y) and algebraic expressions (like 2x + 3).
Equations: You should know how to solve basic algebraic equations for a single variable (e.g., solving for x in the equation 3x + 5 = 14).
Order of Operations: Remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).
The Coordinate Plane: You need to be familiar with the x and y axes, how to plot points (ordered pairs) on the plane, and understand the concept of quadrants.
Basic Arithmetic: Proficiency in addition, subtraction, multiplication, and division of real numbers (including fractions and decimals) is essential.
Integer Operations: Understanding how to add, subtract, multiply, and divide positive and negative numbers.

Quick Review: If you need a refresher on any of these topics, review your previous algebra notes or consult online resources like Khan Academy (www.khanacademy.org). Search for topics like "solving equations," "coordinate plane," or "order of operations."

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## 4. MAIN CONTENT

### 4.1 What is a Linear Function?

Overview: A linear function represents a relationship between two variables where the change in one variable is directly proportional to the change in the other. This relationship can be visualized as a straight line on a graph.

The Core Concept: A linear function is a mathematical relationship where the output (dependent variable, usually denoted as y) changes at a constant rate with respect to the input (independent variable, usually denoted as x). This constant rate of change is called the slope. The general form of a linear function is y = mx + b, where m represents the slope and b represents the y-intercept (the point where the line crosses the y-axis). The key characteristic that distinguishes a linear function from other types of functions (like quadratic or exponential functions) is that its graph is always a straight line. This means that for every equal increase in x, there is an equal increase (or decrease, if the slope is negative) in y. Linear functions can also be represented in other forms, such as standard form (Ax + By = C), but the fundamental relationship remains the same: a constant rate of change between two variables. The power of linear functions lies in their simplicity and their ability to model a wide range of real-world phenomena that exhibit this constant rate of change.

Concrete Examples:

Example 1: Earning Money at a Constant Rate
Setup: Imagine you have a part-time job where you earn $15 per hour. The amount of money you earn depends on the number of hours you work.
Process: Let x represent the number of hours you work and y represent the total amount of money you earn. The equation representing this situation is y = 15x. If you work 2 hours, you earn y = 15(2) = $30. If you work 5 hours, you earn y = 15(5) = $75. The slope, 15, represents the constant rate of change (dollars per hour). The y-intercept is 0, because if you work 0 hours, you earn $0.
Result: This is a linear function because for every additional hour you work, you earn an additional $15. The relationship between hours worked and money earned is directly proportional.
Why this matters: This simple example demonstrates how linear functions can model real-world scenarios involving constant rates, such as hourly wages, constant speed, or steady growth.

Example 2: Decreasing Water Level in a Tank
Setup: A water tank initially contains 500 gallons of water. The tank is draining at a constant rate of 10 gallons per minute.
Process: Let x represent the number of minutes that have passed and y represent the amount of water remaining in the tank. The equation representing this situation is y = -10x + 500. The slope is -10, indicating that the water level is decreasing by 10 gallons per minute. The y-intercept is 500, representing the initial amount of water in the tank. After 10 minutes, the tank contains y = -10(10) + 500 = 400 gallons. After 50 minutes, the tank contains y = -10(50) + 500 = 0 gallons.
Result: This is a linear function because the water level decreases at a constant rate. The negative slope indicates a decreasing relationship.
Why this matters: This example shows how linear functions can model situations involving a constant rate of decrease, like depreciation, cooling temperatures, or declining populations.

Analogies & Mental Models:

Think of it like: A staircase. The slope is like the rise over run of each step. If the steps are consistent (constant slope), it's a linear function. If the steps change height and depth randomly, it's not.
Explain how the analogy maps to the concept: The "rise" represents the change in y, and the "run" represents the change in x. A steeper staircase corresponds to a larger slope (faster rate of change).
Where the analogy breaks down (limitations): A staircase is discrete (you can only be on a step), while a linear function is continuous (you can have values between whole numbers).

Common Misconceptions:

❌ Students often think: That any equation with x and y is a linear function.
✓ Actually: The relationship between x and y must be such that the rate of change is constant. Equations like y = x² are not linear because the rate of change is not constant.
Why this confusion happens: Students may not fully grasp the concept of a constant rate of change and may only focus on the presence of variables.

Visual Description:

Imagine a graph with the x-axis and y-axis. A linear function is represented by a straight line that extends indefinitely in both directions. The line can be horizontal, vertical, or slanted. The steepness of the line represents the slope, and the point where the line crosses the y-axis is the y-intercept. A steeper line indicates a larger slope (faster rate of change). A horizontal line has a slope of zero, indicating no change in y as x changes.

Practice Check:

Which of the following equations represents a linear function?
a) y = 3x + 2
b) y = x² - 1
c) y = 5/x
d) y = √x + 4

Answer with explanation: a) y = 3x + 2 is the only linear function because it is in the form y = mx + b. The other equations involve non-linear terms (x², 1/x, √x).

Connection to Other Sections: This section provides the foundation for understanding all subsequent sections. We will build upon this definition to explore different forms of linear equations, graphing techniques, and real-world applications.

### 4.2 Slope: The Rate of Change

Overview: Slope is a fundamental concept in linear functions that describes the steepness and direction of a line. It represents the rate at which the dependent variable (y) changes with respect to the independent variable (x).

The Core Concept: The slope, often denoted by the letter m, is a measure of how much the y-value changes for every one unit change in the x-value. It's often described as "rise over run," where "rise" is the vertical change (change in y) and "run" is the horizontal change (change in x). Mathematically, the slope between two points (x₁, y₁) and (x₂, y₂) is calculated as:

m = (y₂ - y₁) / (x₂ - x₁)

A positive slope indicates that the line is increasing (going uphill) as you move from left to right. A negative slope indicates that the line is decreasing (going downhill) as you move from left to right. A slope of zero indicates a horizontal line (no change in y as x changes). An undefined slope indicates a vertical line (a change in y with no change in x). The larger the absolute value of the slope, the steeper the line. Understanding slope is crucial for interpreting the behavior of linear functions and their applications in real-world scenarios.

Concrete Examples:

Example 1: Slope of a Line Passing Through Two Points
Setup: Consider a line passing through the points (1, 3) and (4, 9).
Process: Using the slope formula, we have:
m = (9 - 3) / (4 - 1) = 6 / 3 = 2
Result: The slope of the line is 2. This means that for every 1 unit increase in x, the y-value increases by 2 units.
Why this matters: This example demonstrates how to calculate the slope given two points on a line, a fundamental skill for working with linear functions.

Example 2: Interpreting Slope in a Real-World Context
Setup: Suppose a graph shows the distance a car travels (in miles) over time (in hours). The line on the graph has a slope of 60.
Process: The slope represents the rate of change of distance with respect to time, which is the speed of the car.
Result: The car is traveling at a speed of 60 miles per hour.
Why this matters: This example illustrates how slope can be used to interpret real-world data and understand the relationship between variables.

Analogies & Mental Models:

Think of it like: A ski slope. A steeper ski slope has a larger slope value. A downhill slope has a negative slope. A flat ski run has a slope of zero.
Explain how the analogy maps to the concept: The steepness of the ski slope corresponds to the rate of change. The direction (uphill or downhill) corresponds to the sign of the slope.
Where the analogy breaks down (limitations): Ski slopes are often not perfectly linear (they can have curves and bumps), while linear functions are always perfectly straight.

Common Misconceptions:

❌ Students often think: That slope is just a number without any meaning.
✓ Actually: Slope represents the rate of change between two variables and has a specific meaning in the context of the problem.
Why this confusion happens: Students may focus on the calculation of slope without understanding its interpretation.

Visual Description:

Imagine a line on a graph. The slope is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. You can visualize this by drawing a right triangle with the line as the hypotenuse. The vertical side of the triangle represents the rise, and the horizontal side represents the run. The slope is the rise divided by the run.

Practice Check:

A line passes through the points (2, 5) and (6, 13). What is the slope of the line?

Answer with explanation: m = (13 - 5) / (6 - 2) = 8 / 4 = 2. The slope of the line is 2.

Connection to Other Sections: Understanding slope is essential for writing the equation of a line in slope-intercept form (y = mx + b), graphing linear functions, and solving real-world problems involving linear relationships.

### 4.3 Y-Intercept: Where the Line Crosses

Overview: The y-intercept is another key characteristic of a linear function. It's the point where the line intersects the y-axis, representing the value of y when x is equal to zero.

The Core Concept: The y-intercept, often denoted by the letter b, is the value of y when x is 0. Graphically, it is the point where the line crosses the y-axis. In the slope-intercept form of a linear equation (y = mx + b), the y-intercept is explicitly represented by the constant term b. The y-intercept represents the initial value or starting point of the linear function. It's important to remember that any point on the y-axis has an x-coordinate of 0. Therefore, the y-intercept is the ordered pair (0, b). Understanding the y-intercept is crucial for interpreting the initial conditions of a linear relationship and for accurately graphing linear functions.

Concrete Examples:

Example 1: Identifying the Y-Intercept from an Equation
Setup: Consider the equation y = 3x + 5.
Process: The equation is in slope-intercept form (y = mx + b), where b represents the y-intercept.
Result: The y-intercept is 5. This means that the line crosses the y-axis at the point (0, 5).
Why this matters: This example shows how to directly identify the y-intercept from the equation of a line in slope-intercept form.

Example 2: Interpreting the Y-Intercept in a Real-World Context
Setup: A taxi charges a flat fee of $3 plus $2 per mile. The equation representing the total cost (y) as a function of the number of miles (x) is y = 2x + 3.
Process: The y-intercept is 3.
Result: The y-intercept of 3 represents the initial flat fee of $3, which is charged even if the taxi travels 0 miles.
Why this matters: This example illustrates how the y-intercept can represent the initial value or starting point in a real-world scenario.

Analogies & Mental Models:

Think of it like: The starting point of a race. The y-intercept is where the runner begins the race before they start running (before x changes).
Explain how the analogy maps to the concept: The starting point on the track corresponds to the initial value of y when x is zero.
Where the analogy breaks down (limitations): A race usually has a defined finish line, while a linear function extends indefinitely.

Common Misconceptions:

❌ Students often think: That the y-intercept is just another point on the line without any special significance.
✓ Actually: The y-intercept represents the initial value of the function and is a crucial point for graphing and interpreting linear relationships.
Why this confusion happens: Students may not fully understand the concept of the y-axis and its relationship to the value of x.

Visual Description:

Imagine a line on a graph. The y-intercept is the point where the line crosses the vertical y-axis. This point has coordinates (0, b), where b is the y-value at that intersection. You can easily locate the y-intercept by visually inspecting the graph and finding the point where the line crosses the y-axis.

Practice Check:

What is the y-intercept of the line represented by the equation y = -5x + 7?

Answer with explanation: The y-intercept is 7. The equation is in slope-intercept form (y = mx + b), where b is the y-intercept.

Connection to Other Sections: The y-intercept, along with the slope, is essential for writing the equation of a line in slope-intercept form and for graphing linear functions accurately. It also helps to understand the starting value in real-world applications.

### 4.4 Slope-Intercept Form: y = mx + b

Overview: Slope-intercept form is one of the most common and useful ways to represent a linear function. It directly reveals the slope and y-intercept of the line, making it easy to graph and interpret.

The Core Concept: The slope-intercept form of a linear equation is y = mx + b, where m represents the slope of the line and b represents the y-intercept. This form is particularly useful because it explicitly shows the two key characteristics of a linear function: the rate of change (slope) and the initial value (y-intercept). Given an equation in this form, you can immediately identify the slope and y-intercept without any further calculations. Conversely, if you know the slope and y-intercept of a line, you can easily write its equation in slope-intercept form. This form is widely used in graphing linear functions because it allows you to quickly plot the y-intercept and then use the slope to find other points on the line.

Concrete Examples:

Example 1: Writing an Equation in Slope-Intercept Form
Setup: A line has a slope of -2 and a y-intercept of 4.
Process: Using the slope-intercept form (y = mx + b), substitute m = -2 and b = 4.
Result: The equation of the line is y = -2x + 4.
Why this matters: This example demonstrates how to write the equation of a line in slope-intercept form given the slope and y-intercept.

Example 2: Identifying Slope and Y-Intercept from an Equation
Setup: Consider the equation y = (1/2)x - 3.
Process: Compare the equation to the slope-intercept form (y = mx + b).
Result: The slope is 1/2, and the y-intercept is -3.
Why this matters: This example illustrates how to directly identify the slope and y-intercept from an equation in slope-intercept form.

Analogies & Mental Models:

Think of it like: A recipe. The slope (m) is the amount of a key ingredient, and the y-intercept (b) is the starting amount of something else.
Explain how the analogy maps to the concept: Changing the amount of the key ingredient (slope) affects the final result (the line). The starting amount (y-intercept) is the initial condition.
Where the analogy breaks down (limitations): A recipe usually has a specific outcome in mind, while a linear function extends indefinitely.

Common Misconceptions:

❌ Students often think: That the slope and y-intercept are interchangeable in the equation.
✓ Actually: The slope (m) is always the coefficient of x, and the y-intercept (b) is the constant term. Switching them will result in a different line.
Why this confusion happens: Students may not fully understand the roles of m and b in the equation.

Visual Description:

Imagine a line on a graph. The slope-intercept form (y = mx + b) directly tells you two important things: where the line crosses the y-axis (y-intercept, b) and how steep the line is (slope, m). Start by plotting the y-intercept on the y-axis. Then, use the slope (rise over run) to find other points on the line.

Practice Check:

What are the slope and y-intercept of the line represented by the equation y = -4x + 1?

Answer with explanation: The slope is -4, and the y-intercept is 1. The equation is in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

Connection to Other Sections: Understanding slope-intercept form is crucial for graphing linear functions, writing equations of lines given slope and y-intercept, and solving real-world problems involving linear relationships. It also provides a foundation for understanding other forms of linear equations.

### 4.5 Point-Slope Form: y - y₁ = m(x - x₁)

Overview: Point-slope form is another useful way to represent a linear function. It's particularly helpful when you know the slope of the line and a single point that it passes through.

The Core Concept: The point-slope form of a linear equation is y - y₁ = m(x - x₁), where m is the slope of the line and (x₁, y₁) is a known point on the line. This form is derived directly from the definition of slope. If we have two points (x, y) and (x₁, y₁) on a line, the slope is given by m = (y - y₁) / (x - x₁). Multiplying both sides by (x - x₁) gives us the point-slope form. The point-slope form is advantageous when you don't know the y-intercept but you do know the slope and a point. You can easily plug in the values for m, x₁, and y₁ and then simplify the equation to get it into slope-intercept form if desired.

