Navigation

9-12 Integrated Math 3 Unit 3 Lesson 3

Gradeband

K-5 Math 6-8 Math •9-12 Math

Course

Integrated Math 1 Integrated Math 2 •Integrated Math 3

Unit

Unit 1 Unit 2 •Unit 3 Unit 4 Unit 5 Unit 6 Unit 7

Modeling Prompts

Lesson

Lesson 1 Lesson 2 •Lesson 3 Lesson 4 Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 Lesson 10 Lesson 11 Lesson 12 Lesson 13 Lesson 14 Lesson 15 Lesson 16

K-5
6-8
9-12

Integrated Math 1
Integrated Math 2
Integrated Math 3

Algebra 1
Geometry
Algebra 2
Extra Support Materials for Algebra 1

Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7

Modeling Prompts

Lesson 1
Lesson 2
Lesson 3
Lesson 4
Lesson 5
Lesson 6
Lesson 7
Lesson 8
Lesson 9
Lesson 10
Lesson 11
Lesson 12
Lesson 13
Lesson 14
Lesson 15
Lesson 16

Lesson | Loading...

Sign in to access protected content and invite other educators

Sign in using your existing Kendall Hunt account. If you don’t have one, create an educator account.

Curriculum

IM K-5 Math
IM 6-8 Math
IM 9-12 Math

Professional Learning

Illustrative Mathematics® has operated as an independent 501(c)3 non-profit organization since 2013.

©2024 Illustrative Mathematics®. Licensed under CC BY-NC 4.0.

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written permission of Illustrative Mathematics.

This site includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses. See the image attribution section for more information.

Settings

Audience

family

Not all roles available for this page.

Assessment Access

Sign in to view assessments and invite other educators

Sign in using your existing Kendall Hunt account. If you don’t have one, create an educator account.

teacher
student
family

Lesson 3

3.1 Warm-up 3.2 Activity 3.3 Activity Student Lesson Summary Glossary

Unit 3, Lesson 3

Graphs of Rational Functions (Part 2)

$Course Icon$

Integrated Math 3

3.1

Warm-up

Rewritten Equations

Decide if each of these equations is true or false for -values that do not result in a denominator of 0. Be prepared to explain your reasoning.

Are You Ready for More?

3.2

Activity

Publishing a Paperback

Let be the function that gives the average cost per book , in dollars, when using an online store to print copies of a self-published paperback book. Here is a graph of

<p>Coordinate plane, total books printed, 0 to 500 by 50, cost per book, 0 to 20 by 1. Curve beings near 1 comma 20 and drops steeply toward 50 comma 6 point 2, then gently toward 500 comma 4 point 1.</p>

What is the approximate cost per book when 50 books are printed? 100 books?
The author plans to charge $8 per book. About how many should be printed to make a profit?
What is the value of when ? How does this relate to the context?
What does the end behavior of the function say about the context?

Are You Ready for More?

3.3

Activity

Horizontal Asymptotes

Here are four graphs of rational functions.

<p>Graph of a rational function f(x) with horizontal and vertical asymptotes on coordinate plane.</p> — A

<p>Graph of a rational function f(x) with horizontal and vertical asymptotes on coordinate plane.</p> — B

<p>Graph of a rational function f(x) with horizontal and vertical asymptotes on coordinate plane.</p> — C

<p>Graph of a rational function f(x) with horizontal and vertical asymptotes on coordinate plane.</p> — D

Match each function with its graphical representation.
Where do you see the horizontal asymptote of the graph in the expressions for the functions?

Are You Ready for More?

Student Lesson Summary

Consider the rational function . Written this way, we can tell that the graph of the function has a vertical asymptote at by reading the denominator and identifying the value that would cause division by 0. But what can we tell about the value of for values of far away from the vertical asymptote?

One way we can think about these values is to rewrite the expression for by breaking up the fraction:

Written this way, it’s easier to see that as gets larger and larger in either the positive or negative direction, the term will get closer and closer to 0. Because of this, we can say that the value of the function will get closer and closer to 3. Here is a graph of showing values from -40 to 40.

graph of y = f of x. horizontal asymptote at y = 3. vertical asymptote at x = 0.

A dashed line at is included to show how the function approaches this value as inputs are farther and farther from . This is an example of a feature of rational functions: a horizontal asymptote.

The line is a horizontal asymptote for a function if the value of the function gets closer and closer to as the magnitude of increases.

More generally, if a rational function can be rewritten as , where is a constant and and are polynomial expressions in which gets closer and closer to 0 as gets larger and larger in both the positive and negative directions, then will get closer and closer to .

Glossary

horizontal asymptote

The line is a horizontal asymptote of a function if the outputs of the function get closer and closer to as the inputs get larger and larger in either the positive or negative direction. This means the graph gets closer and closer to the line as you move to the right or left along the -axis.