NEW from McGraw-Hill Ryerson Pre-Calculus 12 Student Text Sampler. CONTENTS a Table of Contents a Program Overview a Sample Chapter 1

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1 NEW from McGraw-Hill Rerson Pre-Calculus 1 Student Tet Sampler CONTENTS a Table of Contents a Program Overview a Sample Chapter 1

2 Select from our outstanding Print, Digital, and Online STUDENT RESOURCES STUDENT WORKBOOK STUDENT RESOURCE Student Tet [PRINT] a The onl comprehensive resource designed to provide students and educators with the necessar support to successfull implement Pre-Calculus 1 in all western jurisdictions a Ease-of-navigation for students, parents, and teachers alike a A variet of assessment strategies and tools accommodate the diversit of abilities and learning stles ISBN: Student Workbook [PRINT] a This consumable resource is designed to accompan and support the use of the Pre-Calculus 1 Student Resource a Each section provides additional support for the Ke Ideas, Worked Eamples, Practice questions, and Review to allow students to consolidate their learning and self-assess their understanding a Students can use the workbook during class or as homework ISBN: Student Resource DVD [DVD] a This DVD product provides students with access to the complete digital student tet pages and Interactive activities, animations, video clips, applets, etc. a A convenient organizational sstem that allows students to easil navigate through the resource and its associated digital content ISBN: CONNECTschool for Students [ONLINE: a This online product provides students with /7 access to the complete digital student tet pages and Interactive activities, animations, video clips, applets, web links, etc. a Allows students to personalize and organize their learning eperience using a variet of tools such as: a Post It Notes to create personal reminders and summaries a Bookmark to tag pages a Highlight tet and images a Cop and paste tet and images NOTE: Product will arrive in the form of an access code card which contains a unique pincode ISBN: For pricing please contact our McGraw-Hill Rerson Sales Representative or visit

3 Resources to implement Pre-Calculus 1 Your Wa! TEACHER RESOURCES TEACHER S RESOURCE TEACHER S RESOURCE COMPUTERIZED ASSESSMENT BANK SOLUTIONS CD-ROM Teacher s Resource [PRINT + CD-ROM] A comprehensive resource designed to provide educators with the support needed to successfull implement McGraw-Hill Rerson Pre-Calculus 1. Resource includes: a -page laminated lesson and unit planning charts a Support for differentiated instruction, ELL, and gifted and enrichment a Common errors and suggestions for helping students a Assessment and evaluation support a Editable Blackline Masters provided on a CD-ROM ISBN: Teacher s Resource DVD [DVD] This DVD product allows teachers access in one convenient location to: a The complete digital student resource a The complete digital teacher s resource a Editable Blackline Masters a Interactive lessons (e.g., Smartboard activities, PowerPoint lessons, The Geometer s Sketchpad activities, spreadsheets) ISBN: Teacher s Resource + Teacher s Resource DVD [PRINT + CD-ROM + DVD] a Includes the Teacher s Resource (see description above) a Includes the Teacher s Resource DVD (see description above) ISBN: CONNECTschool for Teachers [ONLINE: a This online product provides teachers with /7 access to the complete digital student tet pages, teacher s resource, blackline masters, Interactive activities, animations, video clips, applets, web links, etc. a Allows teachers to personalize and organize their teaching plans using a variet of tools such as: a Customize lessons for individual needs a Post ke dates for student assignments, tests, etc. a Annotate and highlight student pages and teaching notes NOTE: Product will arrive in the form of an access code card which contains a unique pincode 1-Year Access ISBN: ear Access ISBN: Computerized Assessment Bank [CD-ROM] a Provides hundreds of questions in a searchable database for teachers to use and adapt a Man questions use algorithmic capabilities to allow for generation of multiple variations a Includes all answers ISBN: Solutions CD-ROM [CD-ROM] a Includes full-worked solutions for all eercises in the student resource a Provided in PDF format, broken down b section for ease-of-use ISBN: For pricing please contact our McGraw-Hill Rerson Sales Representative or visit

4 ONLINE RESOURCES CONNECTschool Introducing CONNECTschool available at Q Q Q Q Q Q What is CONNECTschool? a CONNECTschool is the net generation of McGraw-Hill Rerson s digital offering a CONNECTschool is a personalized and interactive learning and teaching hub a CONNECTschool is an online learning and teaching resource that personalizes and deepens students independent self-assessment, stud and practice /7 a For teachers CONNECTschool augments classroom instruction with pre-built lessons that can also be customized. Wh is CONNECTschool unique? aconnectschool allows the student and teacher to personalize their online learning and teaching Can I access CONNECTschool from an computer? ayes, because CONNECTschool is web-based, ou can access our account from an computer with Internet access /7. What do I need to access m CONNECTschool? ayou require a computer with Internet access and a pincode What is included in Pre-Calculus 1 CONNECTschool for Students? acomplete digital student tet pages (in html and PDF format) amedia Assets: Interactive activities, animations, video clips, applets, and web links What is included in Pre-Calculus 1 CONNECTschool for Teachers? acomplete digital student tet pages (in html and PDF format) acomplete digital teacher s resource and blackline masters (in PDF and Word format) ainteractive lessons: Smartboard activities, PowerPoint lessons, The Geometer s Sketchpad activities, and spreadsheets amedia Assets: Interactive activities, animations, video clips, applets, and web links For more information on CONNECTschool please visit For pricing please contact our McGraw-Hill Rerson Sales Representative or visit