Concrete Examples:

Example 1: Writing an Equation in Point-Slope Form
Setup: A line has a slope of 3 and passes through the point (2, 5).
Process: Using the point-slope form (y - y₁ = m(x - x₁)), substitute m = 3, x₁ = 2, and y₁ = 5.
Result: The equation of the line is y - 5 = 3(x - 2).
Why this matters: This example demonstrates how to write the equation of a line in point-slope form given the slope and a point on the line.

Example 2: Converting Point-Slope Form to Slope-Intercept Form
Setup: Consider the equation y + 1 = -2(x - 3).
Process: First, distribute the -2 on the right side: y + 1 = -2x + 6. Then, subtract 1 from both sides: y = -2x + 5.
Result: The equation in slope-intercept form is y = -2x + 5. The slope is -2, and the y-intercept is 5.
Why this matters: This example shows how to convert from point-slope form to slope-intercept form, allowing you to easily identify the slope and y-intercept.

Analogies & Mental Models:

Think of it like: Finding your way on a map. The slope (m) is the direction you need to go, and the point (x₁, y₁) is your current location.
Explain how the analogy maps to the concept: Knowing your direction and current location allows you to determine your path (the line).
Where the analogy breaks down (limitations): A map usually has a limited area, while a linear function extends indefinitely.

Common Misconceptions:

❌ Students often think: That the signs of x₁ and y₁ in the point-slope form are the same as in the original point.
✓ Actually: The point-slope form is y - y₁ = m(x - x₁), so the signs of x₁ and y₁ are reversed when plugged into the equation.
Why this confusion happens: Students may not pay close attention to the minus signs in the equation.

Visual Description:

Imagine a line on a graph. The point-slope form (y - y₁ = m(x - x₁)) tells you the slope of the line (m) and the coordinates of one point on the line (x₁, y₁). You can plot the point (x₁, y₁) on the graph and then use the slope to find other points on the line.

Practice Check:

Write the equation of a line in point-slope form that has a slope of -1 and passes through the point (4, -2).

Answer with explanation: y - (-2) = -1(x - 4), which simplifies to y + 2 = -1(x - 4).

Connection to Other Sections: The point-slope form is a valuable tool for writing the equation of a line when you know the slope and a point. It can be easily converted to slope-intercept form, allowing you to graph the line and identify its slope and y-intercept.

### 4.6 Standard Form: Ax + By = C

Overview: Standard form is yet another way to represent a linear function. While it doesn't directly reveal the slope or y-intercept, it has its own advantages and is often used in specific contexts.

The Core Concept: The standard form of a linear equation is Ax + By = C, where A, B, and C are constants, and A and B are not both zero. In standard form, the x and y terms are on the same side of the equation, and the constant term is on the other side. While the slope and y-intercept are not immediately apparent in standard form, you can easily convert it to slope-intercept form by solving for y. Standard form is particularly useful when dealing with systems of linear equations, especially when using the elimination method to solve for variables. It is also often used in certain applications of linear equations, such as in economics and physics. It's important to note that A, B, and C are typically integers, and A is usually positive.

Concrete Examples:

Example 1: Converting from Standard Form to Slope-Intercept Form
Setup: Consider the equation 2x + 3y = 6.
Process: Subtract 2x from both sides: 3y = -2x + 6. Then, divide both sides by 3: y = (-2/3)x + 2.
Result: The equation in slope-intercept form is y = (-2/3)x + 2. The slope is -2/3, and the y-intercept is 2.
Why this matters: This example demonstrates how to convert from standard form to slope-intercept form to easily identify the slope and y-intercept.

Example 2: Writing an Equation in Standard Form
Setup: A line has a slope of 1/2 and a y-intercept of -1.
Process: First, write the equation in slope-intercept form: y = (1/2)x - 1. Then, multiply both sides by 2 to eliminate the fraction: 2y = x - 2. Finally, rearrange the equation to get the x and y terms on the same side: -x + 2y = -2. Multiply both sides by -1 to make A positive: x - 2y = 2.
Result: The equation in standard form is x - 2y = 2.
Why this matters: This example shows how to write the equation of a line in standard form given its slope and y-intercept.

Analogies & Mental Models:

Think of it like: Balancing a scale. The x and y terms are like objects on one side of the scale, and the constant term is like the weight on the other side.
Explain how the analogy maps to the concept: Changing the values of x and y requires adjusting the weight on the other side to maintain balance (equality).
Where the analogy breaks down (limitations): A scale has a limited capacity, while a linear function extends indefinitely.

Common Misconceptions:

❌ Students often think: That the coefficients A and B directly represent the slope and y-intercept in standard form.
✓ Actually: You need to convert the equation to slope-intercept form to identify the slope and y-intercept. The slope is -A/B and the y-intercept can be found by setting x=0 and solving for y, which gives y=C/B.
Why this confusion happens: Students may not fully understand the relationship between standard form and slope-intercept form.

Visual Description:

While standard form doesn't directly show the slope or y-intercept, you can still visualize the line on a graph. You can find the x and y intercepts by setting y=0 and x=0, respectively, and solving for the other variable. Plot these two points and draw a line through them.

Practice Check:

Convert the equation 3x - 4y = 12 to slope-intercept form.

Answer with explanation: Subtract 3x from both sides: -4y = -3x + 12. Then, divide both sides by -4: y = (3/4)x - 3.

Connection to Other Sections: Understanding standard form is important for working with systems of linear equations and for converting between different forms of linear equations.

### 4.7 Graphing Linear Functions

Overview: Graphing linear functions is a visual way to represent the relationship between two variables. It allows you to see the slope, y-intercept, and other key characteristics of the function.

The Core Concept: Graphing a linear function involves plotting points on the coordinate plane and drawing a straight line through them. There are several methods for graphing linear functions:

1. Plotting Points: Choose several values for x, substitute them into the equation to find the corresponding values for y, and plot the resulting ordered pairs (x, y) on the coordinate plane. Then, draw a straight line through the points. At least two points are needed to define a line.
2. Using Slope-Intercept Form: Identify the slope (m) and y-intercept (b) from the equation in slope-intercept form (y = mx + b). Plot the y-intercept (0, b) on the y-axis. Then, use the slope (rise over run) to find another point on the line. For example, if the slope is 2/3, start at the y-intercept and move up 2 units and right 3 units to find another point. Draw a straight line through the two points.
3. Using X and Y Intercepts: Find the x-intercept (where the line crosses the x-axis) by setting y = 0 and solving for x. Find the y-intercept (where the line crosses the y-axis) by setting x = 0 and solving for y. Plot the x and y intercepts and draw a straight line through the two points.

Concrete Examples:

Example 1: Graphing Using Slope-Intercept Form
Setup

Okay, here is a comprehensive lesson plan on Linear Functions for Algebra I, designed to be both thorough and engaging.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're starting a small business selling custom-designed phone cases. You have some fixed costs, like the cost of your design software and printer, and variable costs, like the cost of each blank phone case and the ink you use. If you sell each phone case for a set price, how can you figure out how many you need to sell to break even, or to reach a certain profit goal? This is where linear functions come in! They are the mathematical tools that help us model and understand situations where things change at a constant rate, like the cost of your business or the distance a car travels at a steady speed. Think of it like this: linear functions are the "straight lines" of the mathematical world, representing consistent, predictable relationships.

Linear functions aren't just abstract math – they're all around us. From calculating the price of a taxi ride based on distance, to predicting the height of a plant growing at a constant rate, to understanding how your phone battery drains over time, linear functions help us make sense of the world and make informed decisions. They are a fundamental building block for more advanced mathematical concepts and are essential for understanding many real-world phenomena.

### 1.2 Why This Matters

Understanding linear functions is crucial for several reasons. First, they provide a foundational understanding for more advanced mathematical concepts such as calculus, statistics, and linear algebra. These concepts are essential for many STEM fields. Second, linear functions have widespread real-world applications in fields like economics (modeling supply and demand), physics (describing motion), engineering (designing structures), and computer science (developing algorithms). Knowing how to work with linear functions can give you a competitive edge in these fields and open doors to exciting career opportunities.

Furthermore, linear functions teach you valuable problem-solving skills that are applicable in many aspects of life. Learning to model real-world situations with mathematical equations, analyze data, and make predictions are skills that will serve you well, regardless of your chosen career path. Whether you're budgeting your personal finances, planning a road trip, or analyzing data in a business setting, linear functions provide a powerful framework for making informed decisions. This knowledge builds upon your understanding of equations and graphing and will be essential for mastering more complex functions like quadratic and exponential functions later on.

### 1.3 Learning Journey Preview

In this lesson, we'll embark on a journey to explore the world of linear functions. We'll start by defining what a linear function is and understanding its different forms, including slope-intercept form, point-slope form, and standard form. We will delve into the concept of slope, learning how to calculate it and interpret its meaning in different contexts. We will then explore how to graph linear functions and how to write equations for linear functions given different pieces of information, such as the slope and a point, or two points. Finally, we will look at real-world applications of linear functions and see how they can be used to solve practical problems. Each concept builds upon the previous one, giving you a solid foundation for understanding and applying linear functions in various situations.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

1. Define a linear function and distinguish it from non-linear functions using its equation and graph.
2. Identify and interpret the slope and y-intercept of a linear function in slope-intercept form (y = mx + b).
3. Calculate the slope of a line given two points on the line using the slope formula.
4. Write the equation of a linear function in slope-intercept form, point-slope form, and standard form given different information (e.g., slope and y-intercept, slope and a point, two points).
5. Graph a linear function using its equation, either by plotting points or using the slope and y-intercept.
6. Translate real-world scenarios into linear equations and interpret the meaning of the slope and y-intercept within the context of the problem.
7. Solve real-world problems involving linear functions, such as finding the equation of a line that models a given situation or predicting the value of a variable based on a linear relationship.
8. Analyze the relationship between parallel and perpendicular lines by comparing their slopes and equations.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into linear functions, you should already be comfortable with the following concepts:

Variables and Expressions: Understanding what variables represent and how to manipulate algebraic expressions.
Solving Equations: Knowing how to solve basic algebraic equations for a single variable (e.g., solving for x in 2x + 3 = 7).
Coordinate Plane: Familiarity with the coordinate plane, including plotting points and understanding the x and y axes.
Ordered Pairs: Knowing how to represent points as ordered pairs (x, y).
Basic Arithmetic: Proficiency in addition, subtraction, multiplication, and division of real numbers, including fractions and decimals.
Graphing Basics: Understanding how to plot points on a coordinate plane.

If you need a refresher on any of these topics, you can review them in your textbook, online resources like Khan Academy, or by asking your teacher for assistance. Having a solid foundation in these areas will make learning about linear functions much easier.

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## 4. MAIN CONTENT

### 4.1 What is a Linear Function?

Overview: A linear function is a mathematical relationship between two variables that can be represented by a straight line on a graph. It describes a consistent rate of change between the variables.

The Core Concept: A linear function establishes a relationship between an input (usually x) and an output (usually y) where the change in y is directly proportional to the change in x. This means that for every unit increase in x, y increases or decreases by a constant amount. This constant amount is called the slope of the line. The general form of a linear function is y = mx + b, where m represents the slope and b represents the y-intercept (the point where the line crosses the y-axis). The equation can also be expressed in other forms, such as point-slope form and standard form, which we will discuss later.

The key characteristic of a linear function is that it forms a straight line when plotted on a graph. This is because the rate of change is constant. Non-linear functions, on the other hand, have a rate of change that varies, resulting in curved graphs. Examples of non-linear functions include quadratic functions (shaped like parabolas), exponential functions (characterized by rapid growth or decay), and trigonometric functions (oscillating patterns).

It's important to note that not all equations involving x and y are linear functions. For an equation to be a linear function, x and y must be raised to the power of 1 (i.e., no exponents other than 1), and they cannot be multiplied or divided by each other. For example, y = x² is not a linear function because x is squared. Similarly, y = 1/x is not a linear function because x is in the denominator.

Concrete Examples:

Example 1: The equation y = 2x + 3 represents a linear function.
Setup: We are given the equation y = 2x + 3.
Process: For every increase of 1 in x, y increases by 2. The line crosses the y-axis at the point (0, 3). If we plot a few points (e.g., when x = 0, y = 3; when x = 1, y = 5; when x = -1, y = 1) and connect them, we will get a straight line.
Result: The resulting graph is a straight line, confirming that this is a linear function.
Why this matters: This example illustrates the basic form of a linear function and how the slope and y-intercept determine the line's position and direction.

Example 2: The cost of renting a bike is $5 per hour plus a $10 initial fee.
Setup: Let y represent the total cost and x represent the number of hours.
Process: The equation representing this situation is y = 5x + 10. The $5 per hour is the slope (rate of change), and the $10 initial fee is the y-intercept (fixed cost).
Result: This is a linear function because the cost increases at a constant rate of $5 per hour.
Why this matters: This example demonstrates how linear functions can be used to model real-world situations involving costs and rates.

Analogies & Mental Models:

Think of it like: A staircase. Each step has the same height (the change in y) and the same width (the change in x). The slope is like the steepness of the staircase.
Explain how the analogy maps to the concept: The constant height and width of each step represent the constant rate of change in a linear function. A steeper staircase corresponds to a larger slope.
Where the analogy breaks down (limitations): A staircase is discrete (you can only be on a step), while a linear function is continuous (you can have values between whole numbers).

Common Misconceptions:

❌ Students often think that any equation with x and y is a linear function.
✓ Actually, only equations where x and y are raised to the power of 1 and are not multiplied or divided by each other are linear functions.
Why this confusion happens: Students may not fully understand the definition of a linear function and may confuse it with other types of equations.

Visual Description:

Imagine a coordinate plane with an x-axis and a y-axis. A linear function is represented by a straight line that extends infinitely in both directions. The line can be slanted upwards (positive slope), downwards (negative slope), horizontal (zero slope), or vertical (undefined slope). The y-intercept is the point where the line crosses the y-axis.

Practice Check:

Which of the following equations represents a linear function?

a) y = x² + 1
b) y = 3x - 2
c) y = 1/x
d) xy = 5

Answer: b) y = 3x - 2. This is the only equation where x and y are raised to the power of 1 and are not multiplied or divided by each other.

Connection to Other Sections:

This section provides the foundation for understanding the different forms of linear equations (Section 4.2) and how to graph them (Section 4.3). It also lays the groundwork for applying linear functions to real-world problems (Section 4.6).

### 4.2 Forms of Linear Equations

Overview: While y = mx + b is the most common form, linear equations can be written in various forms, each highlighting different aspects of the line.

The Core Concept: There are three main forms of linear equations: slope-intercept form, point-slope form, and standard form.

Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept. This form is useful for quickly identifying the slope and y-intercept of a line.
Point-Slope Form: y - y₁ = m( x - x₁), where m is the slope and (x₁, y₁) is a point on the line. This form is helpful when you know the slope and a point on the line.
Standard Form: Ax + By = C, where A, B, and C are constants. This form is often used in systems of equations and can be useful for finding the x and y intercepts.

