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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 www.mcgrawhill.ca/school-----------
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
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
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 (1879 1955) 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
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 www.mcgrawhill.ca/school/learningcentres and follow the links. Chapter 1 MHR 5 1PC1_Chapter1_7th_E.indd 5 1/5/11 3:39:51 PM
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
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
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) -3 3 9-3 3 11 9 - - 3-1 1 1-1 1 3 1-5 1 1 1 3 1 7 3 9 3 11 8 9 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
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 = - + 3. 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 = - + 3. 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 = = = - + 3 = - 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) -. 1.1 Horizontal and Vertical Translations MHR 9
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
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
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
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( + ) - 3 9. 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
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. - 1. 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 8 1 1 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
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 DRAFT5 + - - 5 3 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() = + + 1 a) a vertical translation of units up b) g() = - + 3 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 - - 1 = 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
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
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
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
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 - -3 3 A -1(-) = -3 B - -33 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
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
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() - 8-8 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 - - 8 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
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() - - 1 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
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() ) - - - -1 1 1 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
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 _ ) - -8 - - 8 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 -8 - - - 8 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
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 - 1 1 8 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() ) - 18-1 - 1 18 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() = 3. 1 1 8 g() Wh is this the case? DRAFT -DRAFT f() = -1-8 - 8 1 1. Reflections and Stretches MHR 5
Case Check for a pattern in the -coordinates. = f() = g() -1 1-1 - - 1 1 1 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 -1-8 - 8 1 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
Your Turn The graph of the function = f() has been transformed. Write the equation of the transformed graph, g(). 1 1 8 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
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() ) - -3 3 3 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() = 3 - - b) g() ( ) = + 1 - 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
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 = - - - - - - 1. Reflections and Stretches MHR 9
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
Etend 1. Consider the function f() = ( + )( - 3). Without graphing, determine the zeros of the function after each transformation. a) = f() b) = f(-) c) = f _ ( 1 ) d) = f() 15. The graph of a function = f() is contained completel in the fourth quadrant. Cop and complete each statement. a) If = f() is transformed to = -f(), it will be in quadrant. b) If = f() is transformed to = f(-), it will be in quadrant. c) If = f() is transformed to = f(), it will be in quadrant. d) If = f() ) is transformed to = f _ f( ( 1 1_ ) ),, it will be in quadrant.. 1. Sketch the graph of f() ) = reflected in each line. a) = 3 b) = - C3 A point on the function f() is mapped onto the image point on the function g(). Cop and complete the table b describing a possible transformation of f() to obtain g() for each mapping. f() g() Transformation (5, ) (5, -) (, 8) (-, 8) (, 3) (, 1) 1DRAFT (, -1) (, -) C Sound is a form of energ produced and transmitted b vibrating matter that travels in waves. Pitch is the measure of how high or how low a sound is. The graph of f() demonstrates a normal pitch. Cop the graph, then sketch the graphs of = f(3), indicating a higher pitch, and = f _ ( 1 ), for a lower pitch. Normal Pitch 1 = f() 8 1 1-1 Did You Know? Create Connections The pitch of a sound wave is directl related to C1 Eplain wh the graph of g() ) = f(b) is its frequenc. A high-pitched sound has a high a horizontal stretch about the -ais b a frequenc (a mosquito). A low-pitched sound has a 1 low frequenc (a fog-horn). factor of _, for b >, rather than a factor b A health human ear can hear frequencies in the of b. range of Hz to Hz. C Describe a transformation that results in each situation. Is there more than one possibilit? a) The -intercepts are invariant points. b) The -intercepts are invariant points. C5 a) Write the equation for the general term of the sequence -1, -, -,,,. b) Write the equation for the general term of the sequence 1,,, -, -,. c) How are the graphs of the two sequences related? bdraft 1_DRAFT ofdraft = DRAFT DRAFT= DRAFTf = f ( factor 1. Reflections and Stretches MHR 31
