McGraw-Hill Ryerson Pre-Calculus 12

1 Pre-CalculusMcGraw-Hill Ryerson12 Contents A Tour of Your Textbook ..viiUnit 1 Transformations and Functions ..2 Chapter 1 Function Transformations .. Horizontal and Vertical Translations .. Refl ections and Stretches .. Combining Transformations .. Inverse of a Relation ..44 Chapter 1 Review ..56 Chapter 1 Practice Test ..58 Chapter 2 Radical Functions .. Radical Functions and Transformations .. Square Root of a Function .. Solving Radical Equations Graphically ..90 Chapter 2 Review ..99 Chapter 2 Practice Test ..102 Chapter 3 Polynomial Functions .. Characteristics of Polynomial Functions .. The Remainder Theorem .. The Factor Theorem .. Equations and Graphs of Polynomial Functions ..136 Chapter 3 Review ..153 Chapter 3 Practice Test ..155 Unit 1 Project Wrap-Up ..157 Cumulative Review, Chapters 1 3 ..158 Unit 1 Test ..160 Unit 2 Trigonometry ..162 Chapter 4 Trigonometry and the Unit Circle.

2 Angles and Angle Measure .. The Unit Circle .. Trigonometric Ratios .. Introduction to Trigonometric Equations ..206 Chapter 4 Review ..215 Chapter 4 Practice Test ..218 Chapter 5 Trigonometric Functions and Graphs .. Graphing Sine and Cosine Functions .. Transformations of Sinusoidal Functions .. The Tangent Equations and Graphs of Trigonometric Functions ..266 Chapter 5 Review ..282 Chapter 5 Practice Test ..286 Chapter 6 Trigonometric Reciprocal, Quotient, and Pythagorean Identities .. Sum, Difference, and Double-Angle Identities .. Proving Identities .. Solving Trigonometric Equations Using Identities ..316 Chapter 6 Review ..322 Chapter 6 Practice Test ..324 Unit 2 Project Wrap-Up ..325 Cumulative Review, Chapters 4 6 ..326 Unit 2 Test ..328iv MHR ContentsUnit 3 Exponential and Logarithmic Functions ..330 Chapter 7 Exponential Functions .. Characteristics of Exponential Functions.

3 Transformations of Exponential Functions .. Solving Exponential Equations ..358 Chapter 7 Review ..366 Chapter 7 Practice Test ..368 Chapter 8 Logarithmic Functions .. Understanding Logarithms .. Transformations of Logarithmic Functions .. Laws of Logarithms .. Logarithmic and Exponential Equations ..404 Chapter 8 Review ..416 Chapter 8 Practice Test ..419 Unit 3 Project Wrap-Up ..421 Cumulative Review, Chapters 7 8 ..422 Unit 3 Test ..424 Unit 4 Equations and Functions ..426 Chapter 9 Rational Functions .. Exploring Rational Functions Using Tr ansformations .. Analysing Rational Functions .. Connecting Graphs and Rational Equations ..457 Chapter 9 Review ..468 Chapter 9 Practice Test ..470 Chapter 10 Function Operations .. Sums and Differences of Functions .. Products and Quotients of Functions .. Composite Functions ..499 Chapter 10 Review ..510 Chapter 10 Practice Test ..512 Chapter 11 Permutations, Combinations, and the Binomial Theorem.

4 Permutations .. Combinations .. Binomial Theorem ..537 Chapter 11 Review ..546 Chapter 11 Practice Test ..548 Unit 4 Project Wrap-Up ..549 Cumulative Review, Chapters 9 11 ..550 Unit 4 Test ..552 Answers ..554 Glossary ..638 Index ..643 Credits ..646 Contents MHR vCHAPTER1 CHAPTERM athematical shapes are found in architecture, bridges, containers, jewellery, games, decorations, art, and nature. Designs that are repeated, reflected, stretched, or transformed in some way are pleasing to the eye and capture our this chapter, you will explore the mathematical relationship between a function and its transformed graph. Throughout the chapter, you will explore how functions are transformed and develop strategies for relating complex functions to simpler TransformationsKey Termstransformationmappingtranslationima ge pointrefl ectioninvariant pointstretchinverse of a functionhorizontal line testAlbert Einstein (1879 1955) is often regarded as the father of modern physics.

