Transformation of Rational Functions

Mathematics

Transformation of Rational Functions

About this Lesson

In this lesson, students will apply transformations to the graphs of rational functions, describe the

transformations, and graph the transformed functions. Questions include practice in manipulating

expressions into a form that makes graphing easier. Applications include graphing area and

volume functions in one variable.

Prior to the lesson, students should have experience transforming parent functions and should

know function notation.

Objectives

Students will:

? rewrite rational expressions as sums in order to reveal end behavior.

? apply transformations to the graphs of rational functions.

? sketch the resulting graphs.

T E A C H E R

Level

Algebra 2

Common Core State Standards for Mathematical Content

This lesson addresses the following Common Core State Standards for Mathematical Content.

The lesson requires that students recall and apply each of these standards rather than providing

the initial introduction to the specific skill. The star symbol (¡ï) at the end of a specific standard

indicates that the high school standard is connected to modeling.

Explicitly addressed in this lesson

Code

Standard

F-BF.3

F-IF.7d

Identify the effect on the graph of replacing

f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for

specific values of k (both positive and

negative); find the value of k given the

graphs. Experiment with cases and illustrate

an explanation of the effects on the graph

using technology. Include recognizing even

and odd functions from their graphs and

algebraic expressions for them.

(+) Graph functions expressed symbolically

and show key features of the graph, by hand

in simple cases and using technology for more

complicated cases. Graph rational functions,

identifying zeros and asymptotes when

suitable factorizations are available, and

showing end behavior.¡ï

Level of

Thinking

Analyze

Depth of

Knowledge

III

Analyze

III

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Teacher Overview ¨C Transformation of Rational Functions

Code

Standard

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.

Rearrange formulas to highlight a quantity of

interest, using the same reasoning as in

solving equations. For example, rearrange

Ohm¡¯s law V = IR to highlight resistance R.¡ï

Relate the domain of a function to its graph

and, where applicable, to the quantitative

relationship it describes. For example, if the

function h(n) gives the number of personhours it takes to assemble n engines in a

factory, then the positive integers would be an

appropriate domain for the function.¡ï

A-CED.4

F-IF.5

Level of

Thinking

Apply

Depth of

Knowledge

II

Apply

II

Apply

II

Implicitly addressed in this lesson

Code

Standard

2

5

6

7

Reason abstractly and quantitatively.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

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ii

T E A C H E R

Common Core State Standards for Mathematical Practice

These standards describe a variety of instructional practices based on processes and proficiencies

that are critical for mathematics instruction. LTF incorporates these important processes and

proficiencies to help students develop knowledge and understanding and to assist them in

making important connections across grade levels. This lesson allows teachers to address the

following Common Core State Standards for Mathematical Practice.

Teacher Overview ¨C Transformation of Rational Functions

LTF Content Progression Chart

In the spirit of LTF¡¯s goal to connect mathematics across grade levels, the Content Progression

Chart demonstrates how specific skills build and develop from sixth grade through pre-calculus.

Each column, under a grade level or course heading, lists the concepts and skills that students in

that grade or course should master. Each row illustrates how a specific skill is developed as

students advance through their mathematics courses.

6th Grade

Skills/Objectives

Analyze

characteristics of

graphs.

(200_06.AF_N.02)

7th Grade

Skills/Objectives

Analyze

characteristics of

graphs.

(200_07.AF_N.02)

Algebra 1

Skills/Objectives

Analyze

characteristics of

graphs.

(200_A1.AF_N.02)

Geometry

Skills/Objectives

Analyze

characteristics of

graphs.

(200_GE.AF_N.02)

Algebra 2

Skills/Objectives

Analyze

characteristics of

graphs.

(200_A2.AF_N.02)

Pre-Calculus

Skills/Objectives

Analyze

characteristics of

graphs.

(200_PC.AF_N.02)

Investigate limits

using patterns,

diagrams, geometric

figures, tables,

and/or graphs.