1.3 Combining Transformationss Focus on... sketching the graph of a transformed function b appling translations, reflections, and stretches writing the equation of a function that has been transformed from the function = f() Architects, artists, and craftspeople use transformations in their work. Towers that stretch the limits of architectural technologies, paintings that create futuristic landscapes from ordinar objects, and quilt designs that transform a single shape to create a more comple image are eamples of these transformations. In this section, ou will appl a combination of transformations to base functions to create more comple functions. Investigate the Order of Transformations Materials grid paper National-Nederlanden Building in Prague, Czech Republic DRAFT New graphs can be created b vertical or horizontal translations, vertical or horizontal stretches, or reflections in an ais. When vertical and horizontal translations are applied to the graph of a function, the order in which the occur does not affect the position of the final image. Eplore whether order matters when other combinations of transformations are applied. Consider the graph of = f(). = f() - - A: Stretches 1. a) Cop the graph of = f(). b) Sketch the transformed graph after the following two stretches are performed in order. Write the resulting function equation after each transformation. Stretch verticall about the -ais b a factor of. Stretch horizontall about the -ais b a factor of 3. 3 MHR Chapter 1
c) Sketch the transformed graph after the same two stretches are performed in reverse order. Write the resulting function equation after each transformation. Stretch horizontall about the -ais b a factor of 3. Stretch verticall about the -ais b a factor of.. Compare the final graphs and equations from step 1b) and c). Did reversing the order of the stretches change the final result? B: Combining Reflections and Translations 3. a) Cop the graph of = f(). b) Sketch the transformed graph after the following two transformations are performed in order. Write the resulting function equation after each transformation. Reflect in the -ais. Translate verticall units up. c) Sketch the transformed graph after the same two transformations performed in reverse order. Write the resulting function equation after each transformation. Translate verticall units up. Reflect in the -ais.. Compare the final graphs and equations from step 3b) and c). Did reversing the order of the transformations change the final result? Eplain. 5. a) Cop the graph of = f(). b) Sketch the transformed graph after the following two transformations are performed in order. Write the resulting function equation after each transformation. Reflect in the -ais. Translate horizontall units to the right. c) Sketch the transformed graph after the same two transformations performed in reverse order. Write the resulting function equation after each transformation. Translate horizontall units to the right. Reflect in the -ais.. Compare the final graphs and equations from step 5b) and c). Did reversing the order of the transformations change the final result? Eplain. DRAFT Reflect and Respond 7. a) What do ou think would happen if the graph of a function were transformed b a vertical stretch about the -ais and a vertical translation? Would the order of the transformations matter? b) Use the graph of = to test our prediction. 8. In which order do ou think transformations should be performed to produce the correct graph? Eplain. 1.3 Combining Transformations MHR 33
Link the Ideas Multiple transformations can be applied to a function using the general transformation model - k = af(b( - h)) or = af(b( - h)) + k. To accuratel sketch the graph of a function of the form - k = af(b( - h)), the stretches and reflections (values of a and b) should occur before the translations (h-value and k-value). The diagram shows one recommended sequence for the order of transformations. = f () Horizontal stretch about the -ais b a factor of 1 b Vertical stretch about the -ais b a factor of a Horizontal translation of h units and/or vertical translation of k units DRAFT( - k = af(b( - h)) Reflection in Reflection in the -ais the -ais if b < if a < How does this compare to the usual order of operations? Eample 1 Graph a Transformed Function Describe the combination of transformations (9, 3) 3 that must be applied to the function = f() = f() to obtain the transformed function. Sketch (, ) the graph, showing each step of the 1 transformation. (1, 1) (, ) a) = 3f() 8 b) = f(3 + ) Solution a) Compare the function to = af(b( - h)) + k. For = 3f(), a = 3, b =, h =, and k =. The graph of = f() is horizontall stretched about the -ais b a factor of _ 1 and then verticall stretched about the -ais b a factor of 3. Appl the horizontal stretch b a = f() (.5, 3) (9, 3) factor of _ 1 to obtain the graph 3 (, ) = f() of = f(). (.5, 1) (, ) DRAFT DRAFTaf( DRAFTb( DRAFT( DRAFT DRAFT- DRAFTh)) DRAFT)) - h 1 (, ) (1, 1) 8 3 MHR Chapter 1
Appl the vertical stretch b a factor of 3 to = f() to obtain the graph of = 3f(). Would performing the stretches in reverse order change the final result? 9 8 7 5 (, ) (.5, 9) = 3f() 3 1 (.5, 3) (.5, 1) (, ) (.5, 3) (, ) = f() 8 b) First, rewrite = f(3 + ) in the form = af(b( ( - h)) + k.. This makes it easier to identif specific transformations. = f(3 + ) = f(3( + )) Factor out the coefficient of.. For = f(3( + )), a = 1, b = 3, h = -, and k =. The graph of = f() ) is horizontall stretched about the -ais b a factor of _ 1 and then horizontall translated units to the left. 3 Appl the horizontal stretch = f(3) (3, 3) (9, 3) b a factor of _ 1 1_ to obtain the 3 3 (, 3 ) graph of = f(3). = f() 1 (, ) (, 3 1) 1 (1, 1) (, ) DRAFT Appl the horizontal translation of units to the left to = f(3) to obtain the graph of = f(3( + )). Your Turn Describe the combination of transformations that must be applied to the function = f() to obtain the transformed function. Sketch the graph, showing each step of the transformation. a) = f() - 3 b) = f ( 1 _ - ) = f(3( + )) (-, ) 8 ( -, ) 3 ( - 5, ) 3 (1, 3) (, ) 1 = f(3) 3 1 1 3 (, ) 3 (, 1) 3 1 (, ) (1, 1) (, ) (3, 3) = f() 8 (9, 3) 1.3 Combining Transformations MHR 35