5 Table of Contents UNIT 1: TRANSFORMATIONS AND FUNCTIONS Chapter 1: Function Transformations 1.1 Horizontal and Vertical Translations 1. Reflections and Stretches 1.3 Combining Transformations 1. Inverse of a Relation Chapter 1 Review Chapter 1 Practice Test Chapter : Radical Functions.1 Radical Functions and Transformations. Square Root of a Function.3 Solving Radical Equations Graphicall Chapter Review Chapter Practice Test Chapter 3: Polnomial Functions 3.1 Characteristics of Polnomial Functions 3. The Remainder Theorem 3.3 The Factor Theorem 3. Equations and Graphs of Polnomial Functions Chapter 3 Review Chapter 3 Practice Test Unit 1 Project Wrap-Up Cumulative Review, Chapters 1 3 Unit 1 Test UNIT : TRIGONOMETRY Chapter : Trigonometr and the Unit Circle.1 Angles and Angle Measure. The Unit Circle.3 Trigonometric Ratios. Introduction to Trigonometric Equations Chapter Review Chapter Practice Test Chapter 5: Trigonometric Functions and Graphs 5.1 Graphing Sine and Cosine Functions 5. Transformations of Sinusoidal Functions 5.3 The Tangent Function 5. Equations and Graphs of Trigonometric Functions Chapter 5 Review Chapter 5 Practice Test Chapter : Trigonometric Identities.1 Reciprocal, Quotient, and Pthagorean Identities. Sum, Difference, and Double-Angle Identities..3 Proving Identities. Solving Trigonometric Equations Using Identities Chapter Review Chapter Practice Test Unit Project Wrap-Up Cumulative Review, Chapters Unit Test UNIT 3: EPONENTIAL AND LOGARITHMIC FUNCTIONS Chapter 7: Eponential Functions 7.1 Characteristics of Eponential Functions 7. Transformations of Eponential Functions 7.3 Solving Eponential Equations Chapter 7 Review Chapter 7 Practice Test Chapter 8: Logarithmic Functions 8.1 Understanding Logarithms 8. Transformations of Logarithmic Functions 8.3 Laws of Logarithms 8. Logarithmic and Eponential Equations Chapter 8 Review Chapter 8 Practice Test Unit 3 Project Wrap-Up Cumulative Review, Chapters 7 8 Unit 3 Test UNIT EQUATIONS AND FUNCTIONS Chapter 9: Rational Functions 9.1 Eploring Rational Functions Using Transformations 9. Analsing Rational Functions 9.3 Connecting Graphs and Rational Equations Chapter 9 Review Chapter 9 Practice Test Chapter 1: Function Operations 1.1 Sums and Differences of Functions 1. Products and Quotients of Functions 1.3 Composite Functions Chapter 1 Review Chapter 1 Practice Test Chapter 11: Permutations, Combinations, and the Binomial Theorem 11.1 Permutations 11. Combinations 11.3 Binomial Theorem Chapter 11 Review Chapter 11 Practice Test Unit Project Wrap-Up Cumulative Review, Chapters 9 11 Unit Test For pricing please contact our McGraw-Hill Rerson Sales Representative or visit

6 Unit 1 Transformations and Functions Functions help ou make sense of the world around ou. Man ordinar measuring devices are based on mathematical functions: T F Car odometer: The odometer reading is a function of the number of rotations of the car s transmission drive shaft. Displa on a barcode reader: When the screen displas the data about the object, the reader performs an inverse function b decoding the barcode image. A R Man natural occurrences can be modelled b mathematical functions: Ripples created b a water droplet in a pond: You can model the area spanned b the ripples b a polnomial function. D Eplosion of a supernova: You can model the time the eplosion takes to affect a volume of space b a radical function. In this unit, ou will epand our knowledge of transformations while eploring radical and polnomial functions. These functions and associated transformations are useful in a variet of applications within mathematics. Looking Ahead In this unit, ou will solve problems involving transformations of functions inverses of functions radical functions and equations polnomial functions and equations MHR Unit 1 Transformations and Functions 1PC1_Chapter1_7th_E.indd 1/5/11 3:39:8 PM

7 Unit 1 Project The Art of Mathematics Simone McLeod, a Cree-Ojibwa originall from Winnipeg, Manitoba, now lives in Saskatchewan and is a member of the James Smith Cree Nation. Simone began painting later in life. DRAFT I reall believed that I had to wait until I could find something that had a lot of meaning to me. Each painting contains a piece of m soul. I have a strong faith in humankind and m paintings are silent praers of hope for the future. M Indian name is Earth Blanket (all that covers the earth such as grass, flowers, and trees). The sun, the blankets, and the flowers/rocks are all the same colours to show how all things are equal. Simone s work is collected all over the world, including Europe, India, Asia, South Africa, and New Zealand. In this project, ou will search for mathematical functions in art, nature, and manufactured objects. You will determine equations for the functions or partial functions ou find. You will justif our equations and displa them superimposed on the image ou have selected. Unit 1 Transformations and Functions MHR 3

8 CHAPTER 1 Function Transformations Mathematical shapes are found in architecture, bridges, containers, jeweller, games, decorations, art, and nature. Designs that are repeated, reflected, stretched, or transformed in some wa are pleasing to the ee and capture our imagination. In this chapter, ou will eplore the mathematical relationship between a function and its transformed graph. Throughout the chapter, ou will eplore how functions are transformed and develop strategies for relating comple functions to simpler functions. A R D id Yo u Know? Albert Einstein ( ) is often regarded as the father of modern phsics. He won the Nobel Prize for Phsics in 191 for his services to Theoretical Phsics, and especiall for his discover of the law of the photoelectric effect. The Lorentz transformations are an important part of Einstein s theor of relativit. Ke Terms transformation mapping translation image point reflection T F D invariant point stretch inverse of a function horizontal line test MHR Chapter 1 1PC1_Chapter1_7th_E.indd 1/5/11 3:39:3 PM

9 T F A R D Career Link A phsicist is a scientist who studies the natural world, from sub-atomic particles to matters of the universe. Some phsicists focus on theoretical areas, while others appl their knowledge of phsics to practical areas, such as the development of advanced materials and electronic and optical devices. Some phsicists observe, measure, interpret, and develop theories to eplain celestial and phsical phenomena using mathematics. Phsicists use mathematical functions to make numerical and algebraic computations easier. We b Link To find ind out more about the career of a phsicist, go to and follow the links. Chapter 1 MHR 5 1PC1_Chapter1_7th_E.indd 5 1/5/11 3:39:51 PM

