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9-12 Algebra 2 Unit 3 Lesson 3

Gradeband

K-5 Math 6-8 Math •9-12 Math

Course

Algebra 1 Geometry •Algebra 2 Extra Support Materials for Algebra 1

Unit

Unit 1 Unit 2 •Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8

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

K-5
6-8
9-12

Algebra 1
Geometry
Algebra 2
Extra Support Materials for Algebra 1

Integrated Math 1
Integrated Math 2
Integrated Math 3

Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8

Modeling Prompts

Lesson 1
Lesson 2
Lesson 3
Lesson 4
Lesson 5
Lesson 6
Lesson 7
Lesson 8
Lesson 9
Lesson 10
Lesson 11

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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)

Algebra 2

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.

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3.2

Activity

Publishing a Paperback

Standards Alignment

Building On

Addressing

A-APR.6

Rewrite simple rational expressions in different forms; write ^a(x)/_b(x) in the form q(x) + ^r(x)/_b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

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A-SSE.1

Interpret expressions that represent a quantity in terms of its context

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Building Toward

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

Standards Alignment

Building On

Addressing

A-APR.6

Rewrite simple rational expressions in different forms; write ^a(x)/_b(x) in the form q(x) + ^r(x)/_b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

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F-IF.C

Analyze functions using different representations

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Building Toward

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?

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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.

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Standards Alignment

Building On

Addressing

Building Toward

A-APR.6

Rewrite simple rational expressions in different forms; write ^a(x)/_b(x) in the form q(x) + ^r(x)/_b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

View Full Standard