Eample Combination of Transformations Show the combination of transformations that should be applied to the graph of the function f() = in order to obtain the graph of the transformed function g() = - _ 1 f(( - )) + 1. Write the corresponding equation for g(). Solution For g() = - _ 1 f(( - )) + 1, a = - _ 1, b =, h =, and k = 1. Description Mapping Graph Horizontal stretch (-, ) (-1, ) about the -ais b a (, ) (, ) _ factor of 1 (, ) (1, ) = () _ (, ) ( 1, = ) ) DRAFT = () DRAFT- DRAFT - Vertical stretch about (-1, ) (-1, ) the -ais b a factor (, ) (, ) = 1_ () _ of 1 (1, ) (1, ) = 1 () = () ( ) ) ( 1 1_,, _ ) ( 1 1,, 1 1_ ) ) - Reflection in the -ais (-1, ) (-1, -) = - _ 1 1_ (, ) (, ) ( () ) ) (1, ) (1, -) = 1_ () _ ( 1, _ 1 _ ) ( 1, - _ 1 ) - - = - 1 _() - ( DRAFT DRAFT = DRAFT DRAFT= DRAFT) DRAFT = ( DRAFT DRAFT- Translation of units to the right and 1 unit up = - _ 1 (( - )) + 1 (-1, -) (3, -1) (, ) (, 1) (1, -) (5, -1) ( 1 _, - 1 _ ) ( 1 _ +, - 1 _ = - 1_ (( - )) + 1 = - 1 _ () ) + 1 - - - The equation of the transformed function is g() = - 1 _ (( - )) + 1. 3 MHR Chapter 1
Your Turn Describe the combination of transformations that should be applied to the function f() = in order to obtain the transformed function g() = -f ( 1 _ ( + 8) ) - 3. Write the corresponding equation and sketch the graph of g(). Eample 3 Write the Equation of a Transformed Function Graph The graph of the function = g() represents a transformation of the graph of = f(). Determine the equation of g() in the form = af(b( - h)) + k. Eplain our answer. Solution -88 = g() - = f() ( ) - - 1 8 Locate ke points on the graph of f() ) and their image points on the graph of g(). ( -, ) (-8, 1) (, ) (-7, - ) (, ) (-, 1) The point (, ) on the graph of f() ) is not affected b an stretch, either horizontal or vertical, or an reflection so it can be used to determine the vertical and horizontal translations. The graph of g() has been translated 7 units to the left and units up. h = -7 and k = There is no reflection. units 1 Compare the distances between ke points. In the vertical 8 direction, units becomes 8 units. = g() 8 units There is a vertical stretch b a 8 units factor of. In the horizontal direction, 8 units becomes units. There is also a horizontal stretch = f() units b a factor of _ 1. -1-8 - - - a = and b = DRAFT =DRAFT -DRAFT anddraft kdraft =DRAFT = -77 and k Substitute the values of a, b, h, and k into = af(b( - h)) + k. The equation of the transformed function is g() = f(( + 7)) +. How could ou use the mapping (, ) ( 1 _ b + h, a + k ) to verif this equation? 1.3 Combining Transformations MHR 37
Your Turn The graph of the function = g() represents a transformation of the graph of = f(). State the equation of the transformed function. Eplain our answer. - = f() - Ke Ideas Write the function in the form = af(b( - h)) + k to better identif the transformations. Stretches and reflections ma be performed in an order before translations. The parameters a, b, h, and k in the function = af(b( ( - h)) + k correspond to the following transformations: a corresponds to a vertical stretch about the -ais b a factor of a. If a <, then the function is reflected in the -ais. b corresponds to a horizontal stretch about the -ais b a factor of _ 1 b. If b <, then the function is reflected in the -ais. h corresponds to a horizontal translation. k k corresponds to a vertical translation. Check Your Understanding Practise 1. The function = has been transformed to = af(b). Determine the equation of each transformed function. a) Its graph is stretched horizontall about the -ais b a factor of and then reflected in the -ais. b) Its graph is stretched horizontall about the -ais b a factor of _ 1, reflected in the -ais, and then stretched verticall about the -ais b a factor of _ 1. - - -8 = g() DRAFT. The function = f() is transformed to the function g() = -3f( - 1) - 1. Cop and complete the following statements b filling in the blanks. The function f() is transformed to the function g() b a horizontal stretch about the b a factor of. It is verticall stretched about the b a factor of. It is reflected in the, and then translated units to the right and units down. 38 MHR Chapter 1
3. Cop and complete the table b describing the transformations of the given functions, compared to the function = f(). Reflections Vertical Stretch Factor Horizontal Stretch Factor Vertical Translation Horizontal Translation vertical stretch about the -ais b Function a factor of - = f( - 5) horizontal stretch about the -ais + 5 = f(3) b a factor of _ 1 1_ = _ 1 f _ ( 1 3 ( - ) ) translation of 5 units to the left and 3 units up + = -3f(( + )) b). Using DRAFT( DRAFT DRAFT) the graph of = f(), write the = f() equation of each transformed graph in the form = af(b( - h)) + k. - = f() vertical stretch about the -ais b a factor of _ 3 - - - horizontal stretch about the -ais - b a factor of 3 translation of 3 units to the right a) and units down = k() ). The ke point (-1, 18) is on the graph of = f(). What is its image point under - - each transformation of the graph of f()? - a) + = f( - ) - b) = f(3) c) = -f( - ) + b) - - = m() - - - 5. For each graph of = f(), sketch the graph of the combined transformations. Show each transformation in the sequence. a) - - - d) = -f _ (- 3 - ) + e) + 3 = - _ 1 f(( + )) 3 = f() -DRAFT -DRAFT -DRAFT DRAFTf( DRAFT) 1.3 Combining Transformations MHR 39
