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

Lesson 2: Representing and Interpreting Functions

In this Grade 8 lesson from Illustrative Mathematics Chapter 5, students learn how to represent functions using equations and function diagrams, writing rules that express the output as a function of the input. Students practice identifying independent and dependent variables across real-world contexts such as circle circumference, distance-rate-time, and coin value problems. They also explore how a two-variable equation like 0.1d + 0.25q = 12.5 can be rearranged to express either variable as a function of the other.

Section 1

Functions Defined by Equations

Property

An equation defines a function if for any value substituted for the input variable, a unique value for the output variable can be determined.

Examples

The area of a square is given by the function $A = s^2$ . For any side length $s$ you input, you get exactly one area $A$ . For $s=4$ , the area is always 16.
A phone plan costs 50 dollars plus 5 dollars per gigabyte of data, $g$ . The cost $C$ is a function of data used: $C = 50 + 5g$ . For any amount of data, there is only one possible cost.
The equation $x = y^2$ does not define $y$ as a function of $x$ . If $x=9$ , $y$ could be 3 or $-3$ . Since one input ( $x=9$ ) gives two outputs, it's not a function.

Explanation

An equation acts like a recipe for a function. You provide the input ingredient (the variable, like $x$ ), and the equation gives you a step-by-step process to calculate exactly one result (the output, $y$ ).

Section 2

Rearranging Equations into Function Form

Property

To express a variable (like $y$ ) as a function of another variable (like $x$ ), use inverse operations to algebraically isolate $y$ on one side of the equation. This rewrites the equation into the explicit form $y = f(x)$ .

Examples

Section 3

Analyzing a Functional Relationship

Property

Graphed data can reveal trends, suggest relationships, or uncover anomalies. By analyzing the graph of a function, we can describe its behavior qualitatively.
An increasing function is one where the graph rises from left to right, meaning the $y$ -value increases as the $x$ -value increases.
A decreasing function is one where the graph falls from left to right.
Sudden changes or deviations from an expected pattern, or anomalies, can indicate specific events, such as a thunderstorm affecting temperature data.

Examples

A graph of a bathtub's water level shows the level increasing steadily as it fills, then sharply rising as a person gets in, staying constant, and finally decreasing as it drains. This models the entire process.
The graph of a bouncing ball's height over time starts at a peak, drops to zero, and repeatedly bounces back to progressively lower heights. The curve shows the loss of energy with each bounce.
A graph showing the number of customers in a store over a day would be low in the morning, rise to a peak around lunchtime, dip in the afternoon, and then rise again before closing.

Explanation

The shape of a graph tells a story. Where it goes up, down, flattens out, or suddenly jumps reveals how one variable is affected by another. Analyzing these features helps us understand the real-world situation being modeled.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Functions Defined by Equations

Property

An equation defines a function if for any value substituted for the input variable, a unique value for the output variable can be determined.

Examples

The area of a square is given by the function $A = s^2$ . For any side length $s$ you input, you get exactly one area $A$ . For $s=4$ , the area is always 16.
A phone plan costs 50 dollars plus 5 dollars per gigabyte of data, $g$ . The cost $C$ is a function of data used: $C = 50 + 5g$ . For any amount of data, there is only one possible cost.
The equation $x = y^2$ does not define $y$ as a function of $x$ . If $x=9$ , $y$ could be 3 or $-3$ . Since one input ( $x=9$ ) gives two outputs, it's not a function.

Explanation

Section 2

Rearranging Equations into Function Form

Property

Examples

Section 3

Analyzing a Functional Relationship

Property

Examples

A graph of a bathtub's water level shows the level increasing steadily as it fills, then sharply rising as a person gets in, staying constant, and finally decreasing as it drains. This models the entire process.
The graph of a bouncing ball's height over time starts at a peak, drops to zero, and repeatedly bounces back to progressively lower heights. The curve shows the loss of energy with each bounce.
A graph showing the number of customers in a store over a day would be low in the morning, rise to a peak around lunchtime, dip in the afternoon, and then rise again before closing.

Explanation

Overview

Lesson overview

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

Overview

Section 1

Functions Defined by Equations

Property

An equation defines a function if for any value substituted for the input variable, a unique value for the output variable can be determined.

Examples

The area of a square is given by the function $A = s^2$ . For any side length $s$ you input, you get exactly one area $A$ . For $s=4$ , the area is always 16.
A phone plan costs 50 dollars plus 5 dollars per gigabyte of data, $g$ . The cost $C$ is a function of data used: $C = 50 + 5g$ . For any amount of data, there is only one possible cost.
The equation $x = y^2$ does not define $y$ as a function of $x$ . If $x=9$ , $y$ could be 3 or $-3$ . Since one input ( $x=9$ ) gives two outputs, it's not a function.

Explanation

Section 2

Rearranging Equations into Function Form

Property

Examples

Section 3

Analyzing a Functional Relationship

Property

Examples

A graph of a bathtub's water level shows the level increasing steadily as it fills, then sharply rising as a person gets in, staying constant, and finally decreasing as it drains. This models the entire process.
The graph of a bouncing ball's height over time starts at a peak, drops to zero, and repeatedly bounces back to progressively lower heights. The curve shows the loss of energy with each bounce.
A graph showing the number of customers in a store over a day would be low in the morning, rise to a peak around lunchtime, dip in the afternoon, and then rise again before closing.