Transformation of Rational Functions

Mathematics

TEACHER

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.

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 () 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.

Level of Thinking Analyze

Analyze

Depth of Knowledge III

III

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Code A-APR.6

A-CED.4 F-IF.5

Teacher Overview ? Transformation of Rational Functions

Standard

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. 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.

Level of Thinking Apply

Apply

Apply

Depth of Knowledge II

II

II

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.

Implicitly addressed in this lesson

Code

Standard

2

Reason abstractly and quantitatively.

5

Use appropriate tools strategically.

6

Attend to precision.

7

Look for and make use of structure.

TEACHER

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Teacher Overview ? 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

7th Grade

Algebra 1

Geometry

Algebra 2

Pre-Calculus

Skills/Objectives Skills/Objectives Skills/Objectives Skills/Objectives Skills/Objectives Skills/Objectives

Analyze characteristics of graphs. (200_06.AF_N.02)

Analyze characteristics of graphs. (200_07.AF_N.02)

Analyze characteristics of graphs. (200_A1.AF_N.02)

Analyze

Analyze

Analyze

characteristics of characteristics of characteristics of

graphs.

graphs.

graphs.

(200_GE.AF_N.02) (200_A2.AF_N.02) (200_PC.AF_N.02)

Investigate limits Investigate limits Investigate limits Investigate limits Investigate limits Investigate limits

using patterns,

using patterns,

using patterns,

using patterns,

using patterns,

using patterns,

diagrams, geometric diagrams, geometric diagrams, geometric diagrams, geometric diagrams, geometric diagrams, geometric

figures, tables,

figures, tables,

figures, tables,

figures, tables,

figures, tables,

figures, tables,

and/or graphs.

and/or graphs.

and/or graphs.

and/or graphs.

and/or graphs.

and/or graphs.

(200_06.LI_H.01) (200_07.LI_H.01) (200_A1.LI_H.01) (200_GE.LI_H.01) (200_A2.LI_H.01) (200_PC.LI_H.01)

Identify horizontal, Identify horizontal,

vertical, and/or slant vertical, and/or slant

asymptotes and asymptotes and

removable

removable

discontinuities. discontinuities.

(200_A2.AF_N.04) (200_PC.AF_N.04)

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 ? Algebra 2 Free Response Questions Analysis of Functions: Transformations ? Algebra 2 Multiple Choice Questions

TEACHER

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Teacher Overview ? 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 x2 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 x2 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 x2 . Translate the graph left 3 units, y (x 3)2 . Translate the graph right 1 unit, y (x 1)2 .

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 x2 , 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 ? Physical V ? Verbal A ? Analytical N ? Numerical G ? Graphical

TEACHER

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Answers 1. a. 3x 4

55 2. 4 5

x3

3. L(w) 12 w

L

10

8 6 4

2

w

2 4 6 8 10

4.

Teacher Overview ? Transformation of Rational Functions

b. 2x 8 77

c. 4 2 x

TEACHER

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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