Each form provides a different perspective on the same linear relationship. Knowing how to convert between these forms is a valuable skill. For example, you can convert from point-slope form to slope-intercept form by simplifying the equation and isolating y. Similarly, you can convert from standard form to slope-intercept form by solving for y.

Concrete Examples:

Example 1: Converting from Point-Slope Form to Slope-Intercept Form
Setup: Given the point-slope form equation y - 2 = 3(x + 1).
Process: Distribute the 3 on the right side: y - 2 = 3x + 3. Then, add 2 to both sides: y = 3x + 5.
Result: The equation is now in slope-intercept form: y = 3x + 5. We can see that the slope is 3 and the y-intercept is 5.
Why this matters: This demonstrates how to manipulate equations to reveal key information about the line.

Example 2: Converting from Standard Form to Slope-Intercept Form
Setup: Given the standard form equation 2x + 3y = 6.
Process: Subtract 2x from both sides: 3y = -2x + 6. Then, divide both sides by 3: y = (-2/3)x + 2.
Result: The equation is now in slope-intercept form: y = (-2/3)x + 2. The slope is -2/3 and the y-intercept is 2.
Why this matters: This shows how to rewrite an equation to easily identify the slope and y-intercept.

Analogies & Mental Models:

Think of it like: Different lenses on a camera. Each lens (form) focuses on a different aspect of the same scene (line). Slope-intercept shows the slope and y-intercept clearly. Point-slope focuses on a specific point and the slope. Standard form is like a wider, more general view.
Explain how the analogy maps to the concept: Each form highlights specific features of the line, making it easier to work with in different situations.
Where the analogy breaks down (limitations): Camera lenses are physical objects, while these forms are just different ways of writing the same mathematical relationship.

Common Misconceptions:

❌ Students often think that each form represents a different line.
✓ Actually, all three forms represent the same line; they are just different ways of writing the same equation.
Why this confusion happens: Students may not fully grasp the concept that different forms can represent the same mathematical relationship.

Visual Description:

Imagine a line on a graph. Slope-intercept form directly shows you where the line crosses the y-axis and how steep it is. Point-slope form highlights a specific point on the line and its steepness. Standard form is a more general representation that doesn't immediately reveal the slope or y-intercept but is useful for certain algebraic manipulations.

Practice Check:

Rewrite the equation y - 5 = -2(x + 3) in slope-intercept form.

Answer: y = -2x - 1

Connection to Other Sections:

This section builds on the definition of a linear function (Section 4.1) and prepares you for graphing linear functions (Section 4.3) and writing equations of lines (Section 4.4).

### 4.3 Understanding Slope

Overview: Slope is the heart of a linear function; it describes the rate at which the line rises or falls.

The Core Concept: The slope of a line is a measure of its steepness and direction. It represents the change in y (vertical change, often called "rise") divided by the change in x (horizontal change, often called "run"). The slope is denoted by the letter m. The formula for calculating the slope given two points (x₁, y₁) and (x₂, y₂) on the line is:

m = ( y₂ - y₁) / (x₂ - x₁)

A positive slope indicates that the line is increasing (rising) as you move from left to right. A negative slope indicates that the line is decreasing (falling) as you move from left to right. A slope of zero indicates a horizontal line (no change in y). A vertical line has an undefined slope (change in x is zero, leading to division by zero).

The slope is constant for a linear function. This means that the rate of change is the same no matter where you are on the line. This is what makes linear functions so predictable and useful for modeling real-world situations.

Concrete Examples:

Example 1: Finding the slope given two points.
Setup: Find the slope of the line passing through the points (1, 2) and (4, 8).
Process: Using the slope formula: m = (8 - 2) / (4 - 1) = 6 / 3 = 2.
Result: The slope of the line is 2. This means that for every 1 unit increase in x, y increases by 2 units.
Why this matters: This demonstrates how to apply the slope formula to calculate the slope of a line.

Example 2: Interpreting slope in a real-world context.
Setup: The cost of renting a car is $20 plus $0.30 per mile.
Process: The equation representing this situation is y = 0.30x + 20, where y is the total cost and x is the number of miles driven. The slope is 0.30.
Result: The slope of 0.30 means that for every additional mile driven, the cost increases by $0.30.
Why this matters: This shows how the slope can be interpreted as a rate of change in a real-world scenario.

Analogies & Mental Models:

Think of it like: A hill. The slope is how steep the hill is. A positive slope means you're going uphill, a negative slope means you're going downhill, and a zero slope means you're on flat ground.
Explain how the analogy maps to the concept: The steepness of the hill corresponds to the rate of change in the linear function.
Where the analogy breaks down (limitations): A hill can have varying steepness, while the slope of a linear function is constant.

Common Misconceptions:

❌ Students often confuse the rise and run in the slope formula.
✓ Actually, the slope is the rise (change in y) divided by the run (change in x), not the other way around.
Why this confusion happens: Students may not fully understand the meaning of rise and run or may mix up the order of the variables in the formula.

Visual Description:

Imagine a line on a graph. A positive slope means the line goes up as you move from left to right. A negative slope means the line goes down. A steeper line has a larger absolute value of the slope. A horizontal line has a slope of zero. A vertical line has an undefined slope.

Practice Check:

Find the slope of the line passing through the points (2, 5) and (6, 1).

Answer: m = -1

Connection to Other Sections:

This section is crucial for understanding how to write equations of lines (Section 4.4) and graph linear functions (Section 4.5). It also provides the foundation for understanding parallel and perpendicular lines (Section 4.7).

### 4.4 Writing Equations of Linear Functions

Overview: You can define a line mathematically if you know certain key pieces of information about it.

The Core Concept: You can write the equation of a linear function if you know either:

1. The slope (m) and the y-intercept (b) (using slope-intercept form: y = mx + b)
2. The slope (m) and a point (x₁, y₁) on the line (using point-slope form: y - y₁ = m( x - x₁))
3. Two points (x₁, y₁) and (x₂, y₂) on the line (first calculate the slope using the slope formula, then use point-slope form with either point)

The key is to choose the appropriate form based on the information you are given and then substitute the known values into the formula. Once you have the equation, you can rewrite it in any of the three forms (slope-intercept, point-slope, or standard form) as needed.

Concrete Examples:

Example 1: Writing the equation given the slope and y-intercept.
Setup: Write the equation of a line with a slope of 3 and a y-intercept of -2.
Process: Using slope-intercept form: y = mx + b, substitute m = 3 and b = -2: y = 3x - 2.
Result: The equation of the line is y = 3x - 2.
Why this matters: This demonstrates the direct application of the slope-intercept form.

Example 2: Writing the equation given the slope and a point.
Setup: Write the equation of a line with a slope of -1/2 that passes through the point (4, 1).
Process: Using point-slope form: y - y₁ = m( x - x₁), substitute m = -1/2 and (x₁, y₁) = (4, 1): y - 1 = (-1/2)(x - 4).
Result: The equation of the line in point-slope form is y - 1 = (-1/2)(x - 4). We can rewrite this in slope-intercept form: y = (-1/2)x + 3.
Why this matters: This shows how to use the point-slope form when you don't know the y-intercept.

Example 3: Writing the equation given two points.
Setup: Write the equation of the line passing through the points (2, 3) and (6, 5).
Process: First, calculate the slope: m = (5 - 3) / (6 - 2) = 2 / 4 = 1/2. Then, use point-slope form with either point, for example, (2, 3): y - 3 = (1/2)(x - 2).
Result: The equation of the line in point-slope form is y - 3 = (1/2)(x - 2). We can rewrite this in slope-intercept form: y = (1/2)x + 2.
Why this matters: This demonstrates how to find the equation of a line when you only know two points on the line.

Analogies & Mental Models:

Think of it like: Building a road. You need either the starting point (y-intercept) and the direction (slope), or a point on the road and the direction, or two points on the road to define its path.
Explain how the analogy maps to the concept: The information you need to define the road (line) corresponds to the information you need to write the equation of the line.
Where the analogy breaks down (limitations): Roads are physical objects, while linear functions are mathematical relationships.

Common Misconceptions:

❌ Students often use the wrong form of the equation or substitute the values incorrectly.
✓ Actually, it's important to choose the correct form based on the given information and carefully substitute the values into the formula.
Why this confusion happens: Students may not fully understand the different forms of the equation or may make careless errors when substituting values.

Visual Description:

Imagine a line on a graph. Knowing the slope and y-intercept allows you to directly plot the line. Knowing the slope and a point allows you to draw the line by starting at the point and using the slope to find other points. Knowing two points allows you to draw the line by connecting the two points.

Practice Check:

Write the equation of the line passing through the points (-1, 4) and (3, -2).

Answer: y = (-3/2)x + 5/2

Connection to Other Sections:

This section builds on the understanding of slope (Section 4.3) and different forms of linear equations (Section 4.2) and prepares you for graphing linear functions (Section 4.5) and applying them to real-world problems (Section 4.6).

### 4.5 Graphing Linear Functions

Overview: Visualizing linear functions through graphs is a powerful way to understand their behavior.

The Core Concept: There are several ways to graph a linear function:

1. Using Slope-Intercept Form: Identify the y-intercept (b) and plot it on the y-axis. Then, use the slope (m) to find another point on the line. For example, if the slope is 2/3, start at the y-intercept and move 2 units up and 3 units to the right. Connect the two points to draw the line.
2. Using Two Points: Find two points that satisfy the equation. Plot these points on the coordinate plane and draw a line through them.
3. Using a Table of Values: Create a table of values by choosing several values for x and calculating the corresponding values for y. Plot these points on the coordinate plane and draw a line through them.
4. Using X and Y Intercepts: Find the x-intercept (where the line crosses the x-axis, by setting y = 0 and solving for x) and the y-intercept (where the line crosses the y-axis, by setting x = 0 and solving for y). Plot these two points and draw a line through them.

The graph of a linear function is always a straight line. Understanding how to graph linear functions is essential for visualizing their behavior and solving problems involving linear relationships.

Concrete Examples:

Example 1: Graphing using slope-intercept form.
Setup: Graph the equation y = 2x + 1.
Process: The y-intercept is 1, so plot the point (0, 1). The slope is 2, which can be written as 2/1. Start at (0, 1) and move 2 units up and 1 unit to the right to find another point (1, 3). Connect the two points to draw the line.
Result: The graph is a straight line passing through (0, 1) and (1, 3).
Why this matters: This demonstrates how to use the slope and y-intercept to quickly graph a line.

Example 2: Graphing using two points.
Setup: Graph the equation y = -x + 4.
Process: Choose two values for x, for example, x = 0 and x = 4. When x = 0, y = 4, so plot the point (0, 4). When x = 4, y = 0, so plot the point (4, 0). Connect the two points to draw the line.
Result: The graph is a straight line passing through (0, 4) and (4, 0).
Why this matters: This shows how to graph a line when you don't immediately know the slope and y-intercept.

Analogies & Mental Models:

Think of it like: Drawing a road on a map. You can either start at a known location (y-intercept) and follow a certain direction (slope), or connect two known locations (two points) to draw the road.
Explain how the analogy maps to the concept: The methods for drawing a road correspond to the methods for graphing a linear function.
Where the analogy breaks down (limitations): Maps are physical representations, while graphs are mathematical representations.

Common Misconceptions:

❌ Students often struggle to correctly interpret the slope when graphing.
✓ Actually, the slope tells you how many units to move up or down and how many units to move to the right to find another point on the line.
Why this confusion happens: Students may not fully understand the meaning of slope or may mix up the rise and run.

Visual Description:

Imagine a coordinate plane with an x-axis and a y-axis. A linear function is represented by a straight line on this plane. The line can be drawn using the slope and y-intercept, two points, or a table of values.

Practice Check:

Graph the equation y = (1/2)x - 3.

Answer: The graph is a straight line passing through (0, -3) with a slope of 1/2.

Connection to Other Sections:

This section builds on the understanding of slope (Section 4.3) and writing equations of lines (Section 4.4) and prepares you for applying linear functions to real-world problems (Section 4.6).

### 4.6 Real-World Applications of Linear Functions

Overview: Linear functions are powerful tools for modeling and solving problems in various real-world scenarios.

The Core Concept: Linear functions can be used to model situations where there is a constant rate of change. Some common applications include:

1. Cost Analysis: Modeling the cost of a product or service based on a fixed cost and a variable cost per unit.
2. Distance, Rate, and Time: Calculating the distance traveled by an object moving at a constant speed.
3. Simple Interest: Calculating the amount of interest earned on an investment at a fixed interest rate.
4. Depreciation: Modeling the decrease in value of an asset over time.
5. Supply and Demand: Modeling the relationship between the price of a product and the quantity supplied or demanded.

When applying linear functions to real-world problems, it is important to carefully define the variables, identify the slope and y-intercept, and interpret the results in the context of the problem.

Concrete Examples:

Example 1: Cost Analysis
Setup: A company produces widgets. The fixed cost is $5000, and the variable cost is $10 per widget. Write a linear equation to model the total cost of producing x widgets.
Process: Let y represent the total cost. The equation is y = 10x + 5000.
Result: The equation y = 10x + 5000 models the total cost of producing x widgets. The slope (10) represents the variable cost per widget, and the y-intercept (5000) represents the fixed cost.
Why this matters: This allows the company to predict the cost of producing a certain number of widgets and make informed decisions about pricing and production levels.

Example 2: Distance, Rate, and Time
Setup: A car travels at a constant speed of 60 miles per hour. Write a linear equation to model the distance traveled after x hours.
Process: Let y represent the distance traveled. The equation is y = 60x.
Result: The equation y = 60x models the distance traveled after x hours. The slope (60) represents the speed of the car.
Why this matters: This allows you to calculate how far the car will travel in a given amount of time.

Analogies & Mental Models:

Think of it like: Using a recipe. The linear function is the recipe, and the ingredients (slope and y-intercept) determine the final outcome (the result of the problem).
Explain how the analogy maps to the concept: The recipe provides a set of instructions for combining the ingredients, just like a linear function provides a set of instructions for relating the variables.
Where the analogy breaks down (limitations): Recipes are typically more complex than simple linear functions.

Common Misconceptions:

❌ Students often struggle to identify the slope and y-intercept in real-world problems.
✓ Actually, the slope represents the rate of change, and the y-intercept represents the initial value or fixed cost.
Why this confusion happens: Students may not fully understand the meaning of slope and y-intercept in different contexts.

Visual Description:

Imagine a graph representing a real-world situation. The x-axis represents the independent variable (e.g., time), and the y-axis represents the dependent variable (e.g., distance). A linear function is represented by a straight line on this graph, showing the relationship between the two variables.