10 1.1 Horizontal and Vertical Translations Focus on... determining the effects of h and k in - k = f( - h) on the graph of = f() T F sketching the graph of - k = f( - h) for given values of h and k, given the graph of = f() writing the equation of a function whose graph is a vertical and/or horizontal translation of the graph of = f() A linear frieze pattern is a decorative pattern in which a section of the pattern repeats along a straight line. These patterns often occur in border decorations and tetiles. Frieze patterns are also used b artists, craftspeople, musicians, choreographers, and mathematicians. Can ou think of places where ou have seen a frieze pattern? A R D Lantern Festival in China Investigate Vertical and Horizontal Translations Materials grid paper A: Compare the Graphs of = f() and - k = f() 1. Consider the function f() =. a) Use a table of values to compare the output values for = f(), = f() + 3, and = f () - 3 given input values of -3, -, -1,, 1,, and 3. b) Graph the functions on the same set of coordinate aes.. a) Describe how the graphs of = f() + 3 and = f() - 3 compare to the graph of = f (). b) Relative to the graph of = f(), what information about the graph of = f() + k does k provide? 3. Would the relationship between the graphs of = f() and = f() + k change if f() = or f () =? Eplain. MHR Chapter 1 1PC1_Chapter1_7th_E.indd 1/5/11 3:: PM

11 B: Compare the Graphs of = f() and = f( - h). Consider the function f() =. a) Use a table of values to compare the output values for = f(), = f( + 3), and = f( - 3) given input values of -9, -, -3,, 3,, and 9. b) Graph the functions on the same set of coordinate aes. 5. a) Describe how the graphs of = f( + 3) and = f( - 3) compare to the graph of = f(). b) Relative to the graph of = f(), what information about the graph adraft of = f( - h) does h provide?. Would the relationship between the graphs of = f() and = f( - h) change if f() = or f() =? Eplain. Reflect and Respond 7. How is the graph of a function = f() ) related to the graph of = f() + k when k >? when k <? 8. How is the graph of a function = f() ) related to the graph of = f( - h) when h >? when h <? 9. Describe how the parameters h and k affect the properties of the graph of a function. Consider such things as shape, orientation, -intercepts and -intercept, domain, and range. Link the Ideas A transformation of a function alters the equation and an transformation combination of the location, shape, and orientation of the graph. a change made to a figure or a relation such Points on the original graph correspond to points on the transformed, that the figure or the or image, graph. The relationship between these sets of points can be graph of the relation is called a mapping. shifted or changed in shape Mapping notation can be used to show a relationship between DRAFT the coordinates of a set of points, (, ), and the coordinates of a corresponding set of points, (, + 3), for eample, as (, ) (, + 3). Did You Know? Mapping notation is an alternate notation for function notation. For eample, f() = 3 + can be written as f: 3 +. This is read as f is a function that maps to 3 +. mapping the relating of one set of points to another set of points so that each point in the original set corresponds to eactl one point in the image set 1.1 Horizontal and Vertical Translations MHR 7

12 translation a slide transformation that results in a shift of a graph without changing its shape or orientation vertical and horizontal translations are tpes of transformations with equations of the forms - k = f() and = f( - h), respectivel a translated graph is congruent to the original graph One tpe of transformation is a translation. A translation can move the graph of a function up, down, left, or right. A translation occurs when the location of a graph changes but not its shape or orientation. Eample 1 Graph Translations of the Form - k = f() and = f( - h) a) Graph the functions =, - =, and = ( - 5) on the same set of coordinate aes. b) Describe how the graphs of - = and = ( - 5) compare to the graph of =. Solution a) The notation - k = f() is often used instead of = f() ) + k to emphasize that this is a transformation on.. In this case, the base function is f() = and the value of k is. The notation = f( - h) shows that this is a transformation on.. In this case, the base function is f() ) = and the value of h is 5. Rearrange equations as needed and use tables of values to help ou graph the functions. = = = + = ( - 5) DRAFT 1 8 = + = ( - 5) 8 1 b) The transformed graphs are congruent to the graph of =. For = +, the input values are the same but the output values change. Each point (, ) on the graph of = is transformed to (, + ). DRAFT. In DRAFTh DRAFTis 5. For = ( - 5), to maintain the same output values as the base function table, the input values are different. Ever point (, ) on the graph of = is transformed to ( + 5, ). How do the input changes relate to the translation direction? Each point (, ) on the graph of = is transformed to become the point (, + ) on the graph of - =. Using mapping notation, (, ) (, + ). 8 MHR Chapter 1

13 Therefore, the graph of - = is the graph of = translated verticall units up. Each point (, ) on the graph of = is transformed to become the point ( + 5, ) on the graph of = ( - 5). In mapping notation, (, ) ( + 5, ). Therefore, the graph of = ( - 5) is the graph of = translated horizontall 5 units to the right. Your Turn How do the graphs of + 1 = and = ( + 3) compare to the graph of =? Justif our reasoning. Eample Horizontal and Vertical Translations Sketch the graph of = Solution For = - + 3, h = and k = -3. Start with a sketch of the graph of the base function =,, using ke points. Appl the horizontal translation of units to the right to obtain the graph of = -. To ensure an accurate sketch of a transformed function, translate ke points on the base function first. Appl the vertical translation of 3 units up to = - to obtain the graph of = Would the graph be in the correct location if the order of the translations were reversed? - - = - 8 = =DRAFT DRAFT todraft DRAFT DRAFT =DRAFT -DRAFT up to = = = = - Did You Know? Ke points are points on a graph that give important information, such as the -intercepts, the -intercept, the maimum, and the minimum. The point (, ) on the function = is transformed to become the point (, 3). In general, the transformation can be described as (, ) ( +, + 3). Your Turn Sketch the graph of = ( + 5) Horizontal and Vertical Translations MHR 9

