McGraw-Hill Ryerson Pre-Calculus 12
McGraw-Hill RyersonPre-Calculus12
ContentsA Tour of Your Textbook . viiUnit 2 Trigonometry . 162Unit 1 Transformations andFunctions . 2Chapter 4 Trigonometry and theUnit Circle .1644.1 Angles and Angle Measure . 166Chapter 1 Function Transformations . 41.1 Horizontal and Vertical Translations .61.2 Reflections and Stretches . 164.2 The Unit Circle . 1804.3 Trigonometric Ratios . 1911.3 Combining Transformations . 324.4 Introduction to TrigonometricEquations . 2061.4 Inverse of a Relation . 44Chapter 4 Review . 215Chapter 1 Review . 56Chapter 4 Practice Test . 218Chapter 1 Practice Test . 58Chapter 2 Radical Functions . 602.1 Radical Functions and Transformations . 62Chapter 5 Trigonometric Functions andGraphs .2205.1 Graphing Sine and Cosine Functions . 2222.2 Square Root of a Function .785.2 Transformations of SinusoidalFunctions . 2382.3 Solving Radical Equations Graphically . 905.3 The Tangent Function. 256Chapter 2 Review . 995.4 Equations and Graphs of TrigonometricFunctions . 266Chapter 2 Practice Test . 102Chapter 5 Review . 282Chapter 3 Polynomial Functions .1043.1 Characteristics of PolynomialFunctions . 1063.2 The Remainder Theorem . 1183.3 The Factor Theorem . 1263.4 Equations and Graphs of PolynomialFunctions . 136Chapter 3 Review . 153Chapter 3 Practice Test . 155Unit 1 Project Wrap-Up . 157Cumulative Review, Chapters 1–3 . 158Unit 1 Test . 160Chapter 5 Practice Test . 286Chapter 6 Trigonometric Identities.2886.1 Reciprocal, Quotient, and PythagoreanIdentities. 2906.2 Sum, Difference, and Double-AngleIdentities. 2996.3 Proving Identities. 3096.4 Solving Trigonometric EquationsUsing Identities. 316Chapter 6 Review . 322Chapter 6 Practice Test . 324Unit 2 Project Wrap-Up . 325Cumulative Review, Chapters 4–6 . 326Unit 2 Test . 328iv MHR Contents
Unit 3 Exponential andLogarithmic Functions . 330Chapter 7 Exponential Functions .3327.1 Characteristics of ExponentialFunctions . 334Unit 4 Equations and Functions . 426Chapter 9 Rational Functions .4289.1 Exploring Rational Functions UsingTransformations. 4309.2 Analysing Rational Functions . 4467.2 Transformations of ExponentialFunctions . 3469.3 Connecting Graphs and RationalEquations . 4577.3 Solving Exponential Equations . 358Chapter 9 Review . 468Chapter 7 Review . 366Chapter 9 Practice Test . 470Chapter 7 Practice Test . 368Chapter 8 Logarithmic Functions .3708.1 Understanding Logarithms . 3728.2 Transformations ofLogarithmic Functions . 3838.3 Laws of Logarithms . 3928.4 Logarithmic and ExponentialEquations . 404Chapter 8 Review . 416Chapter 8 Practice Test . 419Chapter 10 Function Operations .47210.1 Sums and Differences of Functions . 47410.2 Products and Quotients of Functions . 48810.3 Composite Functions . 499Chapter 10 Review . 510Chapter 10 Practice Test. 512Chapter 11 Permutations, Combinations,and the Binomial Theorem .51411.1 Permutations . 51611.2 Combinations . 528Unit 3 Project Wrap-Up . 42111.3 Binomial Theorem . 537Cumulative Review, Chapters 7–8 . 422Chapter 11 Review . 546Unit 3 Test . 424Chapter 11 Practice Test. 548Unit 4 Project Wrap-Up . 549Cumulative Review, Chapters 9–11 . 550Unit 4 Test . 552Answers. 554Glossary . 638Index . 643Credits . 646Contents MHR v
CHAPTER1FunctionTransformationsMathematical shapes are found in architecture,bridges, containers, jewellery, games, decorations,art, and nature. Designs that are repeated, reflected,stretched, or transformed in some way are pleasingto the eye and capture our imagination.In this chapter, you will explore the mathematicalrelationship between a function and itstransformed graph. Throughout the chapter, youwill explore how functions are transformed anddevelop strategies for relating complex functionsto simpler functions.Did Yo u Know ?Albert Einstein (1879—1955) is often regarded as the father ofmodern physics. He won the Nobel Prize for Physics in 1921 for“his services to Theoretical Physics, and especially for his discoveryof the law of the photoelectric effect.” The Lorentz transformationsare an important part of Einstein’s theory of relativity.Key Termstransformationinvariant pointmappingstretchtranslationinverse of a functionimage pointhorizontal line testreflection4 MHR Chapter 1