Practice Check:

A phone company charges a monthly fee of $25 plus $0.10 per minute of usage. Write a linear equation to model the total monthly cost of using the phone for x minutes.

Answer: y = 0.10x + 25

Connection to Other Sections:

This section builds on the understanding of slope (Section 4.3), writing equations of lines (Section 4.4), and graphing linear functions (Section 4.5) and provides a practical application of these concepts.

### 4.7 Parallel and Perpendicular Lines

Overview: The relationship between lines can be defined by their slopes. Parallel lines have the same slope and perpendicular lines have slopes that are negative reciprocals of each other.

The Core Concept:

Parallel Lines: Two lines are parallel if they have the same slope but different y-intercepts. This means they will never intersect. If line 1 has a slope of m₁ and line 2 has a slope of m₂, then the lines are parallel if m₁ = m₂.
Perpendicular Lines: Two lines are perpendicular if they intersect at a right

Okay, here is a comprehensive lesson on Algebra I: Linear Functions, designed to meet the stringent requirements outlined.

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## 1. INTRODUCTION
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### 1.1 Hook & Context

Imagine you're starting a small business selling handmade bracelets. You need to figure out how much to charge to make a profit. Each bracelet costs you $3 in materials, and you want to earn $5 profit on each one. How many bracelets do you need to sell to make $100, $500, or even $1000? This seemingly simple question is the foundation of many real-world business decisions, and it can be solved using linear functions. Or perhaps you're planning a road trip. You know your car gets 30 miles per gallon, and the gas costs $4 per gallon. How can you predict your total gas cost based on the distance you drive? Linear functions provide a powerful tool to model these types of situations and make accurate predictions. We will learn how to model these relations.

These scenarios connect to everyday experiences – managing money, planning trips, even understanding the pricing of goods and services. Most of us have encountered situations where a quantity changes at a constant rate. This is the essence of linear functions. By mastering them, you'll gain the ability to analyze and predict outcomes in a wide range of situations, from personal finance to scientific modeling. This lesson will empower you to see the world through a mathematical lens, allowing you to make informed decisions and solve problems more effectively.

### 1.2 Why This Matters

Linear functions are fundamental building blocks in mathematics and have extensive real-world applications. Beyond the examples above, they are used in:

Finance: Modeling loan payments, calculating interest, and tracking investments.
Physics: Describing motion at a constant speed or the relationship between force and distance in simple machines.
Economics: Analyzing supply and demand curves.
Computer Science: Creating simple algorithms and data visualizations.
Engineering: Designing structures and systems with predictable behavior.

Furthermore, understanding linear functions is crucial for success in higher-level math courses like Algebra II, Trigonometry, and Calculus. They serve as a foundation for understanding more complex functions and mathematical models. For example, the concepts of slope and y-intercept are extended and generalized when studying derivatives and integrals in calculus. Mastering linear functions now will make these future topics much easier to grasp.

### 1.3 Learning Journey Preview

In this lesson, we'll embark on a journey to explore the world of linear functions. We'll start with the basics, defining what a linear function is and how to represent it. We'll then delve into the different forms of linear equations (slope-intercept, point-slope, and standard form) and learn how to convert between them. Next, we'll focus on understanding the key characteristics of linear functions, including slope and intercepts, and how they relate to real-world scenarios. We will then learn how to graph linear functions, write equations of linear functions given different pieces of information, and model real-world data with linear functions. Finally, we'll explore applications of linear functions in various fields and discuss career paths that utilize these skills. Each concept will build upon the previous one, creating a cohesive and comprehensive understanding of linear functions.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

1. Define a linear function and identify its key characteristics.
2. Explain the meaning of slope and y-intercept in the context of a linear function.
3. Convert linear equations between slope-intercept form, point-slope form, and standard form.
4. Graph linear functions using slope-intercept form, point-slope form, and by finding intercepts.
5. Write the equation of a linear function given two points, a point and a slope, or a graph.
6. Apply linear functions to model and solve real-world problems, such as calculating costs, predicting outcomes, and analyzing data.
7. Analyze the relationship between two linear functions, including determining if they are parallel or perpendicular.
8. Interpret the meaning of the slope and y-intercept in the context of a real-world problem modeled by a linear function.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into linear functions, it's essential to have a solid grasp of the following concepts:

Variables and Expressions: Understanding what variables represent and how to evaluate algebraic expressions.
Solving Equations: Being able to solve basic algebraic equations for a single variable (e.g., solving for x in 2x + 3 = 7).
Coordinate Plane: Familiarity with the coordinate plane (x-axis, y-axis, quadrants) and how to plot points.
Order of Operations: Knowing the correct order to perform mathematical operations (PEMDAS/BODMAS).
Integers and Rational Numbers: Working comfortably with positive and negative numbers, fractions, and decimals.

Quick Review:

Solving Equations: Remember to perform inverse operations to isolate the variable. For example, to solve x + 5 = 12, subtract 5 from both sides.
Coordinate Plane: The x-coordinate represents the horizontal distance from the origin (0,0), and the y-coordinate represents the vertical distance.
Order of Operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

If you need a refresher on any of these topics, there are numerous online resources available, such as Khan Academy or your textbook. Make sure you feel comfortable with these basics before moving forward.

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## 4. MAIN CONTENT

### 4.1 Defining Linear Functions

Overview: Linear functions are a fundamental type of function in algebra, characterized by a constant rate of change. They represent a straight-line relationship between two variables. Understanding their definition and characteristics is crucial for working with them effectively.

The Core Concept: A linear function is a function whose graph is a straight line. Mathematically, a linear function can be represented by an equation of the form y = mx + b, where:

x is the independent variable (the input).
y is the dependent variable (the output).
m is the slope (the rate of change of y with respect to x).
b is the y-intercept (the value of y when x = 0).

The key characteristic of a linear function is that for every unit increase in x, y changes by a constant amount (m). This constant rate of change is what gives the function its straight-line appearance. If the rate of change is not constant, the function is not linear. Linear functions are often used to model relationships where one quantity changes proportionally with another.

It's important to note that not all equations involving x and y are linear functions. For instance, y = x² is not a linear function because the relationship between x and y is not linear; its graph is a parabola. Similarly, equations with x in the denominator (e.g., y = 1/x) or with x inside a square root (e.g., y = √x) are not linear.

Concrete Examples:

Example 1: The equation y = 2x + 1 represents a linear function.
Setup: This equation states that for every increase of 1 in x, y increases by 2. The y-intercept is 1, meaning that when x is 0, y is 1.
Process: If x = 0, then y = 2(0) + 1 = 1. If x = 1, then y = 2(1) + 1 = 3. If x = 2, then y = 2(2) + 1 = 5.
Result: Plotting these points (0,1), (1,3), and (2,5) on a graph reveals a straight line.
Why this matters: This equation can model various scenarios, such as the cost of renting a tool where there's a base fee of $1 and an hourly charge of $2.

Example 2: The equation y = -x + 5 represents a linear function.
Setup: This equation indicates that for every increase of 1 in x, y decreases by 1. The y-intercept is 5, meaning that when x is 0, y is 5.
Process: If x = 0, then y = -(0) + 5 = 5. If x = 1, then y = -(1) + 5 = 4. If x = 2, then y = -(2) + 5 = 3.
Result: Plotting these points (0,5), (1,4), and (2,3) on a graph reveals a straight line sloping downwards.
Why this matters: This equation can model scenarios like the amount of water remaining in a tank that is being drained at a rate of 1 gallon per minute, starting with 5 gallons.

Analogies & Mental Models:

Think of it like... a ramp. The slope (m) is like the steepness of the ramp. A steeper ramp has a larger slope. The y-intercept (b) is like the starting height of the ramp.
Explain how the analogy maps to the concept: A steeper ramp means a greater change in height for every unit of horizontal distance, just like a larger slope means a greater change in y for every unit change in x. The starting height of the ramp is the height when you're at the beginning (x=0), which corresponds to the y-intercept.
Where the analogy breaks down (limitations): A ramp is usually limited to positive heights and distances. Linear functions can have negative values for both x and y. Also, a ramp has a physical limitation in length, while a linear function extends infinitely in both directions.

Common Misconceptions:

❌ Students often think that any equation with x and y is a linear function.
✓ Actually, only equations that can be written in the form y = mx + b represent linear functions. Equations with exponents on x, y, or where x is in the denominator are not linear.
Why this confusion happens: Students may not fully understand the definition of a linear function and the requirement of a constant rate of change.

Visual Description:

Imagine a graph with an x-axis and a y-axis. A linear function is represented by a straight line drawn on this graph. The line can slope upwards (positive slope), downwards (negative slope), or be horizontal (zero slope). The point where the line crosses the y-axis is the y-intercept. The slope visually represents the steepness and direction of the line.

Practice Check:

Is the equation y = 3x - 2 a linear function? Why or why not?

Answer: Yes, it is a linear function because it can be written in the form y = mx + b, where m = 3 and b = -2.

Connection to Other Sections:

This section lays the foundation for understanding all subsequent sections. Knowing the definition of a linear function is essential for recognizing them, graphing them, and applying them to real-world problems. The next section will delve into different forms of linear equations.

### 4.2 Forms of Linear Equations: Slope-Intercept Form

Overview: The slope-intercept form is one of the most common and useful ways to represent a linear equation. It directly reveals the slope and y-intercept of the line, making it easy to graph and analyze.

The Core Concept: The slope-intercept form of a linear equation is y = mx + b, where:

m represents the slope of the line.
b represents the y-intercept of the line.

The slope (m) indicates the rate of change of y with respect to x. It can be calculated as "rise over run," which is the change in y divided by the change in x between any two points on the line. A positive slope indicates an increasing line (going upwards from left to right), a negative slope indicates a decreasing line (going downwards from left to right), a zero slope indicates a horizontal line, and an undefined slope indicates a vertical line (which is technically not a function).

The y-intercept (b) is the point where the line crosses the y-axis. It is the value of y when x = 0. It's a crucial piece of information for understanding the starting value or initial condition in many real-world applications.

Concrete Examples:

Example 1: Consider the equation y = 3x + 2.
Setup: This equation is already in slope-intercept form.
Process: By comparing it to the general form y = mx + b, we can identify that m = 3 and b = 2.
Result: The slope of the line is 3, and the y-intercept is 2. This means that for every increase of 1 in x, y increases by 3, and the line crosses the y-axis at the point (0, 2).
Why this matters: This equation could represent the cost of a taxi ride, where the initial fare is $2 and the cost per mile is $3.

Example 2: Consider the equation y = -1/2x - 1.
Setup: This equation is also in slope-intercept form.
Process: Comparing it to y = mx + b, we find that m = -1/2 and b = -1.
Result: The slope of the line is -1/2, and the y-intercept is -1. This means that for every increase of 2 in x, y decreases by 1, and the line crosses the y-axis at the point (0, -1).
Why this matters: This could represent the remaining battery life of a device that drains at a rate of 1/2 percent per minute, starting with a full charge (represented as -1 since it's decreasing from 0).

Analogies & Mental Models:

Think of it like... a ski slope. The slope (m) is the steepness of the hill. A positive slope means you're skiing downhill, and a negative slope means you're skiing uphill (or being lifted uphill). The y-intercept (b) is the height at which you start your ski run.
Explain how the analogy maps to the concept: A steeper ski slope means you descend (or ascend) faster for every unit of horizontal distance, just like a larger slope means a greater change in y for every unit change in x. The starting height of the ski run is the height when you're at the beginning (x=0), which corresponds to the y-intercept.
Where the analogy breaks down (limitations): Ski slopes usually have varying steepness, while linear functions have a constant slope. Also, ski slopes are typically limited in length, while linear functions extend infinitely.

Common Misconceptions:

❌ Students often confuse the slope and y-intercept in the equation y = mx + b.
✓ Actually, m is always the coefficient of x and represents the slope, while b is the constant term and represents the y-intercept.
Why this confusion happens: Students may not pay close attention to the position of the variables and constants in the equation.

Visual Description:

Imagine a graph with a straight line. The slope-intercept form tells you two key things about this line: where it crosses the y-axis (the y-intercept) and how steep it is (the slope). If you know these two things, you can easily draw the line.

Practice Check:

What is the slope and y-intercept of the line represented by the equation y = -2x + 7?

Answer: The slope is -2, and the y-intercept is 7.

Connection to Other Sections:

This section builds upon the definition of linear functions and introduces a specific way to represent them. The next section will explore another form of linear equations: point-slope form.

### 4.3 Forms of Linear Equations: Point-Slope Form

Overview: Point-slope form provides a way to write the equation of a line when you know a point on the line and its slope. It's particularly useful when you don't know the y-intercept directly.

The Core Concept: The point-slope form of a linear equation is y - y₁ = m(x - x₁), where:

m is the slope of the line.
(x₁, y₁) is a known point on the line.

This form is derived from the definition of slope: m = (y₂ - y₁) / (x₂ - x₁). By rearranging this formula and replacing (x₂, y₂) with a general point (x, y) on the line, we arrive at the point-slope form. The key is that this form allows you to write the equation of a line using any point on the line, not just the y-intercept.

Concrete Examples:

Example 1: Find the equation of a line that passes through the point (2, 3) and has a slope of 4.
Setup: We know m = 4, x₁ = 2, and y₁ = 3.
Process: Substitute these values into the point-slope form: y - 3 = 4(x - 2).
Result: The equation of the line in point-slope form is y - 3 = 4(x - 2). This can be simplified to slope-intercept form: y = 4x - 5.
Why this matters: This allows you to model scenarios where you know a starting point and a rate of change, even if the initial value (y-intercept) is not immediately apparent.

Example 2: Find the equation of a line that passes through the point (-1, -2) and has a slope of -1/2.
Setup: We know m = -1/2, x₁ = -1, and y₁ = -2.
Process: Substitute these values into the point-slope form: y - (-2) = -1/2(x - (-1)).
Result: The equation of the line in point-slope form is y + 2 = -1/2(x + 1). This can be simplified to slope-intercept form: y = -1/2x - 5/2.
Why this matters: This can model situations where you have a specific data point and a rate of decline.

Analogies & Mental Models:

Think of it like... plotting a course on a map. You know your current location (a point) and the direction you want to travel (the slope). The point-slope form helps you define the entire path.
Explain how the analogy maps to the concept: Your current location is the point (x₁, y₁). The direction you want to travel is the slope m. The point-slope form tells you how to get to any other point (x, y) on your path, given your starting point and direction.
Where the analogy breaks down (limitations): Maps are usually limited in size and scope, while linear functions extend infinitely. Also, real-world travel often involves changes in direction (non-constant slope), while linear functions have a constant slope.