Appl 7. Describe, using an appropriate order, how to obtain the graph of each function from the graph of = f(). Then, give the mapping for the transformation. a) = f( - 3) + b) = -f(3) - c) = - _ 1 f(-( + )) = f() d) - 3 = -f(( - )) e) = -_ 3 f _ (- 3 ) f) 3 - = f(- + 1) - - 8 1 1 8. Given the function = f(), write the - equation of the form - k = af(b( - h)) that would result from each combination b) of transformations. = f() ( ) a) a vertical stretch about the -ais b a factor of 3, a reflection in the -ais, a horizontal translation of units to 8 the left, DRAFT - - - and a vertical translation of 5 units down - b) a horizontal stretch about the -ais b - a factor of _ 1, a vertical stretch about = g() 3 the -ais - b a factor of _ 3 3_, a reflection -8 in both the -ais and the -ais, and a translation of units to the right and c) units up 1 9. The graph of = f() ) is given. Sketch the 8 graph of each of the following functions. = g() = f() ) = f() - - - -8 - - - 8 - a) + = f( - 3) b) = -f(-) c) = f(3( - )) + 1 d) = 3f _ ( 1 3 ) e) + = -3f( + ) f) = 1 _ f (- 1 _ ( + ) ) - 1 1. The graph of the function = g() represents a transformation of the graph of = f(). Determine the equation of g() in the form = af(b( - h)) + k. a) 1 8 - - = g() )DRAFT DRAFT DRAFT f(draft (DRAFT =DRAFT fdraft = 11. Given the function f(), sketch the graph of the transformed function g(). a) f() =, g() = -f(( + )) - b) f() =, g() = -f(-3 + ) + c) f() =, g() = - _ 1 f(-( + 3)) - 3 MHR Chapter 1
1. Alison often sketches her quilt designs on a coordinate grid. The coordinates for a section of one her designs are A(-, ), B(-, -), C(, ), D(1, -1), and E(3, ). She wants to transform the original design b a horizontal stretch about the -ais b a factor of, a reflection in the -ais, and a translation of units up and 3 units to the left. a) Determine the coordinates of the image points, A, B, C, D, and E. b) If the original design was defined b the function = f(), determine the equation of the design resulting from the transformations. 13. Gil is asked to translate the graph of = according to the equation = - +. He decides to do the horizontal translation of 3 units to the right first, then the stretch about the -ais b a factor of _ 1, and lastl the translation of units up. This gives him Graph 1. To check his work, he decides to appl the horizontal stretch about the -ais b a factor of _ 1 1_ first, and then the horizontal translation of units to the right and the vertical translation of units up. This results in Graph. a) Eplain wh the two graphs are in different locations. b) How could Gil have rewritten the equation so that the order in which he did the transformations for Graph resulted in the same position as Graph 1? = - - 8 Graph 1 8 Graph 1. Two parabolic arches are being built. The first arch can be modelled b the function = - + 9, with a range of 9. The second arch must span twice the distance and be translated units to the left and 3 units down. a) Sketch the graph of both arches. b) Determine the equation of the second arch. Etend 15. If the -intercept of the graph of = f() is located at (a, ) and the -intercept is located at (, b), determine the -intercept and -intercept after the following transformations of the graph of = f(). a) = -f(-) - ) b) = f _ f( ( 1 1_ ) ) c) + 3 = f( - ) d) + 3 = _ 1_ f _ - ) f( ( 1 1_ ( ) 1. A rectangle is inscribed between the -ais and the parabola = 9 - with one side along the -ais, as shown. DRAFT b)draft = 9 - - - 8 (, ) (, ) DRAFT = a) Write the equation for the area of the rectangle as a function of. b) Suppose a horizontal stretch b a factor of is applied to the parabola. What is the equation for the area of the transformed rectangle? c) Suppose the point (, 5) is the verte of the rectangle on the original parabola. Use this point to verif our equations from parts a) and b). 1.3 Combining Transformations MHR 1
17. The graph of the function = + + 1 is stretched verticall about the -ais b a factor of, stretched horizontall about the -ais b a factor of _ 1, and translated 3 units to the right and units down. Write the equation of the transformed function. 18. This section deals with transformations in a specific order. Give one or more eamples of transformations in which the order does not matter. Show how ou know that order does not matter. Describe the transformations necessar to create the image. Write the equations necessar to transform the original function. Step The graph shows the function f() = and transformations 1,, 3, and. 3ḌRAFT Create Connections C1 MINI LAB Man designs, Materials Recreate the diagram on a graphing such as this Moroccan carpet, grid paper calculator. Use the window settings are based on transformations. graphing : : [-3, 3, 1] : : [-3, 3, 1]. calculator Describe the transformations necessar to create the image. Write the equations necessar to transform the original function. C Kokitusi`aki (Diana Passmore) and Siksmissi (Kath Anderson) make and sell beaded bracelets such as the one shown representing the bear and the wolf. Work with a partner. Use transformations If the make b bracelets per week at of functions to create designs on a graphing a cost of f(b), what do the following calculator. epressions represent? How do the Step 1 The graph shows the function relate to transformations? f() = - + 3 and transformations a) f(b + 1) b) f(b) + 1 1,, and 3. c) 3f(b) d) f(b) calculator.draft -DRAFT -DRAFT +DRAFT Did You Know? Sisters Diana Passmore and Kath Anderson are descendants of the Little Dog Clan of the Piegan (Pikuni'l') Nation of the Blackfoot Confederac. Recreate the diagram on a graphing calculator. Use the window settings : [-3, 3, 1] : [-3, 3, 1]. C3 Epress the function = - 1 + 19 in the form = a( - h) + k. Use that form to describe how the graph of = can be transformed to the graph of = - 1 + 19. MHR Chapter 1
b) C Musical notes can be repeated (translated H horizontall), transposed (translated verticall), inverted (horizontal mirror), in retrograde (vertical mirror), or in retrograde inversion (18 rotation). If the musical pattern being transformed is the pattern in red, describe a possible transformation to arrive at the patterns H, J, and K. a) H J c) J K H T F A R J K Transformations Around You Project Corner D K What tpe(s) of function(s) do ou see in the image? Describe how each base function has been transformed. - - ( 5 - f() = - 5 3 ) + 5 17-1 + 9 h() = - 3 1 8 1-1 g() = 5 1 1-1.3 Combining Transformations MHR 3 1PC1_Chapter1_7th_E.indd 3 1/5/11 3::5 PM