14 Eample 3 Determine the Equation of a Translated Function Describe the translation that has been applied to the graph of f() to obtain the graph of g(). Determine the equation of the translated function in the form - k = f( - h). a) f() = image point the point that is the result of a transformation of a point on the original graph - - g() b) f() ( ) A - Solution B - C - - D E - B C g() - D E A a) The base function is f() =. Choose ke points on the graph of f() = and locate the corresponding image points on the graph of g(). f() g() (, ) (-, -5) ( -1, 1) (-5, -) (1, 1) (-3, -) ( -, ) (-, -1) (, ) (-, -1) (, ) ( -, - 5) It is a common convention to use a prime ( ) net to each letter representing an image point. DRAFT For a horizontal translation and a vertical translation where ever point (, ) on the graph of = f() is transformed to ( + h, + k), the equation of the transformed graph is of the form - k = f( - h). 1 MHR Chapter 1

15 To obtain the graph of g(), the graph of f() = has been translated units to the left and 5 units down. So, h = - and k = -5. Did You Know? In Pre-Calculus 11, To write the equation in the form - k = f( - h), substitute - ou graphed quadratic functions of the form for h and -5 for k. = ( - p) + q b considering + 5 = f( + ) transformations from b) Begin with ke points on the graph of f(). Locate the corresponding image points. the graph of =. In = ( - p) + q, the parameter p f() g() determines the horizontal translation A(-5, ) A (-1, -7) and the parameter B(-, ) B (, -5) q determines the C(-1, ) C (3, -5) vertical translation of the graph. In this D(1, 3) D (5, -) unit, the parameters E(3, 3) E (7, -) for horizontal and (, ) ( +, - 9) vertical translations DRAFTare represented b h To DRAFTk, obtain the graph of g(), the graph of f() ) has been translated and, respectivel. units to the right and 9 units down. Substitute h = and k = -9 9 into the equation of the form - k = f( - h): + 9 = f( - ) Your Turn Describe the translation that has been applied to the graph of f() ) to obtain the graph of g(). Determine the equation of the translated function in the form - k = f( - h). a) 1 g() ) 8 f() = - - DRAFT DRAFT 8DRAFT DRAFTand DRAFTk k b) A C g() D B A f() C D B 1.1 Horizontal and Vertical Translations MHR 11

16 Ke Ideas Translations are transformations that shift all points on the graph of a function up, down, left, and right without changing the shape or orientation of the graph. The table summarizes translations of the function = f(). Function - k = f() or = f() + k Transformation from = f() Mapping Eample A vertical translation (, ) (, + k) If k >, the translation is up. - k = f(), k > If k <, the translation is down. = f() - k = f(), ( k < = f( - h) A horizontal translation (, ) ( + h, ) If h >, the translation is to the = f( ( - h), h > DRAFT( DRAFT) = f() right. If h <, the translation is to the left. = f( - h), h < A sketch of the graph of - k = f( - h), or = f( - h) + k, can be created b translating ke points on the graph of the base function = f(). Check Your Understanding. Given the graph of = f() and each of the 1. For each function, state the values of following transformations, h and k, the parameters that represent state the coordinates of the image points the horizontal and vertical translations A, B, C, D and E applied to = f(). sketch the graph of the transformed a) - 5 = f() function a) g() = f() + 3 b) h() = f( - ) b) = f() - c) s() = f( + ) d) t() = f() - c) = f( + 1) d) + 3 = f( - 7) = f() e) = f( + ) + B C Practise DRAFT DRAFTf( DRAFT) - A - - D E 1 MHR Chapter 1

17 3. Describe, using mapping notation, how the graphs of the following functions can be obtained from the graph of = f(). a) = f( + 1) b) + = f() c) = f( - 7) + = f( - 3) d) - 3 = f( - 1) = f() - horizontal. Given the graph of = f(), sketch the + 9 = f( + ) and vertical graph of the transformed function. Describe horizontal DRAFT the transformation that can be applied to (, ) ( +, - ) and vertical the graph of f() to obtain the graph of (, ) ( -, + 3) the transformed function. Then, write the horizontal transformation using mapping notation. = f( - h) ) + k and vertical a) r() = f( + ) The graph of the function = is b) s() = f( - ) - translated units to the left and 5 units up c) t() = f( - ) + 5 to form the transformed function = g(). d) v() = f( + 3) + a) Determine the equation of the function = g(). A D E b) What are the domain and range of the = f() image function? B C c) How could ou use the description of - the translation of the function = to determine the domain and range of the image function? Appl 1. The graph of f() = is transformed to 5. For each transformation, identif the the graph of g() = f( - 9) + 5. values of h and k.. Then, write the a) Determine the equation of the equation of the transformed function function g(). in the form - k = f( ( - h). b) Compare the graph of g() to the graph a) f() = _ 1, translated 5 units to the left of the base function f(). and units up b) f() =, translated 8 units to the right and units up c) f() =, translated 1 units to the right and 8 units down d) = f(), translated 7 units to the left and 1 units down. What vertical translation is applied to = if the transformed graph passes through the point (, 19)? 7. What horizontal translation is applied to = if the translation image graph passes through the point (5, 1)? 8. Cop and complete the table. Translation Transformed Function Transformation of Points vertical = f() + 5 (, ) (, + 5) = f( + 7) (, ) ( - 7, ) fdraft f(draft (DRAFT )DRAFT =DRAFT 1DRAFT 1_DRAFT formdraft =DRAFT DRAFT DRAFT DRAFT= DRAFT DRAFT = is DRAFT = in the ) _ c) Determine three points on the graph of f(). Write the coordinates of the image points if ou perform the horizontal translation first and then the vertical translation. d) Using the same original points from part c), write the coordinates of the image points if ou perform the vertical translation first and then the horizontal translation. e) What do ou notice about the coordinates of the image points from parts c) and d)? Is the order of the translations important? 1.1 Horizontal and Vertical Translations MHR 13