Common Misconceptions:

❌ Students often forget to distribute the slope (m) to both terms inside the parentheses when converting from point-slope form to slope-intercept form.
✓ Actually, you must multiply the slope by both x and x₁ to correctly simplify the equation.
Why this confusion happens: Students may rush through the algebraic manipulation without paying attention to the distributive property.

Visual Description:

Imagine a graph with a point marked on it. The point-slope form allows you to draw a line through that point with a specific slope. You can visualize the slope as "rise over run" from that point to any other point on the line.

Practice Check:

Write the equation of a line in point-slope form that passes through the point (1, -4) and has a slope of 2.

Answer: y + 4 = 2(x - 1)

Connection to Other Sections:

This section provides another way to represent linear equations, building upon the understanding of slope and points on a line. The next section will introduce the standard form of a linear equation.

### 4.4 Forms of Linear Equations: Standard Form

Overview: The standard form of a linear equation is a general representation that is particularly useful for solving systems of linear equations and for identifying intercepts.

The Core Concept: The standard form of a linear equation is Ax + By = C, where:

A, B, and C are constants (real numbers).
A and B cannot both be zero.
Ideally, A is a positive integer.

While the standard form doesn't directly reveal the slope and y-intercept like slope-intercept form, it's useful for several reasons:

Finding Intercepts: It's easy to find the x-intercept (where the line crosses the x-axis) by setting y = 0 and solving for x. Similarly, you can find the y-intercept by setting x = 0 and solving for y.
Solving Systems of Equations: Standard form is often used when solving systems of linear equations using methods like elimination.
General Representation: It's a general form that can represent any linear equation (except vertical lines, where B = 0).

Concrete Examples:

Example 1: Consider the equation 2x + 3y = 6.
Setup: This equation is already in standard form.
Process: To find the x-intercept, set y = 0: 2x + 3(0) = 6 => 2x = 6 => x = 3. To find the y-intercept, set x = 0: 2(0) + 3y = 6 => 3y = 6 => y = 2.
Result: The x-intercept is 3 (the point (3, 0)), and the y-intercept is 2 (the point (0, 2)).
Why this matters: Knowing the intercepts allows you to quickly graph the line.

Example 2: Convert the equation y = -1/2x + 4 to standard form.
Setup: We need to rearrange the equation to the form Ax + By = C.
Process: Add 1/2x to both sides: 1/2x + y = 4. Multiply both sides by 2 to eliminate the fraction: x + 2y = 8.
Result: The equation in standard form is x + 2y = 8.
Why this matters: Converting to standard form can be helpful for solving systems of equations or for easily finding intercepts.

Analogies & Mental Models:

Think of it like... organizing ingredients in a recipe. Standard form is like having all the ingredients (x and y terms) on one side of the equation, and the total result (the constant term) on the other side.
Explain how the analogy maps to the concept: The x and y terms are like the different ingredients in the recipe. The coefficients A and B are like the amounts of each ingredient. The constant C is the total amount of the final dish.
Where the analogy breaks down (limitations): Recipes usually involve specific units of measurement, while standard form is a more abstract mathematical representation.

Common Misconceptions:

❌ Students often forget that A, B, and C must be constants when writing an equation in standard form.
✓ Actually, the coefficients of x and y and the constant term must be real numbers.
Why this confusion happens: Students may try to include variables in the coefficients or constant term.

Visual Description:

Imagine a graph with a straight line. The standard form allows you to easily find where the line crosses both the x-axis and the y-axis. These two points are enough to define the line.

Practice Check:

Convert the equation y = 5x - 3 to standard form.

Answer: -5x + y = -3 (or equivalently, 5x - y = 3)

Connection to Other Sections:

This section introduces the standard form of a linear equation, providing another tool for representing and manipulating linear relationships. The next section will focus on understanding slope and intercepts in more detail.

### 4.5 Understanding Slope

Overview: Slope is a fundamental concept in linear functions, representing the rate of change of a line. A thorough understanding of slope is essential for interpreting and applying linear functions effectively.

The Core Concept: The slope (m) of a line measures its steepness and direction. It is defined as the change in y (the rise) divided by the change in x (the run) between any two points on the line:

m = (y₂ - y₁) / (x₂ - x₁)

A positive slope indicates an increasing line (going upwards from left to right). A negative slope indicates a decreasing line (going downwards from left to right). A zero slope indicates a horizontal line (y = constant). An undefined slope indicates a vertical line (x = constant). Vertical lines are not functions.

The slope represents the constant rate of change of the linear function. In real-world applications, the slope often represents a rate, such as speed, cost per unit, or rate of growth.

Concrete Examples:

Example 1: Find the slope of the line passing through the points (1, 2) and (4, 8).
Setup: We have x₁ = 1, y₁ = 2, x₂ = 4, and y₂ = 8.
Process: Apply the slope formula: m = (8 - 2) / (4 - 1) = 6 / 3 = 2.
Result: The slope of the line is 2.
Why this matters: This means that for every increase of 1 in x, y increases by 2.

Example 2: Find the slope of the line passing through the points (-2, 5) and (3, -5).
Setup: We have x₁ = -2, y₁ = 5, x₂ = 3, and y₂ = -5.
Process: Apply the slope formula: m = (-5 - 5) / (3 - (-2)) = -10 / 5 = -2.
Result: The slope of the line is -2.
Why this matters: This means that for every increase of 1 in x, y decreases by 2.

Analogies & Mental Models:

Think of it like... the grade of a road. The slope is like the steepness of the hill. A higher grade means a steeper hill.
Explain how the analogy maps to the concept: A steeper road means a greater change in elevation for every unit of horizontal distance, just like a larger slope means a greater change in y for every unit change in x.
Where the analogy breaks down (limitations): Roads often have varying grades, while linear functions have a constant slope. Also, roads can have curves and bends, while linear functions are straight lines.

Common Misconceptions:

❌ Students often mix up the order of subtraction in the slope formula, calculating (x₂ - x₁) / (y₂ - y₁) instead of (y₂ - y₁) / (x₂ - x₁).
✓ Actually, the change in y must be in the numerator, and the change in x must be in the denominator.
Why this confusion happens: Students may not fully understand the definition of slope as "rise over run."

Visual Description:

Imagine a graph with a line. The slope is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. You can visualize this as a small right triangle where the vertical side is the rise and the horizontal side is the run.

Practice Check:

What is the slope of a horizontal line? What is the slope of a vertical line?

Answer: The slope of a horizontal line is 0. The slope of a vertical line is undefined.

Connection to Other Sections:

This section provides a detailed understanding of slope, which is a key component of linear functions. The next section will focus on understanding intercepts.

### 4.6 Understanding Intercepts

Overview: Intercepts are the points where a line crosses the x-axis and y-axis. They provide important information about the function's behavior and can be easily identified from the equation or graph.

The Core Concept:

Y-intercept: The y-intercept is the point where the line crosses the y-axis. At this point, x = 0. The y-intercept is often denoted as b in the slope-intercept form y = mx + b.
X-intercept: The x-intercept is the point where the line crosses the x-axis. At this point, y = 0. To find the x-intercept, set y = 0 in the equation and solve for x.

Intercepts provide valuable information about the function. The y-intercept represents the initial value or starting point in many real-world applications. The x-intercept represents the value of x when y is zero, which can be a significant point in certain contexts (e.g., the break-even point in a business model).

Concrete Examples:

Example 1: Find the y-intercept and x-intercept of the line y = 2x - 4.
Setup: The equation is in slope-intercept form.
Process: The y-intercept is the constant term, so b = -4. To find the x-intercept, set y = 0: 0 = 2x - 4 => 2x = 4 => x = 2.
Result: The y-intercept is -4 (the point (0, -4)), and the x-intercept is 2 (the point (2, 0)).
Why this matters: The y-intercept tells us the value of y when x is zero, and the x-intercept tells us the value of x when y is zero.

Example 2: Find the y-intercept and x-intercept of the line 3x + 4y = 12.
Setup: The equation is in standard form.
Process: To find the y-intercept, set x = 0: 3(0) + 4y = 12 => 4y = 12 => y = 3. To find the x-intercept, set y = 0: 3x + 4(0) = 12 => 3x = 12 => x = 4.
Result: The y-intercept is 3 (the point (0, 3)), and the x-intercept is 4 (the point (4, 0)).
Why this matters: These intercepts can be used to easily graph the line.

Analogies & Mental Models:

Think of it like... a runner crossing the finish line. The x-intercept is like the finish line (where y = 0), and the y-intercept is like the starting point (where x = 0).
Explain how the analogy maps to the concept: The x-intercept represents the point where the runner completes the race (y = 0). The y-intercept represents the runner's starting position (x = 0).
Where the analogy breaks down (limitations): Runners typically move in a forward direction, while linear functions can have both positive and negative values for x and y.

Common Misconceptions:

❌ Students often confuse the x-intercept and y-intercept, thinking that the y-intercept is always where y = 0.
✓ Actually, the y-intercept is where x = 0, and the x-intercept is where y = 0.
Why this confusion happens: Students may not fully understand the definitions of the intercepts and their relationship to the axes.

Visual Description:

Imagine a graph with a straight line. The y-intercept is the point where the line crosses the vertical y-axis. The x-intercept is the point where the line crosses the horizontal x-axis.

Practice Check:

What is the y-intercept of the line y = -x + 5? What is the x-intercept?

Answer: The y-intercept is 5. The x-intercept is 5.

Connection to Other Sections:

This section provides a detailed understanding of intercepts, which are important characteristics of linear functions. The next section will focus on graphing linear functions.

### 4.7 Graphing Linear Functions

Overview: Graphing linear functions is a visual representation of the relationship between the variables. Understanding how to graph linear functions is crucial for visualizing their behavior and interpreting their properties.

The Core Concept: There are several methods for graphing linear functions:

1. Using Slope-Intercept Form (y = mx + b):
Plot the y-intercept (0, b).
Use the slope (m) to find another point on the line. Remember that slope is "rise over run." From the y-intercept, move up (or down if the slope is negative) by the amount of the rise and then move right by the amount of the run.
Draw a straight line through the two points.

2. Using Two Points:
Find two points that satisfy the equation. This can be done by choosing two values for x and calculating the corresponding values for y.
Plot the two points on the coordinate plane.
Draw a straight line through the two points.

3. Using Intercepts:
* Find the x-intercept (where y = 0) and the y-intercept (

Okay, here is a comprehensive lesson on Linear Functions, designed for Algebra I students (grades 9-12), with the depth, detail, and engagement you requested.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're saving up for a new gaming console that costs $500. You already have $50 saved. You decide to mow lawns in your neighborhood and charge $25 per lawn. How many lawns do you need to mow to reach your goal? This seemingly simple question involves a fundamental concept in mathematics: linear functions. Linear functions aren't just abstract equations; they are powerful tools that help us understand and predict relationships between quantities that change at a constant rate. From calculating earnings to predicting distances, linear functions are all around us, often hidden in plain sight. Think about how much your phone bill costs each month - often a fixed cost plus a charge per gigabyte of data. That's a linear function!

### 1.2 Why This Matters

Understanding linear functions is crucial for several reasons. First, they provide a foundation for more advanced mathematical concepts like calculus and linear algebra. Second, they are used extensively in various fields, including economics (modeling supply and demand), physics (describing motion), computer science (creating algorithms), and engineering (designing structures). If you're interested in becoming an economist, understanding how prices change linearly based on supply is critical. If you want to be a software engineer, you'll use linear functions to design efficient algorithms. Furthermore, mastering linear functions enhances your problem-solving skills, allowing you to analyze data, make predictions, and optimize solutions. Building on your prior knowledge of basic arithmetic and variable manipulation, this lesson will equip you with the tools to confidently tackle real-world problems and prepare you for future mathematical endeavors. Next, you'll encounter quadratic functions, exponential functions, and systems of equations, all of which build upon the principles we'll learn here.

### 1.3 Learning Journey Preview

In this lesson, we will embark on a journey to unravel the mysteries of linear functions. We'll start by defining what a linear function is and exploring its various forms, including slope-intercept form, point-slope form, and standard form. We will then delve into the concept of slope and how it represents the rate of change in a linear relationship. We'll learn how to calculate slope from graphs, tables, and equations. Next, we'll analyze different types of linear functions, such as increasing, decreasing, horizontal, and vertical lines. We'll investigate parallel and perpendicular lines and their unique properties. We will then explore how to write equations of linear functions given different pieces of information, such as slope and y-intercept, two points, or a point and a slope. Finally, we will apply our knowledge to solve real-world problems involving linear functions, such as modeling linear growth, calculating distances, and making predictions. By the end of this lesson, you'll have a solid understanding of linear functions and their applications.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

1. Define a linear function and identify its key characteristics (constant rate of change, straight-line graph).
2. Convert linear equations between slope-intercept form, point-slope form, and standard form.
3. Calculate the slope of a line given two points, an equation, or a graph.
4. Interpret the meaning of slope and y-intercept in real-world contexts.
5. Write the equation of a linear function given a slope and y-intercept, two points, or a point and a slope.
6. Graph linear functions accurately using slope-intercept form and point-slope form.
7. Determine whether two lines are parallel, perpendicular, or neither based on their slopes.
8. Apply linear functions to model and solve real-world problems involving linear relationships.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into linear functions, you should already be familiar with the following concepts:

Basic Arithmetic: Addition, subtraction, multiplication, and division of real numbers (integers, fractions, decimals).
Order of Operations: Following the correct order of operations (PEMDAS/BODMAS) when evaluating expressions.
Variables and Expressions: Understanding what a variable represents and how to manipulate algebraic expressions.
Solving Equations: Solving basic algebraic equations for a single variable (e.g., solving for x in 2x + 3 = 7). This includes using inverse operations.
Coordinate Plane: Understanding the Cartesian coordinate plane, including plotting points and identifying quadrants.
Terminology: Be comfortable with terms like "equation," "variable," "constant," "coefficient," and "intercept."

Quick Review:

Example: Solve for x: 5x - 10 = 25. (Add 10 to both sides: 5x = 35. Divide both sides by 5: x = 7).
Example: Plot the point (3, -2) on the coordinate plane. (Move 3 units to the right on the x-axis and 2 units down on the y-axis).

If you need a refresher on any of these topics, consult your previous algebra notes, online resources like Khan Academy, or ask your teacher for assistance. These foundational skills are essential for success with linear functions.

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## 4. MAIN CONTENT

### 4.1 What is a Linear Function?

Overview: A linear function represents a relationship between two variables where the change in one variable is directly proportional to the change in the other. This proportionality creates a straight-line graph when plotted on a coordinate plane.