(200_06.LI_H.01)

Investigate limits

using patterns,

diagrams, geometric

figures, tables,

and/or graphs.

(200_07.LI_H.01)

Investigate limits

using patterns,

diagrams, geometric

figures, tables,

and/or graphs.

(200_A1.LI_H.01)

Investigate limits

using patterns,

diagrams, geometric

figures, tables,

and/or graphs.

(200_GE.LI_H.01)

Investigate limits

using patterns,

diagrams, geometric

figures, tables,

and/or graphs.

(200_A2.LI_H.01)

Investigate limits

using patterns,

diagrams, geometric

figures, tables,

and/or graphs.

(200_PC.LI_H.01)

Identify horizontal,

vertical, and/or slant

asymptotes and

removable

discontinuities.

(200_A2.AF_N.04)

Identify horizontal,

vertical, and/or slant

asymptotes and

removable

discontinuities.

(200_PC.AF_N.04)

T E A C H E R

Connection to AP*

AP Calculus Topic: Analysis of Functions

*Advanced Placement and AP are registered trademarks of the College Entrance Examination Board.

The College Board was not involved in the production of this product.

Materials and Resources

? Student Activity pages

? Graph paper

Assessments

The following types of formative assessments are embedded in this lesson:

? Students engage in independent practice.

? Students apply knowledge to a new situation.

The following additional assessments are located on the LTF website:

? Analysis of Functions: Transformations ¨C Algebra 2 Free Response Questions

? Analysis of Functions: Transformations ¨C Algebra 2 Multiple Choice Questions

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iii

Teacher Overview ¨C Transformation of Rational Functions

Teaching Suggestions

Questions 1 and 2 allow students to practice the algebraic skills needed to rewrite the expressions

before graphing. Students consider domain and have the opportunity to use vertical asymptotes

to help in graphing the functions. Questions 4 and 6 take the parent graph and extend to both

vertical and horizontal translations. This lesson is an introduction to graphing rational functions.

In questions 4, 7, and 9, students may use the x- and y-intercepts to refine the graph of the

function.

Teachers may scaffold by reviewing the transformation of the quadratic parent function. Students

could graph y ? x 2 and discuss the changes in the equation that would accomplish the following

transformations, each from the original function:

? Translate the graph up 1 unit, y ? x 2 ? 1 .

? Translate the graph down 2 units, y ? x2 ? 2 .

? Reflect the graph across the y-axis, y ? (? x)2 . ( A discussion of symmetry would be

appropriate with this transformation.)

? Reflect the graph across the x-axis, y ? ? x 2 .

? Translate the graph left 3 units, y ? ( x ? 3)2 .

? Translate the graph right 1 unit, y ? ( x ? 1)2 .

T E A C H E R

This lesson could be extended by having students:

? Write the equation of the vertical asymptote for each function and identify this as a

nonremovable discontinuity.

? Write equations that include more than one shift in the transformation. For example,

using y ? x 2 , translate the function left 2 units and up 5 units, y ? ( x ? 2)2 ? 5 .

Modality

LTF emphasizes using multiple representations to connect various approaches to a situation in

order to increase student understanding. The lesson provides multiple strategies and models for

using these representations to introduce, explore, and reinforce mathematical concepts and to

enhance conceptual understanding.

P

V

A

N

G

¨C

¨C

¨C

¨C

¨C

Physical

Verbal

Analytical

Numerical

Graphical

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Teacher Overview ¨C Transformation of Rational Functions

Answers

3x 4

?

5 5

1.

a.

2.

4?

3.

L( w) ?

b.

2x 8

?

7 7

4?

c.

2

x

5

x ?3

12

w

L

10

8

6

4

2

w

2

4

6

8

10

T E A C H E R

4.

a. ? f ( x) is a reflection across the x-axis.

b. f ( x ? 2) is a translation, 2 units to the right.

c. f ( x ? 1) ? 3 is translated 1 unit left, then down 3 units.

a.

b.

c.

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