1. Inverse of a Relation Focus on... sketching the graph of the inverse of a relation determining if a relation and its inverse are functions determining the equation of an inverse An inverse is often thought of as undoing or reversing a position, order, or effect. Whenever ou undo something that ou or someone else did, ou are using an inverse, whether it is unwrapping a gift that someone else wrapped or closing a door that has just been opened, or deciphering a secret code. For eample, when sending a secret message, a ke is used to encode the information. Then, the receiver uses the ke to decode the information. Let each letter in the alphabet be mapped to the numbers to 5. Plain Tet I N V E R S E Numeric Values, 8 13 1 17 18 Cipher, 11 19 15 1 Cipher Tet G L T C P Q C Decrpting is the inverse of encrpting. What decrption function would ou use on GLTCPQC? What other eamples of inverses can ou think of? Investigate the Inverse of a Function Materials grid paper DRAFT 1. Consider the function f() = _ 1-5. a) Cop the table. In the first column, enter the ordered pairs of five points on the graph of f(). To complete the second column of the table, interchange the -coordinates and -coordinates of the points in the first column. Ke Points on the Graph of f() Image Points on the Graph of g() MHR Chapter 1
b) Plot the points for the function f() and draw a line through them. c) Plot the points for the relation g() on the same set of aes and draw a line through them.. a) Draw the graph of = on the same set of aes as in step 1. b) How do the distances from the line = for ke points and corresponding image points compare? c) What tpe of transformation occurs in order for f() to become g()? 3. a) What observation can ou make about the relationship of the coordinates of our ordered pairs between the graphs of f() and g()? b) Determine the equation of g(). How is this equation related to f() = _ 1-5? c) The relation g() is considered to be the inverse of f(). Is the inverse of f() a function? Eplain. Reflect and Respond. Describe a wa to draw the graph of the inverse of a function of a using reflections. 5. Do ou think all inverses of functions are functions? What factors did ou base our decision on?. a) State a hpothesis for writing the equation of the of a linear function. b) Test our hpothesis. Write the equation of the of = 3 +. Check b graphing. 7. Determine the equation of the of = m + b, m. a) Make a conjecture about the relationship between the slope of the and the slope of the original function. b) Make a conjecture about the relationship between the -intercepts and the -intercept of the original and those of the function. 8. Describe DRAFTinverse DRAFTfunction how ou could determine if two relations are inverses of each other. if f is a function with domain A and range B, the inverse function, if it eists, is denoted b f -1 and has domain B and range A b)draft DRAFT -1 f maps to if and onl if f maps to 1. Inverse of a Relation MHR 5
Link the Ideas Did You Know? The 1 in f 1 () does not represent an eponent; that is f 1 1_ () f(). The inverse of a relation is found b interchanging the -coordinates and -coordinates of the ordered pairs of the relation. In other words, for ever ordered pair (, ) of a relation, there is an ordered pair (, ) on the inverse of the relation. This means that the graphs of a relation and its inverse are reflections of each other in the line =. (, ) (, ) The inverse of a function = f() ma be written in the form = f(). The inverse of a function is not necessaril a function. When the inverse of f is itself a function, it is denoted as f -1 and read as f inverse. When the inverse of a function is not a function, it ma be possible to restrict the domain to obtain an inverse function for a portion of the original function. The inverse of a function reverses the processes represented b that function. Functions f() and g() ) are inverses of each other if the operations of f() ) reverse all the operations of g() ) in the opposite order and the operations of g() ) reverse all the operations of f() in the opposite order. For eample, f() ) = + 1 multiplies the input value b and then adds 1. The inverse function subtracts 1 from the input value and then divides b. The inverse function is f -1() 1 = 1-1. Eample 1 Graph an Inverse DRAFT Consider the graph of the relation shown. a) Sketch the graph of the inverse relation. b) State the domain and range of the relation and its inverse. c) Determine whether the relation and its inverse are functions. DRAFTf( DRAFT( DRAFT) DRAFT) in Solution - - - a) To graph the inverse relation, interchange the -coordinates and -coordinates of ke points on the graph of the relation. Points on the Relation Points on the Inverse Relation (-, ) (, -) (-, ) (, -) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) MHR Chapter 1
- - = - - - - b) The domain of the relation becomes the range of the inverse relation and the range of the relation becomes the domain of the inverse relation. Domain Range Relation { -, R} {, R} Inverse Relation {, R} { -, R} c) The relation is a function of because there is onl one value of in the range for each value of in the domain. In other words, the graph of the relation passes the vertical line test. The inverse relation is not a function of because it fails the vertical line test. There is more than one value of in the range for at least one value of in the domain. You can confirm this b using the horizontal line test on the graph of the original relation. - Your Turn - - Consider the graph of the relation shown. a) Determine whether the relation and its inverse are functions. b) Sketch the graph of the inverse relation. - - - c) State the domain, range, and intercepts for the relation and the inverse relation. d) State an invariant points. The graphs are reflections of each other in the line =. The points on the graph of the relation are related to the points on the graph of the inverse relation b the mapping (, ) (, ). What points are invariant after a reflection in the line =? -DRAFT -DRAFT -DRAFT Did You Know? A one-to-one function is a function for which ever element in the range corresponds to eactl one element in the domain. The graph of a relation is a function if it passes the vertical line test. If, in addition, it passes the horizontal line test, it is a one-to-one function. horizontal line test a test used to determine if the graph of an inverse relation will be a function if it is possible for a horizontal line to intersect the graph of a relation more than once, then the inverse of the relation is not a function 1. Inverse of a Relation MHR 7