The Core Concept: At its heart, a linear function describes a constant rate of change. Imagine filling a swimming pool with a hose. If the hose delivers water at a constant rate (e.g., 10 gallons per minute), the amount of water in the pool increases linearly with time. The key characteristic of a linear function is that for every equal increase in the input (usually denoted by x), there is an equal increase (or decrease) in the output (usually denoted by y). This creates a straight line when the points are plotted on a graph. The "straightness" of the line is the visual representation of the constant rate of change. Mathematically, a linear function can be represented by an equation of the form y = mx + b, where m and b are constants. The constant m represents the slope of the line, which indicates the rate of change, and the constant b represents the y-intercept, which is the point where the line crosses the y-axis. Linear functions can also be represented in other forms, such as point-slope form and standard form, which we will explore later. However, the underlying principle of a constant rate of change remains the same.

Concrete Examples:

Example 1: Earning Money at a Fixed Hourly Rate
Setup: You work at a part-time job and earn $15 per hour.
Process: Let x represent the number of hours you work, and let y represent your total earnings. The relationship between hours worked and total earnings can be expressed as y = 15x. For every additional hour you work (x increases by 1), your total earnings increase by $15 (y increases by 15).
Result: This is a linear function because the rate of change (your hourly wage) is constant. If you work 2 hours, you earn $30. If you work 4 hours, you earn $60. The earnings increase proportionally to the number of hours worked.
Why this matters: This is a simple example of how linear functions can be used to model real-world situations involving constant rates of change.

Example 2: The Cost of a Taxi Ride
Setup: A taxi charges a flat fee of $3 plus $2.50 per mile.
Process: Let x represent the number of miles traveled, and let y represent the total cost of the ride. The relationship can be expressed as y = 2.50x + 3. The $2.50 represents the cost per mile (the slope), and the $3 represents the initial fee (the y-intercept).
Result: This is a linear function because the cost increases at a constant rate of $2.50 per mile. If you travel 1 mile, the cost is $5.50. If you travel 3 miles, the cost is $10.50.
Why this matters: This illustrates how linear functions can model situations with a fixed initial cost plus a variable cost that increases at a constant rate.

Analogies & Mental Models:

Think of it like... a ramp. A ramp has a constant slope. For every foot you move horizontally, you rise a certain fixed amount vertically. A linear function is like that ramp, but instead of physical distance, it represents the relationship between two variables.
How the analogy maps to the concept: The steepness of the ramp is analogous to the slope of the line. A steeper ramp means a larger slope, indicating a faster rate of change.
Where the analogy breaks down (limitations): A ramp is limited to positive slopes (going uphill). Linear functions can also have negative slopes (going downhill), representing a decreasing relationship.

Common Misconceptions:

❌ Students often think... that any equation with x and y is a linear function.
✓ Actually... a linear function must have a constant rate of change, which means the variables are raised to the power of 1. Equations like y = x² or y = √x are not linear.
Why this confusion happens: Students might focus solely on the presence of x and y without considering the relationship between them.

Visual Description:

Imagine a straight line drawn on a graph. The line can go up to the right, down to the right, be perfectly horizontal, or perfectly vertical. The key visual element is that it's a straight line, not a curve. If the line goes up to the right, the y values are increasing as the x values increase. If it goes down to the right, the y values are decreasing as the x values increase. A horizontal line means the y value stays the same regardless of the x value. A vertical line means the x value stays the same regardless of the y value.

Practice Check:

Which of the following equations represents a linear function?

a) y = 3x + 5
b) y = x² - 2
c) y = 1/x
d) y = √(x + 1)

Answer: a) y = 3x + 5 is a linear function because it is in the form y = mx + b. The other options involve non-linear terms like x², 1/x, and √x.

Connection to Other Sections: This section lays the groundwork for understanding all subsequent sections. We will build upon this definition to explore different forms of linear equations, calculate slope, and apply linear functions to real-world problems.

### 4.2 Slope: The Rate of Change

Overview: Slope is a measure of how much a line rises or falls for every unit of horizontal change. It's the "steepness" of the line and represents the constant rate of change in a linear function.

The Core Concept: The slope of a line is often described as "rise over run." The "rise" is the vertical change (change in y), and the "run" is the horizontal change (change in x). Mathematically, the slope (m) between two points (x₁, y₁) and (x₂, y₂) is calculated using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

A positive slope indicates that the line is increasing (going uphill from left to right). A negative slope indicates that the line is decreasing (going downhill from left to right). A slope of zero indicates a horizontal line (no vertical change). An undefined slope indicates a vertical line (no horizontal change). The larger the absolute value of the slope, the steeper the line. For instance, a slope of 5 is steeper than a slope of 2. Conversely, a slope of -3 is steeper than a slope of -1.

Concrete Examples:

Example 1: Calculating Slope from Two Points
Setup: Find the slope of the line passing through the points (1, 2) and (4, 8).
Process: Let (1, 2) be (x₁, y₁) and (4, 8) be (x₂, y₂). Using the slope formula:
m = (8 - 2) / (4 - 1) = 6 / 3 = 2
Result: The slope of the line is 2. This means that for every 1 unit increase in x, y increases by 2 units.
Why this matters: This demonstrates how to calculate slope from two given points, which is a fundamental skill in working with linear functions.

Example 2: Interpreting Slope in a Real-World Scenario
Setup: A graph shows the distance a car travels over time. The graph is a straight line passing through the points (0, 0) and (2, 120), where x represents time in hours and y represents distance in miles.
Process: Calculate the slope using the two points:
m = (120 - 0) / (2 - 0) = 120 / 2 = 60
Result: The slope is 60. This means the car is traveling at a speed of 60 miles per hour.
Why this matters: This illustrates how slope can be interpreted as a rate of change in a real-world context, such as speed in this case.

Analogies & Mental Models:

Think of it like... climbing a staircase. The slope is how many steps you go up for every step you go forward. If you go down instead of up, it’s a negative slope.
How the analogy maps to the concept: The height of each step is the "rise," and the depth of each step is the "run." The steeper the staircase, the larger the slope.
Where the analogy breaks down (limitations): Staircases usually have integer values for rise and run. Slopes can be any real number (fractions, decimals, etc.).

Common Misconceptions:

❌ Students often think... that the order of the points doesn't matter when calculating slope.
✓ Actually... the order matters! You must be consistent. If you subtract y₁ from y₂ in the numerator, you must subtract x₁ from x₂ in the denominator. Switching the order will result in the wrong sign for the slope.
Why this confusion happens: Students might forget that the slope formula represents the change in y divided by the change in x.

Visual Description:

Imagine a line on a graph. Pick any two points on the line. Draw a right triangle with the line as the hypotenuse. The vertical side of the triangle is the "rise," and the horizontal side is the "run." The slope is the ratio of the rise to the run. If the line goes up to the right, the rise is positive. If it goes down to the right, the rise is negative. The run is always considered positive (moving from left to right).

Practice Check:

A line passes through the points (2, 5) and (6, 13). What is the slope of the line?

Answer: m = (13 - 5) / (6 - 2) = 8 / 4 = 2. The slope is 2.

Connection to Other Sections: Understanding slope is crucial for writing equations of linear functions (Section 4.4) and determining whether lines are parallel or perpendicular (Section 4.5).

### 4.3 Forms of Linear Equations

Overview: Linear equations can be expressed in different forms, each highlighting specific characteristics of the line. The most common forms are slope-intercept form, point-slope form, and standard form.

The Core Concept: Each form of a linear equation offers a unique perspective on the line it represents.

Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept. This form is particularly useful for quickly identifying the slope and y-intercept of a line and for graphing the line.
Point-Slope Form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line. This form is useful when you know the slope and a point on the line and want to find the equation.
Standard Form: Ax + By = C, where A, B, and C are constants, and A and B are not both zero. While not as directly revealing as the other forms, standard form is useful for certain algebraic manipulations and for representing systems of linear equations.

The ability to convert between these forms is essential for solving various problems involving linear functions. For example, if you are given an equation in standard form, you can convert it to slope-intercept form to easily identify the slope and y-intercept.

Concrete Examples:

Example 1: Converting from Standard Form to Slope-Intercept Form
Setup: Convert the equation 2x + 3y = 6 to slope-intercept form.
Process:
1. Subtract 2x from both sides: 3y = -2x + 6
2. Divide both sides by 3: y = (-2/3)x + 2
Result: The equation in slope-intercept form is y = (-2/3)x + 2. The slope is -2/3, and the y-intercept is 2.
Why this matters: This shows how to manipulate an equation to reveal the slope and y-intercept, making it easier to graph and analyze the line.

Example 2: Using Point-Slope Form to Write an Equation
Setup: Write the equation of the line that passes through the point (3, 5) and has a slope of 2.
Process: Using the point-slope form y - y₁ = m(x - x₁), substitute m = 2, x₁ = 3, and y₁ = 5:
y - 5 = 2(x - 3)
Result: The equation in point-slope form is y - 5 = 2(x - 3). You can leave it in this form or convert it to slope-intercept form by distributing and isolating y: y = 2x - 1.
Why this matters: This demonstrates how to use point-slope form to quickly write the equation of a line when you know a point and the slope.

Analogies & Mental Models:

Think of it like... different languages for describing the same idea. Each form of a linear equation is like a different language for describing the same line. Slope-intercept form is like English, clear and direct. Point-slope form is like Spanish, good for a specific context. Standard form is like Latin, a little more formal.
How the analogy maps to the concept: Just like you can translate between languages, you can convert between different forms of linear equations.
Where the analogy breaks down (limitations): Languages have nuances and cultural context. Linear equation forms are purely mathematical and don't have the same level of complexity.

Common Misconceptions:

❌ Students often think... that they must always convert to slope-intercept form to solve problems.
✓ Actually... the best form to use depends on the problem. Point-slope form is often more convenient when you're given a point and a slope. Standard form is useful for certain types of algebraic manipulations.
Why this confusion happens: Students might focus on memorizing one form without understanding the advantages of each.

Visual Description:

Slope-Intercept Form: You can immediately see the slope (how steep the line is) and where the line crosses the y-axis.
Point-Slope Form: You can see one specific point the line passes through and the slope of the line. Imagine "anchoring" the line at that point and then rotating it based on the slope.
Standard Form: It's harder to "see" the slope and y-intercept directly. However, it's useful for visualizing systems of equations where you want to find the intersection of two lines.

Practice Check:

Convert the equation y = -3x + 7 to standard form.

Answer: Add 3x to both sides: 3x + y = 7. This is the equation in standard form.

Connection to Other Sections: This section provides the tools for writing and manipulating linear equations, which are essential for graphing lines (Section 4.6) and solving real-world problems (Section 4.8).

### 4.4 Writing Equations of Linear Functions

Overview: Given different pieces of information about a line (slope and y-intercept, two points, or a point and a slope), you can write its equation using the appropriate form.

The Core Concept: The key to writing the equation of a linear function is to identify the slope (m) and a point on the line. Once you have this information, you can use either slope-intercept form or point-slope form to write the equation.

Given Slope and Y-Intercept: Use slope-intercept form directly: y = mx + b.
Given a Point and a Slope: Use point-slope form: y - y₁ = m(x - x₁). Then, if desired, convert to slope-intercept form.
Given Two Points: First, calculate the slope using the slope formula: m = (y₂ - y₁) / (x₂ - x₁). Then, choose either point and use point-slope form to write the equation.

Concrete Examples:

Example 1: Writing an Equation Given a Slope and Y-Intercept
Setup: Write the equation of a line with a slope of 4 and a y-intercept of -2.
Process: Use slope-intercept form: y = mx + b. Substitute m = 4 and b = -2: y = 4x - 2.
Result: The equation of the line is y = 4x - 2.
Why this matters: This is the most straightforward case, demonstrating the direct application of slope-intercept form.

Example 2: Writing an Equation Given a Point and a Slope
Setup: Write the equation of a line that passes through the point (2, 7) and has a slope of -3.
Process: Use point-slope form: y - y₁ = m(x - x₁). Substitute m = -3, x₁ = 2, and y₁ = 7: y - 7 = -3(x - 2). You can also convert to slope-intercept form: y - 7 = -3x + 6 => y = -3x + 13.
Result: The equation of the line is y - 7 = -3(x - 2) (point-slope form) or y = -3x + 13 (slope-intercept form).
Why this matters: This demonstrates the usefulness of point-slope form when you're given a point and a slope.

Example 3: Writing an Equation Given Two Points
Setup: Write the equation of the line that passes through the points (1, 3) and (4, 9).
Process:
1. Calculate the slope: m = (9 - 3) / (4 - 1) = 6 / 3 = 2.
2. Use point-slope form with the point (1, 3): y - 3 = 2(x - 1).
3. Convert to slope-intercept form: y - 3 = 2x - 2 => y = 2x + 1.
Result: The equation of the line is y = 2x + 1.
Why this matters: This shows how to combine the slope formula and point-slope form to write an equation when you're given two points.

Analogies & Mental Models:

Think of it like... building a road. You need to know the direction (slope) and a starting point to build the road in a straight line.
How the analogy maps to the concept: The slope is like the direction of the road, and the point is like a marker that tells you where the road starts.
Where the analogy breaks down (limitations): Roads can curve. Linear functions are always straight lines.

Common Misconceptions:

❌ Students often think... that they need to memorize a separate formula for each case (slope and y-intercept, two points, etc.).
✓ Actually... you only need to understand the slope formula and point-slope form. Slope-intercept form is just a special case of point-slope form where the point is the y-intercept.
Why this confusion happens: Students might try to memorize formulas without understanding the underlying concepts.

Visual Description:

Imagine plotting the given information on a graph. If you have a slope and y-intercept, you can immediately draw the line. If you have a point and a slope, you can start at the point and use the slope to "walk" along the line. If you have two points, you can draw a line through them.

Practice Check:

Write the equation of the line that passes through the point (-2, 1) and is parallel to the line y = 3x - 5. (Hint: Parallel lines have the same slope).

Answer: The slope of the parallel line is 3. Use point-slope form: y - 1 = 3(x + 2), or in slope-intercept form: y = 3x + 7.

Connection to Other Sections: This section is essential for graphing linear functions (Section 4.6) and solving real-world problems (Section 4.8).

### 4.5 Parallel and Perpendicular Lines

Overview: Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other.

The Core Concept: The relationship between the slopes of parallel and perpendicular lines is a fundamental concept in geometry and algebra.

Parallel Lines: Two lines are parallel if and only if they have the same slope. This means they have the same steepness and will never intersect. If y = m₁x + b₁ and y = m₂x + b₂ are parallel, then m₁ = m₂.
Perpendicular Lines: Two lines are perpendicular if and only if their slopes are negative reciprocals of each other. This means that the product of their slopes is -1. If y = m₁x + b₁ and y = m₂x + b₂ are perpendicular, then m₁ m₂ = -1, or m₂ = -1/m₁. Vertical and horizontal lines are also perpendicular. A vertical line has an undefined slope and a horizontal line has a slope of 0.