18 11. The graph of the function drawn in red is a translation of the original function drawn in blue. Write the equation of the translated function in the form - k = f( - h). a) 13. Architects and designers often use translations in their designs. The image shown is from an Italian roadwa. - a) Use the coordinate plane overla with BḊRAFT - f() = 1 the base semicircle shown to describe _ the approimate transformations of - the semicircles. b) If the semicircle at the bottom left of b) the image is defined b the function = f(), state the approimate = f() equations of three other semicircles. 1. This Pow Wow belt shows a frieze - pattern where a particular image has - been translated throughout the length of the belt Janine is an avid cclist. After ccling to a lake and back home, she graphs her distance versus time (graph A). a) If she left her house at 1 noon, briefl describe a possible scenario for Janine s trip. a) With or without technolog, create a design using a pattern that is a function. b) Describe the differences it would make Use a minimum of four translations to Janine s ccling trip if the graph of of our function to create our own the function were translated, as shown frieze pattern. in graph B. b) Describe the translation of our design c) The equation for graph A could be in words and in an equation of the form written as = f(). Write the equation = f( - h). for graph B. Did You Know? Distance From Home (km) 3 1 A B DRAFT In First Nations communities toda, Pow Wows have evolved into multi-tribal festivals. Traditional dances are performed b men, women, and children. The dancers wear traditional regalia specific to their dance stle and nation of origin. 8 1 Time (h) 1 MHR Chapter 1

19 15. Michelle Lake and Coral Lake, located near the Columbia Ice Fields, are the onl two lakes in Alberta in which rare golden trout live. a) Determine the equation of the image function. b) Describe the translations on the graph of =. c) Determine the -intercept of the translated function. 18. Use translations to describe how the graph of = _ 1 compares to the graph of each function. Suppose the graph represents the number a) - = _ 1_ = 1 b) + of golden trout in Michelle Lake in the ears since 197. d) = c) - 3 = 1 1 DRAFT f(t) 19. a) Predict the relationship between the DRAFT- graph of = 3 - and the graph of + 3 = ( - ) 3 - ( - ). 1 b) Graph each function to verif our 1 prediction. 8 Create Connections C1 The graph of the function = f() is transformed to the graph of 8 1 t = f( - h) + k. Time Since 197 (ears) a) Show that the order in which ou appl translations does not matter. Eplain Let the function f(t) ) represent the number wh this is true. of fish in Michelle Lake since 197. b) How are the domain and range affected Describe an event or a situation for the b the parameters h and k? fish population that would result in the C Complete the square and eplain how to following transformations of the graph. transform the graph of = to the graph Then, use function notation to represent of each function. the transformation. a) f() = a) a vertical translation of units up b) g() = b) a horizontal translation of 3 units to the right Number of Trout (hundreds) 1. Paul is an interior house painter. He determines that the function n = f(a) gives the number of gallons, n, of paint needed to cover an area, A, in square metres. Interpret n = f(a) + 1 and n = f(a + 1) in this contet. Etend 17. The graph of the function = is translated to an image parabola with zeros 7 and 1. DRAFT = + DRAFTd) DRAFT DRAFT= DRAFT1 DRAFT = DRAFT+ + 3 DRAFT DRAFT DRAFTand the DRAFT( DRAFT DRAFT) C3 The roots of the quadratic equation = are -3 and. Determine the roots of the equation ( - 5) - ( - 5) - 1 =. C The function f() = + could be a vertical translation of units up or a horizontal translation of units to the left. Eplain wh. 1.1 Horizontal and Vertical Translations MHR 15

20 1. Reflections and Stretches Focus on... developing an understanding of the effects of reflections on the graphs of functions and their related equations developing an understanding of the effects of vertical and horizontal stretches on the graphs of functions and their related equations Reflections, smmetr, as well as horizontal and vertical stretches, appear in architecture, tetiles, science, and works of art. When something is smmetrical or stretched in the geometric sense, its parts have a one-to-one correspondence. How does this relate to the stud of functions? Investigate Reflections and Stretches of Functions Materials grid paper graphing technolog Ndebele artist, South Africa A: Graph Reflections in the -Ais and the -Ais 1. a) Draw a set of coordinate aes on grid paper. In quadrant I, plot a point A. Label point A with its coordinates. b) Use the -ais as a mirror line, or line of reflection, and plot point A, the mirror image of point A in the -ais. c) How are the coordinates of points A and A related? d) If point A is initiall located in an of the other quadrants, does the relationship in part c) still hold true?. Consider the graph of the function = f(). DRAFT = f() - a) Eplain how ou could graph the mirror image of the function in the -ais. b) Make a conjecture about how the equation of f() changes to graph the mirror image. 1 MHR Chapter 1

21 3. Use graphing technolog to graph the function = +, -5 5, and its mirror image in the -ais. What equation did ou enter to graph the mirror image?. Repeat steps 1 to 3 for a mirror image in the -ais. Reflect and Respond 5. Cop and complete the table to record our observations. Write concluding statements summarizing the effects of reflections in the aes. Function = f() Reflection in Verbal Description Mapping -ais (, ) (, ) -ais (, ) (, ) B: Graph Vertical and Horizontal Stretches Equation of Transformed Function. a) Plot a point A on a coordinate grid and label it with its coordinates. b) Plot and label a point A with the same -coordinate as point A, but with the -coordinate equal to times the -coordinate of point A. c) Plot and label a point A with the same -coordinate as point A, but with the -coordinate equal to _ 1_ the -coordinate of point A. d) Compare the location of points A and A to the Has the distance location of the original point A. Describe how to the -ais or the -ais changed? multipling the -coordinate b a factor of or a factor of _ 1 1_ affects the position of the image point. 7. Consider the graph of the function = f() in step. Sketch the graph of the function when the -values have been a) multiplied b DRAFT 7.DRAFT b) multiplied b _ 1 8. What are the equations of the transformed functions in step 7 in the form = af()? 9. For step 7a), the graph has been verticall stretched about the -ais b a factor of. Eplain the statement. How would ou describe the graph in step 7b)? 1. Consider the graph of the function = f() in step. a) If the -values were multiplied b or multiplied b _ 1, describe what would happen to the graph of the function = f(). b) Determine the equations of the transformed functions in part a) in the form = f(b). 1. Reflections and Stretches MHR 17