1.1Horizontal andVertical TranslationsFocus on . . . determining the effects of h and k in y - k f(x - h)on the graph of y f(x) sketching the graph of y - k f(x - h) for given valuesof h and k, given the graph of y f(x) writing the equation of a function whose graph is avertical and/or horizontal translation of the graph ofy f(x)A linear frieze pattern is a decorative patternin which a section of the pattern repeatsalong a straight line. These patterns oftenoccur in border decorations and textiles.Frieze patterns are also used by artists,craftspeople, musicians, choreographers,and mathematicians. Can you think ofplaces where you have seen a frieze pattern?Lantern Festival in ChinaInvestigate Vertical and Horizontal TranslationsMaterials grid paperA: Compare the Graphs of y f(x) and y - k f(x)1. Consider the function f(x) x .a) Use a table of values to compare the output values for y f(x),y f(x) 3, and y f (x) - 3 given input values of -3, -2, -1, 0,1, 2, and 3.b) Graph the functions on the same set of coordinate axes.2. a) Describe how the graphs of y f(x) 3 and y f(x) - 3 compareto the graph of y f (x).b) Relative to the graph of y f(x), what information about the graphof y f(x) k does k provide?3. Would the relationship between the graphs of y f(x) andy f(x) k change if f(x) x or f (x) x2? Explain.6 MHR Chapter 1
B: Compare the Graphs of y f (x) and y f(x - h)4. Consider the function f(x) x .a) Use a table of values to compare the output values for y f(x),y f(x 3), and y f(x - 3) given input values of -9, -6, -3, 0,3, 6, and 9.b) Graph the functions on the same set of coordinate axes.5. a) Describe how the graphs of y f(x 3) and y f(x - 3) compareto the graph of y f(x).b) Relative to the graph of y f(x), what information about the graphof y f(x - h) does h provide?6. Would the relationship between the graphs of y f(x) andy f(x - h) change if f(x) x or f (x) x2? Explain.Reflect and Respond7. How is the graph of a function y f (x) related to the graph ofy f (x) k when k 0? when k 0?8. How is the graph of a function y f(x) related to the graph ofy f(x - h) when h 0? when h 0?9. Describe how the parameters h and k affect the properties of thegraph of a function. Consider such things as shape, orientation,x-intercepts and y-intercept, domain, and range.Link the IdeasA transformation of a function alters the equation and anycombination of the location, shape, and orientation of the graph.Points on the original graph correspond to points on the transformed,or image, graph. The relationship between these sets of points can becalled a mapping.Mapping notation can be used to show a relationship betweenthe coordinates of a set of points, (x, y), and the coordinatesof a corresponding set of points, (x, y 3), for example, as(x, y) (x, y 3).D id Yo u K n ow?Mapping notation is an alternate notation for function notation. For example,f(x) 3x 4 can be written as f: x 3x 4. This is read as “f is a functionthat maps x to 3x 4.”transformation a change made to afigure or a relation suchthat the figure or thegraph of the relation isshifted or changed inshapemapping the relating of one setof points to another setof points so that eachpoint in the original setcorresponds to exactlyone point in the imageset1.1 Horizontal and Vertical Translations MHR 7
translation a slide transformationthat results in a shiftof a graph withoutchanging its shape ororientation vertical and horizontaltranslations are typesof transformations withequations of the formsy - k f(x) andy f(x - h), respectively a translated graphis congruent to theoriginal graphOne type of transformation is a translation. A translation can move thegraph of a function up, down, left, or right. A translation occurs whenthe location of a graph changes but not its shape or orientation.Example 1Graph Translations of the Form y - k f(x) and y f (x - h)a) Graph the functions y x2, y - 2 x2, and y (x - 5)2 on the sameset of coordinate axes.b) Describe how the graphs of y - 2 x2 and y (x - 5)2 compare to thegraph of y x2.Solutiona) The notation y - k f(x) is often used instead of y f(x) k toemphasize that this is a transformation on y. In this case, the basefunction is f(x) x2 and the value of k is 2.The notation y f (x - h) shows that this is a transformation on x. Inthis case, the base function is f (x) x2 and the value of h is 5.Rearrange equations as needed and use tables of values to help yougraph the functions.xy x2xy x2 2xy (x - 189yFor y x2 2, the input values are thesame but the output values change.Each point (x, y) on the graph of y x 2is transformed to (x, y 2).y x2 210864y x2y (x - 5)22-2 0246810xFor y (x - 5)2, to maintainthe same output values as thebase function table, the inputvalues are different. Every point(x, y) on the graph of y x 2 istransformed to (x 5, y). How dothe input changes relate to thetranslation direction?b) The transformed graphs are congruent to the graph of y x2.Each point (x, y) on the graph of y x2 is transformed to become thepoint (x, y 2) on the graph of y - 2 x2. Using mapping notation,(x, y) (x, y 2).8 MHR Chapter 1