Concrete Examples:

Example 1: Determining if Lines are Parallel
Setup: Are the lines y = 2x + 3 and y = 2x - 1 parallel?
Process: Both lines are in slope-intercept form. The slope of the first line is 2, and the slope of the second line is 2.
Result: Since the slopes are equal, the lines are parallel.
Why this matters: This demonstrates the direct application of the parallel line condition.

Example 2: Determining if Lines are Perpendicular
Setup: Are the lines y = (1/3)x + 4 and y = -3x + 2 perpendicular?
Process: The slope of the first line is 1/3, and the slope of the second line is -3. The product of the slopes is (1/3) (-3) = -1.
Result: Since the product of the slopes is -1, the lines are perpendicular.
Why this matters: This demonstrates the application of the perpendicular line condition.

Example 3: Finding the Equation of a Line Perpendicular to Another Line
Setup: Find the equation of a line that passes through the point (5, 1) and is perpendicular to the line y = -2x + 6.
Process:
1. The slope of the given line is -2. The slope of a perpendicular line is the negative reciprocal of -2, which is 1/2.
2. Use point-slope form with the point (5, 1) and the slope 1/2: y - 1 = (1/2)(x - 5).
3. Convert to slope-intercept form: y = (1/2)x - 5/2 + 1 => y = (1/2)x - 3/2
Result: The equation of the line is y = (1/2)x - 3/2.
Why this matters: This combines the concept of perpendicular lines with writing equations of linear functions.

Analogies & Mental Models:

Think of it like... train tracks (parallel) and the corner of a square (perpendicular). Train tracks run side by side and never intersect. The corner of a square forms a perfect right angle.
How the analogy maps to the concept: Parallel lines are like train tracks, always running in the same direction. Perpendicular lines are like the sides of a square, meeting at a 90-degree angle.
Where the analogy breaks down (limitations): Train tracks can curve and eventually intersect. Parallel lines on a flat plane never intersect.

Common Misconceptions:

❌ Students often think... that perpendicular lines must have negative slopes.
✓ Actually... one line can have a positive slope, and the other must have a negative slope. The key is that they are negative reciprocals of each other. Also, a horizontal and vertical line are perpendicular, and one has a slope of 0 while the other is undefined.
Why this confusion happens: Students might focus on the "negative" part of "negative reciprocal" without understanding the "reciprocal" part.

Visual Description:

Imagine two lines on a graph. If they are parallel, they look like they are running side by side and never getting closer or further apart. If they are perpendicular, they intersect at a perfect right angle (90 degrees).

Practice Check:

Line A has the equation y = 4x - 2. Line B passes through the point (1, 5) and is perpendicular to Line A. What is the equation of Line B?

Answer: The slope of Line A is 4. The slope of Line B is -1/4. Using point-slope form: y - 5 = (-1/4)(x - 1), or in slope-intercept form: y = (-1/4)x + 21/4.

Connection to Other Sections: This section builds upon the understanding of slope and equation writing, and it is relevant to geometry and other areas of mathematics.

### 4.6 Graphing Linear Functions

Overview: Graphing linear functions involves plotting points on a coordinate plane and drawing a straight line through them. Understanding slope-intercept form and point-slope form makes graphing much easier.

The Core Concept: Graphing a linear function is the visual representation of the relationship between x and y.

Using Slope-Intercept Form: To graph y = mx + b, start by plotting the y-intercept (0, b). Then, use the slope m to find another point on the line. Remember that slope is "rise over run." From the y-intercept, move "run" units horizontally and "rise" units vertically to find another point. Draw a straight line through the two points.
Using Point-Slope Form: To graph y - y₁ = m(x - x₁), start by plotting the point (x₁, y₁). Then, use the slope m to find another point on the line, as described above. Draw a straight line through the two points.
Using Two Points: Plot the two points on the coordinate plane and draw a straight line through them.

Concrete Examples:

Example 1: Graphing Using Slope-Intercept Form
Setup: Graph the line y = (2/3)x + 1.
Process:
1. The y-intercept is 1, so plot the point (0, 1).
2. The slope is 2/3. From (0, 1), move 3 units to the right (run) and 2 units up (rise) to find another point (3, 3).
3. Draw a straight line through the points (0, 1) and (3, 3).
Result: You have graphed the line y = (2/3)x + 1.
Why this matters: This demonstrates the ease of graphing when the equation is in slope-intercept form.

Example 2: Graphing Using Point-Slope Form
Setup: Graph the line y - 2 = -1(x + 1).
Process:
1. The point is (-1, 2), so plot the point (-1, 2).
2. The slope is -1 (or -1/1). From (-1, 2), move 1 unit to the right (run) and 1 unit down (rise) to find another point (0, 1).
3. Draw a straight line through the points (-1, 2) and (0, 1).
Result: You have graphed the line y - 2

Okay, here is a comprehensive lesson on Algebra I: Linear Functions, designed to be deeply structured, engaging, and suitable for high school students. It emphasizes real-world applications, career connections, and a clear progression of concepts.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're starting a small business selling custom-designed phone cases. You need to figure out how much to charge for each case to make a profit. The materials cost you $5 per case, and you want to earn $10 in profit for each one you sell. How would you calculate the selling price? This simple scenario introduces the concept of a linear relationship: the selling price is directly related to the cost of materials plus your desired profit. Think about other situations where things change at a steady rate: the speed of a car, the amount of water filling a pool, or even the growth of a plant. Linear functions are the mathematical tools that allow us to understand and predict these kinds of changes.

Let's say you are saving up for a new video game. You have $20 already, and you plan to save $5 each week. Can you predict how much money you'll have after 6 weeks? After 10 weeks? After a whole year? Linear functions provide a framework for answering such questions, allowing you to model your savings progress and make informed decisions about your spending. This is not just abstract math; it's about understanding and controlling the world around you.

### 1.2 Why This Matters

Linear functions are fundamental building blocks in mathematics and are essential for understanding more complex concepts in algebra, calculus, and beyond. In the real world, linear models are used extensively in finance (calculating loan payments, tracking investments), physics (describing motion at a constant speed), engineering (designing structures and systems), and computer science (developing algorithms and data models). Understanding linear functions opens doors to a wide range of careers, from data analysis and financial modeling to engineering and software development.

This lesson builds upon your existing knowledge of basic arithmetic, variables, and equations. It provides a foundation for understanding other types of functions (quadratic, exponential, trigonometric) and introduces key concepts like slope, intercepts, and rates of change that are crucial for higher-level mathematics. Mastery of linear functions will empower you to analyze data, make predictions, and solve problems in various contexts, both inside and outside the classroom. After this, you can move on to solving systems of linear equations, which will allow you to solve even more complex, real-world problems.

### 1.3 Learning Journey Preview

In this lesson, we'll explore the world of linear functions step-by-step. We'll start by defining what a linear function is and how to represent it in different forms (equations, graphs, tables). Then, we'll delve into the key characteristics of linear functions, such as slope and intercepts, and learn how to calculate and interpret them. We'll also learn how to write equations of lines given different information (slope and a point, two points) and how to graph linear functions accurately. Finally, we'll apply our knowledge to solve real-world problems and see how linear functions are used in various fields. The concepts will build upon each other: Understanding slope will lead to writing equations, and writing equations will allow us to model real-world scenarios. We'll finish by looking at how linear functions are used in different careers, such as financial analysis, engineering, and data science.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

1. Define a linear function and identify whether a given equation, graph, or table represents a linear function.
2. Explain the meaning of slope and y-intercept in the context of a linear function and relate them to the graph of the line.
3. Calculate the slope of a line given two points on the line or its equation.
4. Write the equation of a line in slope-intercept form (y = mx + b) given the slope and y-intercept, a point and the slope, or two points on the line.
5. Graph a linear function accurately using slope-intercept form or by plotting points.
6. Apply linear functions to model and solve real-world problems involving constant rates of change.
7. Analyze the relationship between the slope of a line and its steepness and direction.
8. Compare and contrast different representations of linear functions (equations, graphs, tables) and translate between them.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into linear functions, you should have a solid understanding of the following:

Basic Arithmetic: Addition, subtraction, multiplication, and division of real numbers (including fractions and decimals).
Variables and Expressions: Understanding what a variable represents and how to evaluate algebraic expressions.
Solving Equations: Solving simple one- and two-step equations for a variable.
Coordinate Plane: Familiarity with the coordinate plane, including x-axis, y-axis, and plotting points.
Order of Operations (PEMDAS/BODMAS): Knowing the correct order to perform mathematical operations.
Integers: Understanding positive and negative numbers.

Quick Review: If you need a refresher on any of these topics, consult your previous math notes, online resources like Khan Academy (www.khanacademy.org), or your textbook. Make sure you are comfortable with these foundational concepts before proceeding.

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## 4. MAIN CONTENT

### 4.1 What is a Linear Function?

Overview: A linear function is a special type of mathematical relationship where the change in one variable (usually 'y') is directly proportional to the change in another variable (usually 'x'). This means that for every consistent change in 'x', there is a consistent, predictable change in 'y'.

The Core Concept: A linear function can be thought of as a straight line when plotted on a graph. The key characteristic is that the rate of change between the two variables is constant. This constant rate of change is called the slope of the line. The slope tells us how much 'y' changes for every one unit change in 'x'. A linear function can be represented in various forms, the most common being the slope-intercept form: y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept (the point where the line crosses the y-axis).

Not every equation with 'x' and 'y' is a linear function. For example, y = x^2 (a quadratic function) is not linear because the rate of change is not constant. As 'x' increases, 'y' increases at an increasing rate. Similarly, equations with 'x' in the denominator (e.g., y = 1/x) or with 'x' under a radical (e.g., y = √x) are not linear. The key is to look for a constant rate of change between the two variables. Linear functions are defined by the fact that the variables are raised to the power of 1.

The equation y = mx + b is the standard way to represent linear functions. However, it is important to note that other forms exist, such as standard form (Ax + By = C) and point-slope form (y - y1 = m(x - x1)), which we will discuss in later sections. Each form has its advantages depending on the information you are given and the type of problem you are trying to solve.

Concrete Examples:

Example 1: Cost of T-shirts

Setup: A company charges $10 per t-shirt plus a $5 setup fee for an order. Let 'x' be the number of t-shirts and 'y' be the total cost.
Process: The equation representing this situation is y = 10x + 5. If you buy 1 t-shirt (x=1), the cost is y = 10(1) + 5 = $15. If you buy 5 t-shirts (x=5), the cost is y = 10(5) + 5 = $55. Notice that for each additional t-shirt you buy, the cost increases by $10.
Result: This equation represents a linear function because the cost increases at a constant rate of $10 per t-shirt (the slope). The $5 setup fee is the y-intercept, representing the cost even if you buy zero t-shirts.
Why this matters: Businesses use linear functions to model costs, pricing, and profits. Understanding these relationships helps them make informed decisions about production and sales.

Example 2: Distance Traveled

Setup: A car is traveling at a constant speed of 60 miles per hour. Let 'x' be the time in hours and 'y' be the distance traveled in miles.
Process: The equation representing this situation is y = 60x. After 1 hour (x=1), the car has traveled y = 60(1) = 60 miles. After 3 hours (x=3), the car has traveled y = 60(3) = 180 miles. For each additional hour of driving, the distance increases by 60 miles.
Result: This is a linear function because the distance increases at a constant rate of 60 miles per hour (the slope). The y-intercept is 0, meaning that at time zero, the car has traveled zero miles.
Why this matters: Linear functions are used to model motion and calculate distances, times, and speeds. This is crucial in fields like transportation, logistics, and physics.

Analogies & Mental Models:

Think of it like... a water faucet filling a tank at a constant rate. The amount of water in the tank increases linearly with time. The rate at which the water flows is the slope, and the initial amount of water in the tank (if any) is the y-intercept.
Explain how the analogy maps to the concept: The constant flow rate of water is analogous to the constant rate of change in a linear function. The water level represents the 'y' value, and time represents the 'x' value.
Where the analogy breaks down (limitations): The analogy doesn't account for situations where the flow rate changes, which would result in a non-linear function.

Common Misconceptions:

❌ Students often think that any equation with 'x' and 'y' is a linear function.
✓ Actually, a linear function must have a constant rate of change. Equations with exponents on the variables or variables in the denominator are not linear.
Why this confusion happens: Students may not fully understand the concept of a constant rate of change and may focus only on the presence of variables.

Visual Description:

Imagine a straight line drawn on a graph. The line can slope upwards (positive slope), downwards (negative slope), or be horizontal (zero slope). The steeper the line, the larger the absolute value of the slope. The y-intercept is the point where the line crosses the vertical axis.

Practice Check:

Which of the following equations represents a linear function?

a) y = 2x + 3
b) y = x^2 - 1
c) y = 5/x
d) y = √x + 4

Answer: a) y = 2x + 3 is a linear function because it has a constant rate of change (2). The other equations are not linear because they involve exponents, division by a variable, or a square root.

Connection to Other Sections:

This section provides the foundation for understanding the rest of the lesson. It defines what a linear function is and introduces the key concepts of slope and y-intercept, which will be explored in more detail in subsequent sections. The ability to identify linear functions is crucial for applying them to real-world problems.

### 4.2 Slope: The Rate of Change

Overview: The slope of a line is a measure of its steepness and direction. It quantifies how much the 'y' value changes for every one unit change in the 'x' value. It is a crucial concept for understanding the behavior of linear functions.

The Core Concept: Slope is often referred to as "rise over run." The "rise" is the vertical change (change in 'y'), and the "run" is the horizontal change (change in 'x'). Mathematically, the slope ('m') is calculated as:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two distinct points on the line.

A positive slope indicates that the line is increasing (going upwards) as you move from left to right. A negative slope indicates that the line is decreasing (going downwards) as you move from left to right. A slope of zero indicates a horizontal line (no change in 'y' as 'x' changes). An undefined slope indicates a vertical line (infinite change in 'y' for no change in 'x').

The slope is the single most important characteristic of a linear function. It is the constant rate of change that defines the linear relationship. A steeper slope (larger absolute value) means a faster rate of change, while a shallower slope (smaller absolute value) means a slower rate of change.

Concrete Examples:

Example 1: Climbing a Hill

Setup: Imagine climbing a hill. For every 10 feet you walk horizontally (run), you gain 2 feet in elevation (rise).
Process: The slope of the hill is rise/run = 2/10 = 1/5 = 0.2. This means that for every 1 foot you walk horizontally, you gain 0.2 feet in elevation.
Result: The slope represents the steepness of the hill. A larger slope would mean a steeper hill.
Why this matters: Civil engineers use slope calculations to design roads, bridges, and other infrastructure projects, ensuring they are safe and efficient.