22 Link the Ideas reflection a transformation where each point of the original graph has an image point resulting from a reflection in a line ma result in a change of orientation of a graph while preserving its shape Reflect and Respond 11. Cop and complete the table to record our observations. Write concluding statements summarizing the effects of stretches about the aes. Function = f() Stretch About Verbal Description Mapping -ais (, ) (, ) -ais (, ) (, ) Equation of Transformed Function A reflection of a graph creates a mirror image in a line called the line of reflection. Reflections, like translations, do not change the shape of the graph. However, unlike translations, reflections ma change the orientation of the graph. When the output of a function = f() ) is multiplied b -1, the result, = -f(), is a reflection of the graph in the -ais. When the input of a function = f() ) is multiplied b -1, the result, = f(-), is a reflection of the graph in the -ais. DRAFT Eample 1 Compare the Graphs of = f(), = -f(), and = f(-) a) Given the graph of = f(), graph the functions = -f() and = f(-). b) How are the graphs of = -f() and = f(-) related to the graph of = f()? DRAFT-1, DRAFT1, the D - - A - = f() B - C E 18 MHR Chapter 1

23 Solution a) Use ke points on the graph of = f() to create tables of values. The image points on the graph of = -f() have the same -coordinates but different -coordinates. Multipl the -coordinates of points on the graph of = f() b -1. = f() = -f() A - -3 A - -1(-3) = 3 B - -3 B - -1(-3) = 3 C 1 C 1-1() = D 3 D 3-1() = - E 5 - E 5-1(-) = = -f() A - B - - A = f() B - C C D D E E The image points on the graph of = f(-) ) have the same -coordinates but different -coordinates. Multipl the -coordinates of points on the graph of = f() ) b -1. = f() ) = f(-) A A -1(-) = -3 B B -1(-) = -3 C 1 C -1(1) = -1 D 3 D -1(3) = -3 E 5 - E -1(5) = -5 - = f(-) D D Each image point is the same distance from the line of reflection as the corresponding ke point. A line drawn perpendicular to the line of reflection contains both the ke point and its image point. The negative sign can be interpreted as a change in sign of one of the coordinates. BDRAFT ADRAFT -DRAFT -DRAFT -DRAFT 5DRAFT DRAFT A B D 3 E - - C C E - A B = f() - B A E 1. Reflections and Stretches MHR 19

24 invariant point a point on a graph that remains unchanged after a transformation is applied to it an point on a curve that lies on the line of reflection is an invariant point stretch a transformation in which the distance of each -coordinate or -coordinate from the line of reflection is multiplied b some scale factor scale factors between and 1 result in the point moving closer to the line of reflection; scale factors greater than 1 result in the point moving farther awa from the line of reflection b) The transformed graphs are congruent to the graph of = f(). The points on the graph of = f() relate to the points on the graph of = -f() b the mapping (, ) (, -). The graph of = -f() is a reflection of the graph of = f() in the -ais. Notice that the point C(1, ) maps to itself, C (1, ). This point is an invariant point. The points on the graph of = f() relate to the points on the graph of = f(-) b the mapping (, ) (-, ). The graph of = f(-) is a reflection of the graph of = f() in the -ais. The point (, -1) is an invariant point. Your Turn a) Given the graph of = f(), graph the functions = -f() ) and = f(-). b) Show the mapping of ke points on the graph of = f() to image points on the graphs of = -f() ) and = f(-). c) Describe how the graphs of = -f() ) and = f(-) ) are related to the graph of = f(). State an invariant points = f() ( ) Vertical and Horizontal Stretches What is another invariant point? DRAFT DRAFTf( DRAFT( DRAFT) DRAFT) to A stretch, unlike a translation or a reflection, changes the shape of the graph. However, like translations, stretches do not change the orientation of the graph. When the output of a function = f() is multiplied b a non-zero constant a, the result, = af() or _ a = f(), is a vertical stretch of the graph about the -ais b a factor of a. If a <, then the graph is also reflected in the -ais. When the input of a function = f() is multiplied b a non-zero constant b, the result, = f(b), is a horizontal stretch of the graph about the -ais b a factor of _ 1. If b <, then the graph is also b reflected in the -ais. MHR Chapter 1

25 Eample Graph = af() Given the graph of = f(), transform the graph of f() to sketch the graph of g() describe the transformation state an invariant points state the domain and range of the functions a) g() = f() b) g() = _ 1 f() Solution = f() a) Use ke points on the graph of = f() ) to create a table of values. The image points on the graph of g() = f() have the same -coordinates but different -coordinates. Multipl the -coordinates of points on the graph of = f() b. = f() = g() = f() The vertical distances of the transformed graph have been changed b a factor of a,, where a > 1. The points on the graph of = af() ) are farther awa from the -ais than the corresponding points of the graph of = f(). Since a =, the points on the graph of = g() relate to the points on the graph of = f() b the mapping (, ) (, ). Therefore, each point on the graph of g() is twice as far from the -ais as the corresponding point on the graph of f(). The graph of g() = f() is a vertical stretch of the graph of = f() about the -ais b a factor of. What is unique about The invariant points are (-, ) and (, ). the invariant points? For f(), the domain is { -, R}, or [-, ], and the range is {, R}, or [, ]. = f() ( ) For g(), the domain is { -, R}, or [-, ], and the range is { 8, R}, or [, 8]. - = g() ) SinceDRAFT f(draft (DRAFT )ḊRAFT DRAFT =DRAFT fdraft adraft =DRAFT of = Since a How can ou determine the range of the new function, g(), using the range of f() and the parameter a? Did You Know? There are several was to epress the domain and range of a function. For eample, ou can use words, a number line, set notation, or interval notation. 1. Reflections and Stretches MHR 1