Therefore, the graph of y - 2 x2 is the graph of y x2 translatedvertically 2 units up.Each point (x, y) on the graph of y x2 is transformed to become thepoint (x 5, y) on the graph of y (x - 5)2. In mapping notation,(x, y) (x 5, y).Therefore, the graph of y (x - 5)2 is the graph of y x2 translatedhorizontally 5 units to the right.Your TurnHow do the graphs of y 1 x2 and y (x 3)2 compare to the graphof y x2? Justify your reasoning.Example 2Horizontal and Vertical TranslationsSketch the graph of y x - 4 3.SolutionFor y x - 4 3, h 4 and k -3. Start with a sketch of the graph of thebase function y x , using key points. Apply the horizontal translation of4 units to the right to obtain the graph ofy x - 4 .yD i d You K n ow?64y x - 4 y x 262To ensure an accurate sketch of atransformed function, translate keypoints on the base function first.-2 0 Apply the vertical translation of 3 unitsup to y x - 4 to obtain the graphof y x - 4 3.y864xKey points arepoints on a graphthat give importantinformation, suchas the x-intercepts,the y-intercept, themaximum, and theminimum.y x - 4 34Would the graph be in the correctlocation if the order of thetranslations were reversed?2-2 0y x - 4 246xThe point (0, 0) on the function y x is transformed to becomethe point (4, 3). In general, the transformation can be described as(x, y) (x 4, y 3).Your TurnSketch the graph of y (x 5)2 - 2.1.1 Horizontal and Vertical Translations MHR 9
Example 3Determine the Equation of a Translated FunctionDescribe the translation that has been applied to the graph of f(x)to obtain the graph of g(x). Determine the equation of the translatedfunction in the form y - k f (x - h).a)yf(x) x2642-6-4-2 024x6-2-4g(x)-6yb)6f(x)A-6CB4-4ED2-2 024x6-2-4g(x)B -6A It is a commonconvention to use aprime ( ) next to eachletter representing animage point.C D E Solutiona) The base function is f (x) x2. Choose key points on the graph ofimage point the point that isthe result of atransformation of apoint on the originalgraph10 MHR Chapter 1f(x) x2 and locate the corresponding image points on the graphof g(x).f(x)(0, 0)(-1, 1)(1, 1)(-2, 4)(2, 4)(x, y) g(x)(-4, -5)(-5, -4)(-3, -4)(-6, -1)(-2, -1)(x - 4, y - 5)For a horizontal translation anda vertical translation whereevery point (x, y) on the graphof y f(x) is transformed to(x h, y k), the equation of thetransformed graph is of the formy - k f(x - h).
To obtain the graph of g(x), the graph of f(x) x2 has been translated4 units to the left and 5 units down. So, h -4 and k -5.To write the equation in the form y - k f(x - h), substitute -4for h and -5 for k.y 5 f(x 4)b) Begin with key points on the graph of f(x). Locate the correspondingimage points.f(x)g(x)A(-5, 2) A (-1, -7)B(-4, 4) B (0, -5)C(-1, 4) C (3, -5)D(1, 3) D (5, -6)E(3, 3) E (7, -6)(x, y) (x 4, y - 9)To obtain the graph of g(x), the graph of f(x) has been translated4 units to the right and 9 units down. Substitute h 4 and k -9into the equation of the form y - k f(x - h):D i d You K n ow?In Pre-Calculus 11,you graphed quadraticfunctions of the formy (x - p)2 qby consideringtransformations fromthe graph of y x2.In y (x - p)2 q,the parameter pdetermines thehorizontal translationand the parameterq determines thevertical translationof the graph. In thisunit, the parametersfor horizontal andvertical translationsare represented by hand k, respectively.y 9 f(x - 4)Your TurnDescribe the translation that has been applied to the graph of f(x)to obtain the graph of g(x). Determine the equation of the translatedfunction in the form y - k f(x - h).ya)10g(x)864f(x) x 2-4b)-2 0A B -446xy6C g(x)-62D 42-2 0f(x)A2CB4D6x1.1 Horizontal and Vertical Translations MHR 11