Example 2: Losing Weight

Setup: You are on a diet and lose 2 pounds per week. Let 'x' be the number of weeks and 'y' be your weight loss in pounds.
Process: The slope is -2 (negative because you are losing weight). This means that for every week that passes, you lose 2 pounds. The equation representing this situation is y = -2x.
Result: The negative slope indicates a decreasing trend.
Why this matters: Linear functions can be used to model weight loss or gain, allowing you to track progress and make adjustments to your diet and exercise plan.

Analogies & Mental Models:

Think of it like... a ramp. The steeper the ramp, the larger the slope. A very shallow ramp has a small slope, and a vertical wall has an undefined slope.
Explain how the analogy maps to the concept: The steepness of the ramp directly corresponds to the slope of the line. The rise and run of the ramp correspond to the change in 'y' and change in 'x', respectively.
Where the analogy breaks down (limitations): The analogy doesn't account for negative slopes, which would correspond to a ramp going downwards.

Common Misconceptions:

❌ Students often confuse the rise and run when calculating the slope.
✓ Actually, the slope is always rise (change in 'y') divided by run (change in 'x'). Make sure to subtract the y-values and x-values in the same order.
Why this confusion happens: Students may not pay attention to the order of subtraction or may mix up the x and y coordinates.

Visual Description:

Imagine a line on a graph. Pick two points on the line. Draw a vertical line segment connecting the two points. This is the "rise" (change in y). Draw a horizontal line segment connecting the two points. This is the "run" (change in x). The slope is the ratio of the rise to the run.

Practice Check:

Find the slope of the line passing through the points (1, 3) and (4, 9).

Answer: m = (9 - 3) / (4 - 1) = 6 / 3 = 2. The slope is 2.

Connection to Other Sections:

Understanding slope is essential for writing equations of lines (Section 4.4) and graphing linear functions (Section 4.5). The slope is the key parameter that determines the direction and steepness of the line.

### 4.3 Y-Intercept: Where the Line Crosses the Y-Axis

Overview: The y-intercept is the point where a line intersects the y-axis. It represents the value of 'y' when 'x' is equal to zero. The y-intercept is a critical component of a linear function and provides valuable information about its behavior.

The Core Concept: In the slope-intercept form of a linear equation (y = mx + b), the y-intercept is represented by the constant term 'b'. This value tells us where the line crosses the y-axis. The coordinates of the y-intercept are always (0, b).

The y-intercept can be interpreted as the initial value of the linear function. It is the starting point before any change in 'x' occurs. In real-world scenarios, the y-intercept often represents a fixed cost, an initial investment, or a starting value.

The y-intercept is important because it anchors the line on the coordinate plane. Knowing the slope and y-intercept allows you to accurately graph the linear function.

Concrete Examples:

Example 1: Taxi Fare

Setup: A taxi charges a flat fee of $3 plus $2 per mile. Let 'x' be the number of miles traveled and 'y' be the total fare.
Process: The equation representing this situation is y = 2x + 3. The y-intercept is 3.
Result: This means that even if you travel zero miles (x=0), you still have to pay the $3 flat fee. The $3 is the initial cost.
Why this matters: Understanding the y-intercept helps you calculate the cost of a taxi ride, even if you only travel a short distance.

Example 2: Saving Money

Setup: You start with $50 in your savings account and deposit $10 per week. Let 'x' be the number of weeks and 'y' be the total amount in your account.
Process: The equation representing this situation is y = 10x + 50. The y-intercept is 50.
Result: This means that you initially have $50 in your account before you start making any deposits.
Why this matters: The y-intercept tells you your starting point for saving money.

Analogies & Mental Models:

Think of it like... the starting point of a race. The y-intercept is where you begin the race before you start running (changing your position).
Explain how the analogy maps to the concept: The starting point represents the initial value of the linear function.
Where the analogy breaks down (limitations): This analogy doesn't directly represent negative y-intercepts.

Common Misconceptions:

❌ Students often confuse the y-intercept with the x-intercept (where the line crosses the x-axis).
✓ Actually, the y-intercept is where the line crosses the y-axis, which occurs when x = 0. The x-intercept is where the line crosses the x-axis, which occurs when y = 0.
Why this confusion happens: Students may not fully understand the coordinate system and the definitions of the x and y axes.

Visual Description:

Imagine a line on a graph. The y-intercept is the point where the line crosses the vertical (y) axis. It is the 'y' value when 'x' is zero.

Practice Check:

What is the y-intercept of the line y = -3x + 7?

Answer: The y-intercept is 7.

Connection to Other Sections:

The y-intercept, along with the slope, is essential for writing equations of lines in slope-intercept form (Section 4.4) and graphing linear functions (Section 4.5).

### 4.4 Writing Equations of Lines

Overview: Being able to write the equation of a line is a fundamental skill in algebra. It allows you to model and analyze linear relationships between variables. There are several ways to write the equation of a line, depending on the information you are given.

The Core Concept: There are three main forms for writing the equation of a line:

1. Slope-Intercept Form: y = mx + b (where 'm' is the slope and 'b' is the y-intercept). This form is useful when you know the slope and y-intercept.
2. Point-Slope Form: y - y1 = m(x - x1) (where 'm' is the slope and (x1, y1) is a point on the line). This form is useful when you know the slope and a point on the line.
3. Standard Form: Ax + By = C (where A, B, and C are constants). This form is less common but can be useful in certain situations.

To write the equation of a line, you need to know either the slope and y-intercept, the slope and a point, or two points on the line. If you are given two points, you can first calculate the slope using the formula m = (y2 - y1) / (x2 - x1) and then use the point-slope form to write the equation. Finally, rearrange the equation into slope-intercept form if desired.

Concrete Examples:

Example 1: Given Slope and Y-Intercept

Setup: Write the equation of a line with a slope of 2 and a y-intercept of -3.
Process: Using the slope-intercept form (y = mx + b), substitute m = 2 and b = -3.
Result: The equation of the line is y = 2x - 3.
Why this matters: This is the simplest case and directly applies the definition of slope-intercept form.

Example 2: Given Slope and a Point

Setup: Write the equation of a line with a slope of -1 and passing through the point (4, 5).
Process: Using the point-slope form (y - y1 = m(x - x1)), substitute m = -1, x1 = 4, and y1 = 5. This gives y - 5 = -1(x - 4). Simplify the equation to slope-intercept form: y - 5 = -x + 4 => y = -x + 9.
Result: The equation of the line is y = -x + 9.
Why this matters: This demonstrates how to use the point-slope form when you don't know the y-intercept directly.

Example 3: Given Two Points

Setup: Write the equation of a line passing through the points (1, 2) and (3, 8).
Process: First, calculate the slope: m = (8 - 2) / (3 - 1) = 6 / 2 = 3. Then, use the point-slope form with one of the points (e.g., (1, 2)): y - 2 = 3(x - 1). Simplify to slope-intercept form: y - 2 = 3x - 3 => y = 3x - 1.
Result: The equation of the line is y = 3x - 1.
Why this matters: This shows how to find the equation of a line when you only know two points on the line.

Analogies & Mental Models:

Think of it like... building a road. You need to know the slope (steepness) and either a starting point (y-intercept) or a specific location on the road to define its path.
Explain how the analogy maps to the concept: The slope represents the steepness of the road, and the y-intercept or a specific point represents a fixed location on the road.
Where the analogy breaks down (limitations): This analogy doesn't directly represent the algebraic manipulation needed to solve for the equation.

Common Misconceptions:

❌ Students often forget to simplify the equation after using the point-slope form.
✓ Actually, you need to distribute the slope and isolate 'y' to get the equation in slope-intercept form (y = mx + b).
Why this confusion happens: Students may focus on plugging in the values but forget to complete the algebraic steps.

Visual Description:

Imagine a line on a graph. Knowing the slope and y-intercept allows you to uniquely define the line. Similarly, knowing the slope and a point on the line also allows you to uniquely define the line.

Practice Check:

Write the equation of a line passing through the point (2, -1) with a slope of 4.

Answer: Using point-slope form: y - (-1) = 4(x - 2) => y + 1 = 4x - 8 => y = 4x - 9.

Connection to Other Sections:

This section builds upon the concepts of slope and y-intercept (Sections 4.2 and 4.3) and is essential for graphing linear functions (Section 4.5) and solving real-world problems (Section 4.6).

### 4.5 Graphing Linear Functions

Overview: Graphing linear functions is a visual way to represent the relationship between two variables. It allows you to quickly understand the behavior of the function and identify key features like slope and y-intercept.

The Core Concept: There are two main methods for graphing linear functions:

1. Using Slope-Intercept Form: Start by plotting the y-intercept (0, b) on the graph. Then, use the slope (m) to find another point on the line. Remember that slope is rise over run. From the y-intercept, move up (or down if the slope is negative) by the amount of the rise, and then move right by the amount of the run. Plot the new point and draw a straight line through the two points.
2. Plotting Points: Choose two or more values for 'x' and substitute them into the equation to find the corresponding 'y' values. Plot these points on the graph and draw a straight line through them.

It is important to use a ruler or straight edge to draw accurate lines. Also, be sure to label the axes and include a scale.

Concrete Examples:

Example 1: Graphing using Slope-Intercept Form

Setup: Graph the line y = 2x + 1.
Process: The y-intercept is 1, so plot the point (0, 1). The slope is 2, which can be written as 2/1. From the y-intercept, move up 2 units and right 1 unit to plot the point (1, 3). Draw a line through (0, 1) and (1, 3).
Result: You have a line that crosses the y-axis at 1 and has a slope of 2.
Why this matters: This is the most efficient way to graph a line when you know the slope and y-intercept.

Example 2: Graphing by Plotting Points

Setup: Graph the line y = -x + 4.
Process: Choose two values for 'x', for example, x = 0 and x = 4. When x = 0, y = -0 + 4 = 4, so plot the point (0, 4). When x = 4, y = -4 + 4 = 0, so plot the point (4, 0). Draw a line through (0, 4) and (4, 0).
Result: You have a line that crosses the y-axis at 4 and the x-axis at 4.
Why this matters: This method is useful when the equation is not in slope-intercept form or when you want to verify your graph.

Analogies & Mental Models:

Think of it like... connecting the dots. You need at least two points to draw a straight line. The slope and y-intercept help you find those points.
Explain how the analogy maps to the concept: The points represent coordinates on the graph, and the line represents the linear function.
Where the analogy breaks down (limitations): The analogy doesn't directly represent the concept of slope.

Common Misconceptions:

❌ Students often struggle to accurately plot points on the graph.
✓ Actually, make sure you understand the coordinate system and the scale of the axes. Use a ruler to draw straight lines.
Why this confusion happens: Students may not pay attention to the details of the coordinate plane or may rush the process.

Visual Description:

Imagine a coordinate plane with x and y axes. A linear function is represented by a straight line on this plane. The y-intercept is the point where the line crosses the y-axis, and the slope determines the direction and steepness of the line.

Practice Check:

Graph the line y = (1/2)x - 2 using the slope-intercept form.

Answer: The y-intercept is -2, so plot (0, -2). The slope is 1/2, so move up 1 unit and right 2 units to plot (2, -1). Draw a line through these two points.

Connection to Other Sections:

This section utilizes the concepts of slope and y-intercept (Sections 4.2 and 4.3) and builds upon the ability to write equations of lines (Section 4.4). Graphing linear functions provides a visual representation of the relationships described by these equations.

### 4.6 Applying Linear Functions to Real-World Problems

Overview: Linear functions are powerful tools for modeling and solving real-world problems that involve constant rates of change. By identifying the slope and y-intercept in a given scenario, you can write a linear equation and use it to make predictions and solve for unknown values.

The Core Concept: To apply linear functions to real-world problems, follow these steps:

1. Identify the variables: Determine what quantities are changing and assign variables to them (e.g., 'x' for time, 'y' for distance).
2. Find the slope: Identify the constant rate of change in the problem. This is the slope ('m') of the linear function.
3. Find the y-intercept: Identify the initial value or starting point. This is the y-intercept ('b') of the linear function.
4. Write the equation: Use the slope-intercept form (y = mx + b) to write the equation of the linear function.
5. Solve for unknowns: Use the equation to solve for unknown values by substituting known values and solving for the remaining variable.

Concrete Examples:

Example 1: Cell Phone Plan

Setup: A cell phone plan charges $20 per month plus $0.10 per minute of usage. What is the total cost for a month with 150 minutes of usage?
Process: Let 'x' be the number of minutes of usage and 'y' be the total cost. The slope is 0.10 (cost per minute), and the y-intercept is 20 (monthly fee). The equation is y = 0.10x + 20. Substitute x = 150: y = 0.10(150) + 20 = 15 + 20 = 35.
Result: The total cost for the month is $35.
Why this matters: This helps you understand and compare different cell phone plans.

Example 2: Car Rental

Setup: A car rental company charges $30 per day plus $0.20 per mile. How much will it cost to rent a car for 3 days and drive 200 miles?
Process: Let 'x' be the number of miles driven and 'y' be the total cost. The cost for the 3 days is a fixed cost of 3 $30 = $90. The slope is 0.20 (cost per mile), and the y-intercept is 90 (fixed cost). The equation is y = 0.20x + 90. Substitute x = 200: y = 0.20(200) + 90 = 40 + 90 = 130.
Result: The total cost for the car rental is $130.
Why this matters: This helps you estimate the cost of renting a car for a trip.

Analogies & Mental Models:

Think of it like... calculating the total cost of a project. You have a fixed cost (y-intercept) and a variable cost per unit (slope). The linear function allows you to calculate the total cost based on the number of units.
Explain how the analogy maps to the concept: The fixed cost represents the y-intercept, and the variable cost per unit represents the slope.
Where the analogy breaks down (limitations): This analogy assumes that the variable cost is constant, which may not always be the case in real-world scenarios.

Common Misconceptions:

❌ Students often struggle to identify the slope and y-intercept in word problems.
✓ Actually, look for keywords that indicate a constant rate of change (slope) and an initial value (y-intercept). Pay attention to the units of measurement.
Why this confusion happens: Students may not be able to translate the words into mathematical concepts.

Visual Description:

Imagine a graph representing a real-world scenario. The x-axis represents one variable (e.g., time), and the y-axis represents another variable (e.g., distance). The linear function is a line that models the relationship between these variables.

Practice Check:

A plumber charges a $50 service fee plus $75 per hour. How much will it cost to hire the plumber for 3 hours?

Answer: Let 'x' be the number of hours and 'y' be the total cost. The slope is 75, and the y-intercept is 50. The equation is y = 75x + 50. Substitute x = 3: y = 75(3) + 50 = 225 + 50 = 275. The total cost is $275.

Connection to Other Sections:

This section applies all the concepts learned in previous sections (slope, y-intercept, writing equations, graphing) to solve real-world problems. It demonstrates the practical value of linear functions in various fields.

### 4.7 Parallel and Perpendicular Lines