26 b) The image points on the graph of g() = _ 1 f() have the same -coordinates but different -coordinates. Multipl the -coordinates of points on the graph of = f() b _ 1. = f() = g() = 1 _ f() Did You Know? = f() The vertical distances of the transformed Translations and reflections are called graph have been changed b a factor rigid transformations a,, where < a < 1. The points on the because the shape of = g() graph of = af() ) are closer to the -ais the graph does not than the corresponding points of the change. Stretches graph of = f(). are called non-rigid because the shape of Since a = _ 1 the graph can change., the points on the graph of = g() ) relate to the points on the graph of = f() ) b the mapping (, ) ) (,(, _ 1 1_ ). Therefore, each point on the graph of g() ) one half as far from the as the corresponding point on the graph of f(). The graph of g() = _ 1 f() a vertical stretch of the graph of = f() about the b a factor of _ 1 1_. The invariant points are (-, ) and (, ). What conclusion can ou make about the invariant For f(), the domain points after a vertical stretch? { -, R}, or [-, ], and the range {, R}, or [, ]. For g(), the domain { -, R}, or [-, ], DRAFT-ais and the range is {, R}, or [, ]. DRAFT Your Turn Given the function f() =, transform the graph of f() to sketch the graph of g() describe the transformation state an invariant points state the domain and range of the functions a) g() = f() b) g() = _ 1 3 f() MHR Chapter 1

27 Eample 3 Graph = f(b) Given the graph of = f(), transform the graph of f() to sketch the graph of g() describe the transformation state an invariant points state the domain and range of the functions a) g() = f() b) g() = f _ ( 1 ) Solution - - = f() a) Use ke points on the graph of = f() ) to create a table of values. The image points on the graph of g() = f() ) have the same -coordinates but different -coordinates. Multipl the -coordinates of points on the graph of = f() b _ 1. = f() = g() ) = f() ) SinceDRAFT bdraft -DRAFT -DRAFT -DRAFT DRAFT - = g() ) - = f() ( ) Since b =, the points on the graph of = g() relate to the points on the graph of = f() b the mapping (, ) _ ( 1, ). Therefore, each point on the graph of g() is one half as far from the -ais as the corresponding point on the graph of f(). The graph of g() = f() is a horizontal stretch about the -ais b a factor of _ 1 of the graph of f(). The invariant point is (, ). For f(), the domain is { -, R}, or [-, ], and the range is {, R}, or [, ]. For g(), the domain is { -, R}, or [-, ], and the range is {, R}, or [, ]. The horizontal distances of the transformed graph have been changed b _ a factor of 1, where b > 1. The points b on the graph of = f(b) are closer to the -ais than the corresponding points of the graph of = f(). Since bdraft How can ou determine the domain of the new function, g(), using the domain of f() and the parameter b? 1. Reflections and Stretches MHR 3

28 b) The image points on the graph of g() = f ( 1 _ ) have the same -coordinates but different -coordinates. Multipl the -coordinates of points on the graph of = f() b. = f() = g() = f ( 1 _ ) The horizontal distances of the = g() = f() transformed graph have been _ changed b a factor 1 1_, b where < b < 1. The points on the graph of = f(b) ) are farther awa from the than the corresponding points of the = f(). Since b = _ 1, the points on the graph of = g() ) relate to the points on the graph of = f() ) b the mapping (, ) ) (, ). Therefore, each point on the graph of g() ) twice as far from the as the corresponding point on the graph of f(). The graph of g() = f _ ( 1 ) a horizontal stretch about the b a factor of of the graph of f(). The invariant point (, ). How do ou know which points will be invariant points after a horizontal stretch? For f(), the domain { -, R}, or [-, ], and the range {, R}, or [, ]. For g(), the domain { -8 8, R}, or [-8, 8], and the range {, R}, or [, ]. DRAFT-ais DRAFT-ais DRAFT= DRAFTf DRAFT( graph of Your Turn Given the function f() =, transform the graph of f() to sketch the graph of g() describe the transformation state an invariant points state the domain and range of the functions a) g() = f(3) b) g() = f _ ( 1 ) DRAFT, DRAFTf( DRAFT(b DRAFTb) are DRAFT DRAFT( DRAFT). DRAFT). = MHR Chapter 1

29 Eample Write the Equation of a Transformed Function The graph of the function = f() has been transformed b either a stretch or a reflection. Write the equation of the transformed graph, g(). a) -8 Solution g() f() = 8 b) f() = a) Notice that the V-shape has changed, so the graph has been transformed b a stretch. g() Since the original function is f() ) =,,, a stretch can be described in two was. Choose ke points on the graph of = f() ) and determine their image points on the graph of the transformed function, g(). Case 1 Check for a pattern in the -coordinates. = f() ) = g() ) A vertical stretch results when the vertical distances of the transformed graph are a constant multiple of those of the original graph with respect to the -ais. The transformation can be described b the mapping (, ) (, 3). This is of the form = af(), indicating that there is a vertical stretch about the -ais b a factor of 3. The equation of the transformed function is g() = 3f() or g() = g() Wh is this the case? DRAFT -DRAFT f() = Reflections and Stretches MHR 5