Eample Restrict the Domain Consider the function f() = -. a) Graph the function f(). Is the inverse of f() a function? b) Graph the inverse of f() on the same set of coordinate aes. c) Describe how the domain of f() could be restricted so that the inverse of f() is a function. Solution a) The graph of f() = - is a f() translation of the graph of = b units down. Since the graph of the function fails the horizontal line test, the inverse of f() - - is not a function. DRAFT- DRAFT - DRAFT b) Use ke points on the graph of f() DRAFT f() ) to help ou sketch the graph of the inverse of f(). Notice that the graph of the - - inverse of f() ) does not pass the vertical line test. The inverse of - f() ) is not a function. Inverse of f() - = c) The inverse of f() ) is a function if the graph of f() passes the horizontal line test. One possibilit is to restrict the Inverse of f() domain of f() so that the resulting - - graph is onl one half of the parabola. - Since the equation of the ais of f(), smmetr is =, restrict the domain - to {, R}. = DRAFT DRAFT DRAFT DRAFT- How else could the domain of f() be restricted? Your Turn Consider the function f() = ( + ). a) Graph the function f(). Is the inverse of f() a function? b) Graph the inverse of f() on the same set of coordinate aes. c) Describe how the domain of f() could be restricted so that the inverse of f() is a function. 8 MHR Chapter 1
Eample 3 Determine the Equation of the Inverse Algebraicall determine the equation of the inverse of each function. Verif graphicall that the relations are inverses of each other. a) f() = 3 + b) f() = - Solution a) Let = f(). To find the equation of the inverse, = f(), interchange and, and then solve for. f() = 3 + = 3 + Replace f() with. = 3 + Interchange and to determine the inverse. - = 3 Solve for. - = 3 f -1 () = - Replace with f -1 (), since the inverse of a linear 3 function is also a function. Graph = 3 + and = - on the same set of coordinate aes. 3 - = 3 + - - - - - = DRAFT = - 3 Notice that the -intercept and -intercept of = 3 + become the -intercept and -intercept, respectivel, of = -. Since the 3 functions are reflections of each other in the line =, the functions are inverses of each other. 1. Inverse of a Relation MHR 9
b) The same method applies to quadratic functions. f() = - = - Replace f() with. = - Interchange and to determine the inverse. + = Solve for. ± + = Wh is this not replaced with f = ± + -1 ()? What could be done so that f -1 () could be used? Graph = - and = ± + on the same set of coordinate aes. = - = ± + -3 5 5 ±3 - ± -1-3 -3 ±1 - - 1-3 3 5 = - - - = - - - - = ± + How could ou use the tables of values to verif that the relations are inverses of each other? DRAFT Notice that the -intercepts and -intercept of = - become the -intercepts and -intercept, respectivel, of = ± +. The relations are reflections of each other in the line =. While the relations are inverses of each other, = ± + is not a function. Your Turn Write the equation for the inverse of the function f() = + 8 3. Verif our answer graphicall. 5 MHR Chapter 1
Ke Ideas You can find the inverse of a relation b interchanging the -coordinates and -coordinates of the graph. The graph of the inverse of a relation is the graph of the relation reflected in the line =. The domain and range of a relation become the range and domain, respectivel, of the inverse of the relation. Use the horizontal line test to determine if an inverse will be a function. You can create an inverse that is a function over a specified interval b restricting the domain of a function. When the inverse of a function f() ) is itself a function, it is denoted b f -1 1 (). You can verif graphicall whether two functions are inverses of each other. Check Your Understanding Practise 1. Cop each graph. Use the reflection line = to sketch the graph of = f() ) on the same set of aes. a) - - = f() ( ) b) = f() ) - - -. Cop the graph of each relation and sketch the graph of its inverse relation. DRAFT =DRAFT f(draft fdraft DRAFT -DRAFT -DRAFT DRAFT a) b) - - - - - =DRAFT fdraft DRAFT )DRAFT = f( - - - 1. Inverse of a Relation MHR 51
3. State whether or not the graph of the relation is a function. Then, use the horizontal line test to determine whether the inverse relation will be a function. a) c) = ( - ) + - - b) - - - - c) - -. For each graph, identif a restricted domain for which the function has an inverse that is also a function. a) b) - - = ( + ) - = - 1 - d) = -( + ) + - - - - - 5. Algebraicall determine the equation of the inverse of each function. a) f() ) = 7 b) f() ) = -3 + c) f() ) = + f() 3 d) ) = - 5 3 e) f() ) = 5 - DRAFT DRAFT =DRAFT DRAFT f) f() = _ 1 ( + ). Match the function with its inverse. Function a) = + 5 b) = _ 1 - c) = - 3 d) = - 1, =DRAFT DRAFT DRAFT = e) = _ 1 ( + 1), -1 Inverse A = + 1 B = - 3 C = + 8 D = - - 1 E = - 5 5 MHR Chapter 1