Key IdeasTranslations are transformations that shift all points on the graph of a functionup, down, left, and right without changing the shape or orientation of the graph.The table summarizes translations of the function y f (x).Functiony - k f(x) ory f(x) kTransformationfrom y f(x)A vertical translationMappingExample(x, y) (x, y k)If k 0, thetranslation is up.If k 0, thetranslation is down.y - k f(x), k 00y f(x - h)A horizontal translationy f(x)y - k f(x), k 0(x, y) (x h, y)If h 0, thetranslation is to theright.If h 0, thetranslation is to theleft.y f(x - h), h 0y f(x)0 y f(x - h), h 0A sketch of the graph of y - k f(x - h), or y f(x - h) k, can be created bytranslating key points on the graph of the base function y f (x).Check Your UnderstandingPractise1. For each function, state the values ofh and k, the parameters that representthe horizontal and vertical translationsapplied to y f(x).a) y - 5 f(x)b) y f(x) - 4c) y f(x 1)d) y 3 f(x - 7)e) y f (x 2) 42. Given the graph of y f(x) and each of thefollowing transformations, state the coordinates of the image pointsA , B , C , D and E sketch the graph of the transformedfunctiona) g(x) f (x) 3b) h(x) f(x - 2)c) s(x) f(x 4)yy f(x)2BC-4A12 MHR Chapter 1d) t(x) f(x) - 2-2 0-2D2Ex
3. Describe, using mapping notation, how thegraphs of the following functions can beobtained from the graph of y f(x).a) y f(x 10)8. Copy and complete the table.TranslationTransformedFunctionTransformation ofPointsy f(x) 5(x, y) (x, y 5)b) y 6 f(x)y f(x 7)(x, y) (x - 7, y)c) y f(x - 7) 4y f(x - 3)d) y - 3 f(x - 1)y f(x) - 6vertical4. Given the graph of y f(x), sketch thegraph of the transformed function. Describethe transformation that can be applied tothe graph of f(x) to obtain the graph ofthe transformed function. Then, write thetransformation using mapping notation.a) r(x) f(x 4) - 3horizontaland verticaly 9 f(x 4)horizontaland vertical(x, y) (x 4, y - 6)(x, y) (x - 2, y 3)horizontaland verticaly f(x - h) k9. The graph of the function y x2 isb) s(x) f(x - 2) - 4c) t(x) f(x - 2) 5translated 4 units to the left and 5 units upto form the transformed function y g(x).d) v(x) f(x 3) 2a) Determine the equation of the functionAy f(x)-6y g(x).y2BC-4 -2 0DEb) What are the domain and range of theimage function?2x-2c) How could you use the description ofthe translation of the function y x2 todetermine the domain and range of theimage function?10. The graph of f(x) x is transformed toApply5. For each transformation, identify thevalues of h and k. Then, write theequation of the transformed functionin the form y - k f(x - h).1a) f(x) x , translated 5 units to the leftand 4 units upb) f(x) x2, translated 8 units to the rightand 6 units upc) f(x) x , translated 10 units to theright and 8 units downd) y f(x), translated 7 units to the leftand 12 units down6. What vertical translation is applied toy x2
Therefore, the graph of y- 2 2 2x is the graph of y x translated vertically 2 units up. Each point (x, y) on the graph of y x2 is transformed to become the point (x 5, y) on the graph of y (x - 5)2.In mapping notation, (x, y) (x 5, y).Therefore, the graph of y (x - 5)2 is the graph of y x2 translated horizontally 5 units to the right.
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Pre-Calculus . The Pre-Calculus End-of-Course assessment is designed to measure student proficiency of the Common Core State Standards pertaining to Pre-Calculus. This course-level assessment is provided to all students who have completed Pre-Calculus or related courses. EOC Assessment Aligns to the Following Course Codes: 2053 - Pre-Calculus
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The entire body of McGraw-Hill Ryerson Mathematics of Data Management, Solutions was created on a home computer in Microsoft Word 2000. Graphics for the solutions were created with the help of spreadsheets (Microsoft Excel 2000 and Corel Quattro Pro 8), Fathom 1.1, and TI-83 Plus graphing calculator screens captured to the computer.
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The entire body of McGraw-Hill Ryerson Mathematics of Data Management, Solutions was created on a home computer in Microsoft Word 2000. Graphics for the solutions were created with the help of spreadsheets (Microsoft Excel 2000 and Corel Quattro Pro 8), Fathom 1.1, and TI-83 Plus graphing calculator screens captured to the computer.