30 Case Check for a pattern in the -coordinates. = f() = g() g() 1 A horizontal stretch results when the horizontal distances of the 1 transformed graph are a constant multiple of those of the original 8 f() = graph with respect to the The transformation can be described b the mapping (, ) _ ( 1 3, ). This of the form = f(b), indicating that there a horizontal stretch about the b a factor of _ 1_. The equation of the 3 transformed function g() ) = f(3) or g() = 3. b) Notice that the shape of the graph has not changed, so the graph has been transformed b a reflection. Choose ke points on the graph of f() = and determine their image points on the graph of the transformed function, g(). = f() = g() - - DRAFT-ais. DRAFT-ais. - - DRAFT - - The transformation can be described b the mapping (, ) (, -). This is of the form = -f(), indicating a reflection in the -ais. The equation of the transformed function is g() = -. MHR Chapter 1

31 Your Turn The graph of the function = f() has been transformed. Write the equation of the transformed graph, g() g() f() = - - Ke Ideas An point on a line of reflection is an invariant point. Transformation from Function = f() Mapping Eample = -f() A reflection in the -ais (, ) ) (, -) = f() ( ) = -f() = f(-) ) A reflection in the -ais (, ) ) (-, ) ) = f() = f(-) = af() ) A vertical stretch about the (, ) (, a) -ais b a factor of a ; if a <, then the graph is = af(), a > 1 DRAFTample also reflected in the -ais = f() DRAFT =DRAFT af(draft (DRAFT = afdraft = f(b) A horizontal stretch about _ the -ais b a factor of 1 b ; if b <, then the graph is also reflected in the -ais (, ) ( _ b, ) = f() = f(b), b > 1. Reflections and Stretches MHR 7

32 Check Your Understanding Practise 1. a) Cop and complete the table of values for the given functions. f() = + 1 g() = -f() h() = f(-) - - b) Sketch the graphs of f(), g(), and h() on the same set of coordinate aes. c) Eplain how the points on the graphs of g() and h() relate to the transformation of the function f() = + 1. List an invariant points. d) How is each function related to the graph of f() = + 1?. a) Cop and complete the table of values for the given functions. f() = _ g() ) = 3f() h() ) = 1 1_ 3 f() ) b) Sketch the graphs of f(), g(), and h() on the same set of coordinate aes. c) Eplain how the points on the graphs of g() and h() relate to the transformation of the function f() =. List an invariant points. d) How is each function related to the graph of f() =? 3. Consider each graph of a function. Cop the graph of the function and sketch its reflection in the -ais on the same set of aes. State the equation of the reflected function in simplified form. State the domain and range of each function. a) f() = b) g() ( ) = c) DRAFT h() = 1 _. Consider each function in #3. Cop the graph of the function and sketch its reflection in the -ais on the same set of aes. State the equation of the reflected function. State the domain and range for each function. 8 MHR Chapter 1

33 5. Use words and mapping notation to describe how the graph of each function can be found from the graph of the function = f(). a) = f() b) = f(3) c) = -f() d) = f(-). The graph of the function = f() is verticall stretched about the -ais b a factor of. = f() d) - - DRAFT( DRAFT DRAFT) g() ( ) f() ( ) - - a) Determine the domain and range of - the transformed function. b) Eplain the effect that a vertical - stretch has on the domain and range of a function. 7. Describe the transformation that Appl must be applied to the graph of f() ) 8. A weaver sets up a pattern on a computer to obtain the graph of g(). Then, using the graph shown. A new line of determine the equation of g() in the merchandise calls for the design to be form = af(b). altered to = f(.5). Sketch the graph of a) the new design. 8 g() f() b) f() g() c) g() f() formdraft DRAFT af(draft (bdraft DRAFT =DRAFT afdraft DRAFT DRAFT( DRAFT) form = Reflections and Stretches MHR 9

34 9. Describe what happens to the graph of a function = f() after the following changes are made to its equation. a) Replace with. b) Replace with 1 _. c) Replace with. d) Replace with 1 _. e) Replace with -3. in miles per hour, of the vehicle before f) Replace with - _ 1 3. braking b the function D = 1_ ddraft DRAFT 3fn S, where 1. Thomas and Sharn discuss the order f is the drag factor of the road surface and of the transformations of the graph of n is the braking efficienc as a decimal. = -3 compared to the graph of =. Suppose the braking efficienc is 1% Thomas states that the reflection must or 1. be applied first. Sharn claims that the a) Sketch the graph of the length of the vertical stretch should be applied first. skid mark as a function of speed for a a) Sketch the DRAFT1_ DRAFT_ graph of = -3 b DRAFTS DRAFT. drag factor of 1, or D = _ 1 appling the reflection first. S. 3 b) The drag factor for asphalt is.9, for b) Sketch the graph of = -3 b gravel is.8, for snow is.55, and for appling the stretch first. ice is.5. Compare the graphs of the c) Eplain our conclusions. Who is functions for these drag factors to the correct? graph in part a). 11. An object falling in a vacuum is affected onl b the gravitational force. An Did You Know? equation that can model a free-falling A technical accident investigator or reconstructionist object on Earth is d = -.9t t, where d is is a speciall trained police officer who investigates serious traffic accidents. These officers use the distance travelled, in metres, and t is photograph, measurements of skid patterns, and the time, in seconds. An object free falling other information to determine the cause of the on the moon can be modelled b the collision and if an charges should be laid. equation d = -1.t. a) Sketch the graph of each function. b) Compare each function equation to the base function d = t. Did You Know? The actual strength of Earth s gravit varies depending on location. On March 17, 9, the European Space Agenc launched a gravit-mapping satellite called Gravit and Ocean Circulation Eplorer (GOCE). The data transmitted from GOCE are being used to build a model of Earth s shape and a gravit map of the plant. 1. Eplain the differences that occur in transforming the graph of the function = f() to the graph of the function = f(b) as compared to transforming = f() to = af(). 13. The speed of a vehicle the moment the brakes are applied can be determined b its skid marks. The length, D, in feet, of the skid mark is related to the speed, S, -1.DRAFT 1.tDRAFT =DRAFT -DRAFT DRAFT.DRAFT d = t DRAFT= DRAFTS DRAFT DRAFT3. 3 MHR Chapter 1