Appl 7. For each table, plot the ordered pairs (, ) and the ordered pairs (, ). State the domain of the function and its inverse. a) c) = f() - - -1 1 9. For each of the following functions, 1 7 determine the equation for the 1 inverse, f -1 () b) graph f() and f -1 () determine the domain and range - of f() ) and f -1 1 () - a) f() ) = 3 + -1 5 b) f() ) = - 5 5 3 c) f() ) = _ 1 1_ - 8. Cop each graph of = f() and then d) f() ) = +, sketch DRAFT the graph of its inverse. Determine e) f() ) = -, if the inverse is a function. Give a reason 1. For each function f(), for our answer. i) determine the equation of the inverse of a) f() ) b first rewriting the function in the form = a( - h) + k ii) graph f() and the inverse of f() - a) f() = + 8 + 1 - = f() ( ) b) f() = - + - 11. Joceln and Gerr determine that the - inverse of the function f() = - 5,, is f -1 () = + 5. Does the graph verif that these functions are inverses of each b) other? Eplain wh. 8 = f() -DRAFT -DRAFT DDRAFT -DRAFT RAFT DRAFT - - = f -1 () - - - - - - - - = f() - 1. Inverse of a Relation MHR 53
1. For each of the following functions, determine the equation of the inverse graph f() and the inverse of f() restrict the domain of f() so that the inverse of f() is a function with the domain of f() restricted, sketch the graphs of f() and f -1 () a) f() = + 3 1. The function for converting the temperature from degrees Fahrenheit,, to degrees Celsius,, is = _ 5 ( - 3). 9 a) Determine the equivalent temperature in degrees Celsius for 9 F. b) Determine the inverse of this function. What does it represent? What do the variables represent? b) f() = _ 1 c) Determine the equivalent temperature c) f() = - in degrees Fahrenheit for 3 C. d) f() = ( + 1) d) Graph both functions. What does e) f() = -( - 3) the invariant point represent in this situation? f) f() = ( - 1) - 17. A forensic specialist can estimate the 13. Determine graphicall whether the height of a person from the lengths of their functions in each pair are inverses of bones. One function relates the length,,, each other. of the femur to the height,, of the person, a) f() = - and g() = + both in centimetres. b) f() = 3 + 5 and g() = - 5 For a male: =.3 + 5.53 3 c) f() = - 7 and g() = 7 - For a female: =.7 + 5.13 d) f() = - a) Determine the height of a male and of a and g() = + female with a femur length of 5.7 cm. e) f() = 8-7 and g() ) = 8 b) Use inverse functions to determine the + 7 femur length of 1. For each function, state two was to restrict the domain so that the inverse is a i) a male whose height is 187.9 cm function. ii) a female whose height is 175. cm a) f() ) = + 18. In Canada, ring sizes are specified using b) f() ) = - a numerical scale. The numerical ring size,, is approimatel related to finger c) f() = ( - 3) circumference,, in millimetres, b d) f() = ( + ) - = - 3.5 15. functiondraft Given the function f() = -,.55. determine each of the following. a) What whole-number ring size corresponds to a finger circumference a) f -1 () of 9.3 mm? b) f -1 (-) b) Determine an equation for the inverse c) f -1 (8) of the function. What do the variables d) f -1 () represent? DRAFT =DRAFT (DRAFT (DRAFT (DRAFT -DRAFT +DRAFT DRAFT 3)DRAFT )DRAFT DRAFT -DRAFT DRAFT DRAFT, DRAFT, DRAFT, of - = - 3) + ) - c) What finger circumferences correspond to ring sizes of, 7, and 9? 5 MHR Chapter 1
Etend 19. When a function is constantl increasing or decreasing, its inverse is a function. For each graph of f(), i) choose an interval over which the function is increasing and sketch the inverse of the function when it is restricted to that domain ii) choose an interval over which the function is decreasing and sketch the inverse of the function when it is restricted to that domain c) Determine the value of a if f -1 (a) = 1 and f() = + 5 + 3, -1.5. 1. If the point (1, 8) is on the graph of the function = f(), what point must be on the graph of each of the following? a) = f -1 ( + ) b) = f -1 () + 3 c) = -f -1 (-) + 1 Create Connections C1 Describe the inverse sequence of operations for each of the following. a) f() = + 1 b) f() = ( + 3) - 1 C a) Sketch the graphs of the function f() = - + 3 and its inverse, f -1 (). b) Eplain wh f() = f -1 (). c) If a function and its inverse are the same, how are the related to the line fdraft =? a) C3 Two students are arguing about whether or not a given function and its inverse are f() functions. Eplain how the students could verif who is correct. f() = + 5 C MINI LAB Two functions, and 3 - - - g() ) = 3-5, are inverses of each other. - Step 1 Evaluate output values for f() for = 1, =, = -8, and = a. Use - the results as input values for g(). What do ou notice about the output b) values for g()? Eplain wh this happens. State a hpothesis that could f() ( ) be used to verif whether or not two functions are inverses of each other. Step Reverse the order in which ou used -1-88 - 8 the functions. Start with using the - input values for g(), and then use the - outputs in f(). What conclusion can ou make about inverse functions?. Suppose a function f() has an inverse Step 3 Test our conclusions and hpothesis function, f -1 1(). b selecting two functions of our own. a) Determine f -1 (5) if f(17) = 5. Step Eplain how our results relate to the b) Determine f(-) if f -1 ( 3 ) = -. statement if f(a) = b and f -1 (b) = a, then the two functions are inverses of each other. Note that this must also be true when the function roles are switched. -1DRAFT function,draft -DRAFT DRAFTf( DRAFT( DRAFT) DRAFT) DRAFT= + = DRAFT 3 DRAFT3 DRAFT-8, DRAFT8, and f 1. Inverse of a Relation MHR 55
