Transformation of Rational Functions

Transcription

1 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 epressions into a form that makes graphing easier. Applications include graphing area and volume functions in one variable. Prior to the lesson, students should have eperience transforming parent functions and should know function notation. Objectives Students will: rewrite rational epressions as sums in order to reveal end behavior. apply transformations to the graphs of rational functions. sketch the resulting Level Algebra 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. Eplicitly addressed in this lesson Code Standard Level of Thinking F-BF.3 Identify the effect on the graph of replacing f() by f() + k, k f(), f(k), and f( + k) for specific values of k (both positive and negative); find the value of k given the Eperiment with cases and illustrate an eplanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic epressions for them. F-IF.7d (+) Graph functions epressed 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. Depth of Knowledge III III Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at i

2 Teacher Overview Transformation of Rational Functions Code Standard Level of Thinking A-APR.6 Rewrite simple rational epressions in Apply different forms; write a()/b() in the form q() + r()/b(), where a(), b(), q(), and r() are polynomials with the degree of r() less than the degree of b(), using inspection, long division, or, for the more complicated eamples, a computer algebra system. A-CED.4 Rearrange formulas to highlight a quantity of Apply interest, using the same reasoning as in solving equations. For eample, rearrange Ohm s law V = IR to highlight resistance R. F-IF.5 Relate the domain of a function to its graph Apply and, where applicable, to the quantitative relationship it describes. For eample, 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. 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 Reason abstractly and quantitatively. 5 Use appropriate tools strategically. 6 Attend to precision. 7 Look for and make use of structure. Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at ii

3 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 sith 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 (00_06.AF_N.0) (00_06.LI_H.0) 7th Grade (00_07.AF_N.0) (00_07.LI_H.0) Algebra (00_A.AF_N.0) (00_A.LI_H.0) Geometry (00_GE.AF_N.0) (00_GE.LI_H.0) Algebra (00_A.AF_N.0) (00_A.LI_H.0) Identify horizontal, vertical, and/or slant asymptotes and removable discontinuities. (00_A.AF_N.04) Pre-Calculus (00_PC.AF_N.0) (00_PC.LI_H.0) Identify horizontal, vertical, and/or slant asymptotes and removable discontinuities. (00_PC.AF_N.04) Connection to AP* AP Calculus Topic: Analysis of Functions *Advanced Placement and AP are registered trademarks of the College Entrance Eamination 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 Free Response Questions Analysis of Functions: Transformations Algebra Multiple Choice Questions Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at iii

4 Teacher Overview Transformation of Rational Functions Teaching Suggestions Questions and allow students to practice the algebraic skills needed to rewrite the epressions 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 etend 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 - 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 and discuss the changes in the equation that would accomplish the following transformations, each from the original function: Translate the graph up unit, y. Translate the graph down units, y. Reflect the graph across the y-ais, y( ). ( A discussion of symmetry would be appropriate with this transformation.) Reflect the graph across the -ais, y. Translate the graph left 3 units, Translate the graph right unit, y y ( 3). ( ). This lesson could be etended 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 eample, using y, translate the function left units and up 5 units, y ( ) 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, eplore, and reinforce mathematical concepts and to enhance conceptual understanding. P Physical V Verbal A Analytical N Numerical G Graphical Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at iv

5 Answers. a Teacher Overview Transformation of Rational Functions b. c Lw ( ) L w w 4. a. f( ) is a reflection across the -ais. b. f( ) is a translation, units to the right. c. f( ) 3 is translated unit left, then down 3 units. a. b. c. Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at v

6 36 5. hr () r h 0 Teacher Overview Transformation of Rational Functions r a. f( ) is a reflection across the -ais. b. f( ) is translated units right. c. f( ) 3 is translated 3 units up. a. b. c. Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at vi

7 7. Teacher Overview Transformation of Rational Functions Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at vii

8 Mathematics Transformation of Rational Functions. Write each of these epressions as a sum or a difference. a. b. c Divide the epression 4 7 3, and then use that answer to rewrite the epression as a sum. 3. The area of a rectangle is square inches. Write the length l of this rectangle as a function of its width w. Sketch a graph of this function over an appropriate domain. 4. Sketch the graph of f ( ). In parts (a) (c), write each function as a transformation of f( ) using function notation. Describe the transformation, and then use the graph of f( ) to help you sketch the graph of the transformed function. a. y b. y c. y 3 Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at

9 Student Activity Transformation of Rational Functions 5. If the volume of a cone is cubic feet, write the formula for the height h of this cone in terms of its radius r. Sketch a graph of this function over an appropriate domain. 6. Sketch the graph of f ( ). In parts (a) (c), write each function as a transformation of f( ) using function notation. Describe each transformation, and then use the graph of f( ) to help you sketch the graphs of the transformed functions. a. f ( ) b. f ( ) ( ) c. f ( ) 3 Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at

10 Student Activity Transformation of Rational Functions 7. Write as a sum. Use your answer to help you sketch a graph of the function f ( ). 8. Write as a sum. Use your answer to help you sketch a graph of the function f ( ). 9. Write 9 as a sum. 5 Use your answer to help you sketch a graph of the function 9 f( ) 5. Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at 3