Learn on PengiIllustrative Mathematics, Grade 8Chapter 5: Functions and Volume

Lesson 3: Linear Functions and Rates of Change

In this Grade 8 Illustrative Mathematics lesson, students explore linear functions by identifying rates of change and initial values in equations of the form y = mx + b, where m represents the rate of change and b represents the starting value. Students work with real-world contexts — including filling and draining water tanks, savings accounts, and unit conversions — to determine whether a linear function is increasing or decreasing based on the sign of the slope. They also practice comparing two linear functions represented in different forms, such as an equation and a graph, to draw conclusions about rates of change and initial values.

Section 1

Linear Function

Property

A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line

f(x) = mx + b

where $b$ is the initial or starting value of the function (when input, $x = 0$ ), and $m$ is the constant rate of change, or slope of the function. The $y$ -intercept is at $(0, b)$ .

Examples

A car travels at a constant speed of 50 miles per hour. Its distance $D$ from a starting point after $t$ hours can be modeled by $D(t) = 50t$ .
A phone plan costs 20 dollars a month plus 5 cents for each text message. The monthly cost $C$ for $x$ messages is $C(x) = 0.05x + 20$ .
For the function $f(x) = 3x + 2$ , the value when $x=4$ is $f(4) = 3(4) + 2 = 14$ . The point $(4, 14)$ is on the line.

Explanation

Think of a linear function as a rule for anything that changes at a steady rate. The 'm' is the rate of change (how steep the line is), and 'b' is the starting point on the vertical axis before any change happens.

Section 2

Finding Rate of Change and Initial Value from Graphs and Tables

Property

To find the rate of change ( $m$ ) and initial value ( $b$ ) from a representation:

From a Graph: The rate of change is the slope, $m = \frac{{\text{rise}}}{{\text{run}}} = \frac{{\Delta y}}{{\Delta x}}$ . The initial value, $b$ , is the y-coordinate of the point where the line crosses the y-axis, $(0, b)$ .
From a Table: Select any two points $(x_1, y_1)$ and $(x_2, y_2)$ to find the rate of change, $m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$ . The initial value, $b$ , is the value of $y$ when $x=0$ .

Examples

Section 3

Interpreting the Equation in Context

Property

In a real-world linear model written in the form $y = mx + b$ , the mathematical variables have specific, practical meanings:

The slope ( $m$ ) represents the rate of change. It describes how much the dependent variable ( $y$ ) changes for every one-unit increase in the independent variable ( $x$ ).
The y-intercept ( $b$ ) represents the initial value or starting point. It is the value of the dependent variable ( $y$ ) when the independent variable ( $x$ ) is 0.

Examples

An equation for manatee deaths is $y = 4.7 + 2.6t$ , where $t$ is years since 1975. This line models an increasing trend, with deaths increasing by about 2.6 per year.
A plumber charges a fee based on $C = 75h + 50$ , where $C$ is total cost and $h$ is hours worked. The slope is 75, meaning the cost increases by 75 dollars for each hour of work. The y-intercept is 50, meaning there is a 50 dollar initial fee before any work begins.
The amount of water $V$ in a tank after $t$ minutes is modeled by $V = -10t + 300$ . The slope is -10, meaning the water decreases by 10 gallons each minute. The y-intercept is 300, meaning the tank initially contained 300 gallons.

Explanation

When a real-world situation is modeled by a linear function, the slope and y-intercept are no longer just abstract numbers. The slope tells you the exact rate at which a quantity is changing over time or per unit. The y-intercept tells you the starting amount or fixed fee before that change even begins.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Linear Function

Property

A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line

f(x) = mx + b

where $b$ is the initial or starting value of the function (when input, $x = 0$ ), and $m$ is the constant rate of change, or slope of the function. The $y$ -intercept is at $(0, b)$ .

Examples

A car travels at a constant speed of 50 miles per hour. Its distance $D$ from a starting point after $t$ hours can be modeled by $D(t) = 50t$ .
A phone plan costs 20 dollars a month plus 5 cents for each text message. The monthly cost $C$ for $x$ messages is $C(x) = 0.05x + 20$ .
For the function $f(x) = 3x + 2$ , the value when $x=4$ is $f(4) = 3(4) + 2 = 14$ . The point $(4, 14)$ is on the line.