Chapter 1 Review 1.1 Horizontal and Vertical Translations, 1. Reflections and Stretches, pages 1 31 pages 15 5. Name the line of reflection when the graph 1. Given the graph of the function = f(), of = f() is transformed as indicated. sketch the graph of each transformed Then, state the coordinates of the image function. point of (3, 5) on the graph of each reflection. a) = -f() C D b) = f(-) -. Cop each graph of = f(). Then, A B - = f() sketch the reflection indicated state the domain and range of the transformed function a) - 3 = f() list an invariant points b) h() = f( + 1) a) = f(-) ) b) = -f() c) + 1 = f( - ). Describe how to translate the graph f() of = to obtain the graph of the function shown. Write the equation of the transformed function in the form - k = - h. - = f() f() =, g() = f(), and h() = f _ ( 1 ) - - - on the same set of coordinate aes. - b) Describe how the value of the - coefficient of for g() and h() affects the graph of the function f() =. 8. 3. The range of the function = f() is Consider the graphs of the functions f() and g(). { - 5, R}. What is the functiondraft DRAFT = ) DRAFT - - 7. a) Sketch the graphs of the functions ) range of the function = f( - ) +? f() (, ) g(). James wants to eplain vertical and horizontal translations b describing (, ) the effect of the translation on the (, ) coordinates of a point on the graph of a - - function. He sas, If the point (a, b) is on the graph of = f(), then the point a) Is the graph of g() a horizontal or a (a - 5, b + ) is the image point on the vertical stretch of the graph of f()? graph of + = f( - 5). Do ou agree Eplain our reasoning. with James? Eplain our reasoning. b) Write the equation that models the graph of g() as a transformation of the graph of f(). DRAFT DRAFT DRAFT 5,DRAFT DRAFT DRAFT 5, DRAFT DRAFT= DRAFT-f DRAFTf( DRAFT( DRAFT) DRAFT) = - 5 MHR Chapter 1
1.3 Combining Transformations, pages 3 3 9. Given the graph of = f(), sketch the graph of each transformed function. - - - = f() 1. Inverse of a Relation, pages 55 13. a) Cop the graph of = f() and sketch the graph of = f(). b) Name the line of reflection and list an invariant points. c) State the domain and range of the two functions. a) = f _ ( 1 ) b) = _ 1 f(3) - 1. Eplain how the transformations described - = f() ) b = f(( + 1)) and = f( + 1) are similar and how the are different. - 11. Write the equation for the graph of g() as a transformation of the equation for the 1. Cop and complete the table. graph of f(). DRAFT= = f() ) = f -1 () g() 8-3 3 7 1-1 f() 15. Sketch the graph of the inverse relation for each graph. State whether the relation and - 8 its inverse are functions. - a) b) 1. Consider the graph of = f(). Sketch the graph of each transformation. - - - - = f() 8 1. Algebraicall determine the equation of the inverse of the function = ( - 3) + 1. Determine a restriction on the domain of the function in order for its inverse to be a function. Show our thinking. 8 17. Graphicall determine if the functions are inverses of each other. a) = _ 1 f(-( + )) a) f() = - + 5 and g() = + 5 b) - = -f(( - 3)) b) f() = - 3 and g() = 8 + 3 8 c) - 1 = 3f( + ) DRAFT DRAFT = Chapter 1 Review MHR 57
Chapter 1 Practice Test Multiple Choice For #1 to #7, choose the best answer. 1. What is the effect on the graph of the function = when the equation is changed to = ( + 1)? A The graph is stretched verticall. B The graph is stretched horizontall. C The graph is the same shape but translated up. D The graph is the same shape but translated to the left.. The graph shows a transformation of the graph of =. Which equation models the graph? -8 - - -8 A + = - B - = - C - = + D + = + 3. If (a, b) ) is a point on the graph of = f(), which of the following points is on the graph of = f( + )? A (a +, b) ) B (a -, b) C (a, b + ) D (a, b - ). Which equation represents the image of = + after a reflection in the -ais? A = - - B = + C = - + D = - 5. The effect on the graph of = f() if it is transformed to = _ 1 f(3) is A a vertical stretch b a factor of _ 1 and a horizontal stretch b a factor of 3 B a vertical stretch b a factor of _ 1 and a horizontal stretch b a factor of _ 1 3 C a vertical stretch b a factor of and a horizontal stretch b a factor of 3 D a vertical stretch b a factor of and a horizontal stretch b a factor of _ 1 1_ 3. Which of the following transformations of f() ) produces a graph that has the same -intercept as f()? Assume that (, ) is not a point on f(). A -9f() 9 f ) B f() ) - 9 C f(-9) ) D f( - 9) 7. Given the graphs of = f() and = g(), what is the equation for g() in terms of f()? )DRAFT =DRAFT f(draft (DRAFT DRAFT DRAFT fdraft +DRAFT DRAFT +DRAFT + = - - - - - A g() = f _ (- 1 ) B g() = f(-) C g() = -f() D g() = -f _ ( 1 ) f() g() 58 MHR Chapter 1
Short Answer 8. The domain of the function = f() is { -3, R}. What is the domain of the function = f( + ) - 1? 9. Given the graph of = f(), sketch the a) Describe the transformation. graph of - = -_ 1 f _ ( 1 b) Write the equation of the function g(). ( + 3) ). c) Determine the minimum value of g(). d) The domain of the function f() is the set of real numbers. The domain of the function g() ) is also the set of real numbers. Does this impl that all of = f() the points are invariant? Eplain our answer. - - - 1. The function g() ) is a transformation of the function f(). - DRAFT) - f() ( ) g() ( 1. Consider the graph of the function = f(). - - a) Write the equation of the function f(). - - b) Write the equation of the function g() - = f() ( ) in the form g() = af(), and describe - the transformation. c) Write the equation of the function g() in the form g() = f(b), and describe a) Sketch the graph of the inverse. the transformation. b) Eplain how the coordinates of ke d) Algebraicall prove that the two points are transformed. equations from parts b) and c) are c) State an invariant points. equivalent. 11. Write the equation of the inverse function 15. Consider the function h() = -( + 3) - 5. of = 5 +. Verif graphicall that the functions are inverses of each other. 1. A transformation of the graph of = f() results in a horizontal stretch about the -ais b a factor of, a horizontal reflection in the -ais, a vertical stretch about the -ais b a factor of 3, and a horizontal translation of units to the right. Write the equation for the transformed function. Etended Response 13. The graph of the function f() = is transformed to the graph of g() = f( + ) - 7. a)draft DRAFT a) Eplain how ou can determine whether or not the inverse of h() is a function. b) Write the equation of the inverse relation in simplified form. c) What restrictions could be placed on the domain of the function so that the inverse is also a function? Chapter 1 Practice Test MHR 59
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