Explanation

Section 2

Finding Rate of Change and Initial Value from Graphs and Tables

Property

To find the rate of change ( $m$ ) and initial value ( $b$ ) from a representation:

From a Graph: The rate of change is the slope, $m = \frac{{\text{rise}}}{{\text{run}}} = \frac{{\Delta y}}{{\Delta x}}$ . The initial value, $b$ , is the y-coordinate of the point where the line crosses the y-axis, $(0, b)$ .
From a Table: Select any two points $(x_1, y_1)$ and $(x_2, y_2)$ to find the rate of change, $m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$ . The initial value, $b$ , is the value of $y$ when $x=0$ .

Examples

Section 3

Interpreting the Equation in Context

Property

In a real-world linear model written in the form $y = mx + b$ , the mathematical variables have specific, practical meanings:

The slope ( $m$ ) represents the rate of change. It describes how much the dependent variable ( $y$ ) changes for every one-unit increase in the independent variable ( $x$ ).
The y-intercept ( $b$ ) represents the initial value or starting point. It is the value of the dependent variable ( $y$ ) when the independent variable ( $x$ ) is 0.

Examples

An equation for manatee deaths is $y = 4.7 + 2.6t$ , where $t$ is years since 1975. This line models an increasing trend, with deaths increasing by about 2.6 per year.
A plumber charges a fee based on $C = 75h + 50$ , where $C$ is total cost and $h$ is hours worked. The slope is 75, meaning the cost increases by 75 dollars for each hour of work. The y-intercept is 50, meaning there is a 50 dollar initial fee before any work begins.
The amount of water $V$ in a tank after $t$ minutes is modeled by $V = -10t + 300$ . The slope is -10, meaning the water decreases by 10 gallons each minute. The y-intercept is 300, meaning the tank initially contained 300 gallons.

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Linear Function

Property

A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line

f(x) = mx + b

where $b$ is the initial or starting value of the function (when input, $x = 0$ ), and $m$ is the constant rate of change, or slope of the function. The $y$ -intercept is at $(0, b)$ .

Examples

A car travels at a constant speed of 50 miles per hour. Its distance $D$ from a starting point after $t$ hours can be modeled by $D(t) = 50t$ .
A phone plan costs 20 dollars a month plus 5 cents for each text message. The monthly cost $C$ for $x$ messages is $C(x) = 0.05x + 20$ .
For the function $f(x) = 3x + 2$ , the value when $x=4$ is $f(4) = 3(4) + 2 = 14$ . The point $(4, 14)$ is on the line.

Explanation

Section 2

Finding Rate of Change and Initial Value from Graphs and Tables

Property

To find the rate of change ( $m$ ) and initial value ( $b$ ) from a representation:

From a Graph: The rate of change is the slope, $m = \frac{{\text{rise}}}{{\text{run}}} = \frac{{\Delta y}}{{\Delta x}}$ . The initial value, $b$ , is the y-coordinate of the point where the line crosses the y-axis, $(0, b)$ .
From a Table: Select any two points $(x_1, y_1)$ and $(x_2, y_2)$ to find the rate of change, $m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$ . The initial value, $b$ , is the value of $y$ when $x=0$ .

Examples

Section 3

Interpreting the Equation in Context

Property

In a real-world linear model written in the form $y = mx + b$ , the mathematical variables have specific, practical meanings:

The slope ( $m$ ) represents the rate of change. It describes how much the dependent variable ( $y$ ) changes for every one-unit increase in the independent variable ( $x$ ).
The y-intercept ( $b$ ) represents the initial value or starting point. It is the value of the dependent variable ( $y$ ) when the independent variable ( $x$ ) is 0.

Examples

An equation for manatee deaths is $y = 4.7 + 2.6t$ , where $t$ is years since 1975. This line models an increasing trend, with deaths increasing by about 2.6 per year.
A plumber charges a fee based on $C = 75h + 50$ , where $C$ is total cost and $h$ is hours worked. The slope is 75, meaning the cost increases by 75 dollars for each hour of work. The y-intercept is 50, meaning there is a 50 dollar initial fee before any work begins.
The amount of water $V$ in a tank after $t$ minutes is modeled by $V = -10t + 300$ . The slope is -10, meaning the water decreases by 10 gallons each minute. The y-intercept is 300, meaning the tank initially contained 300 gallons.