5 He won the Nobel Prize for Physics in 1921 for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect. The Lorentz transformations are an important part of Einstein s theory of You Know?4 MHR Chapter 1 Lantern Festival in and Vertical TranslationsFocus on ..determining the effects of h and k in y - k = f(x - h) on the graph of y = f(x)sketching the graph of y - k = f(x - h) for given values of h and k, given the graph of y = f(x)writing the equation of a function whose graph is a vertical and/or horizontal translation of the graph of y = f(x)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 textiles. Frieze patterns are also used by artists, craftspeople, musicians, choreographers, and mathematicians. Can you think of places where you have seen a frieze pattern?

6 A: Compare the Graphs of y = f(x) and y - k = f(x) 1. Consider the function f(x) = |x|.a) Use a table of values to compare the output values for y = f(x), y = f(x) + 3, and y = f(x) - 3 given input values of -3, -2, -1, 0, 1, 2, and ) Graph the functions on the same set of coordinate axes. 2. a) Describe how the graphs of y = f(x) + 3 and y = f(x) - 3 compare to the graph of y = f(x).b) Relative to the graph of y = f(x), what information about the graph of y = f(x) + k does k provide? 3. Would the relationship between the graphs of y = f(x) and y = f(x) + k change if f(x) = x or f(x) = x2? Vertical and Horizontal TranslationsMaterialsgrid paper 6 MHR Chapter 1B: Compare the Graphs of y = f(x) and y = f(x - h) 4. Consider the function f(x) = |x|.a) Use a table of values to compare the output values for y = f(x), y = f(x + 3), and y = f(x - 3) given input values of -9, -6, -3, 0, 3, 6, and ) Graph the functions on the same set of coordinate axes.

7 5. a) Describe how the graphs of y = f(x + 3) and y = f(x - 3) compare to the graph of y = f(x).b) Relative to the graph of y = f(x), what information about the graph of y = f(x - h) does h provide? 6. Would the relationship between the graphs of y = f(x) and y = f(x - h) change if f(x) = x or f(x) = x2? and Respond 7. How is the graph of a function y = f(x) related to the graph of y = f(x) + k when k > 0? when k < 0? 8. How is the graph of a function y = f(x) related to the graph of y = f(x - h) when h > 0? when h < 0? 9. Describe how the parameters h and k affect the properties of the graph of a function. Consider such things as shape, orientation, x-intercepts and y-intercept, domain, and transformation of a function alters the equation and any combination of the location, shape, and orientation of the on the original graph correspond to points on the transformed, or image, graph. The relationship between these sets of points can be called a notation can be used to show a relationship between the coordinates of a set of points, (x, y), and the coordinates of a corresponding set of points, (x, y + 3), for example, as (x, y) (x, y + 3).

8 Link the Ideastransformationa change made to a figure or a relation such that the figure or the graph of the relation is shifted or changed in shapemappingthe relating of one set of points to another set of points so that each point in the original set corresponds to exactly one point in the image setMapping notation is an alternate notation for function notation. For example, f(x) = 3x + 4 can be written as f: x 3x + 4. This is read as f is a function that maps x to 3x + 4. Did You Know? Horizontal and Vertical Translations MHR 7 One type 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. Graph Translations of the Form y - k = f(x) and y = f(x - h)a) Graph the functions y = x2, y - 2 = x2, and y = (x - 5)2 on the same set of coordinate ) Describe how the graphs of y - 2 = x2 and y = (x - 5)2 compare to the graph of y = ) The notation y - k = f(x) is often used instead of y = f(x) + k to emphasize that this is a transformation on y.

9 In this case, the base function is f(x) = x2 and the value of k is 2. The notation y = f(x - h) shows that this is a transformation on x. In this case, the base function is f(x) = x2 and the value of h is 5. Rearrange equations as needed and use tables of values to help you graph the = x2xy = x2 + 2xy = (x - 5)2-39-3112 9-24-263 4-11-134 10002501113612426743931189yx246810-22468 100y = x2y = (x - 5)2y = x2 + 2b) The transformed graphs are congruent to the graph of y = x2. Each point (x, y) on the graph of y = x2 is transformed to become the point (x, y + 2) on the graph of y - 2 = x2. Using mapping notation, (x, y) (x, y + 2).translationa slide transformation that results in a shift of a graph without changing its shape or orientationvertical and horizontal translations are types of transformations with equations of the forms y - k = f(x) and y = f(x - h), respectivelya translated graph is congruent to the original graphExample 1 For y = x2 + 2, the input values are the same but the output values change.

10 Each point (x, y) on the graph of y = x2 is transformed to (x, y + 2).For y = (x - 5)2, to maintain the same output values as the base function table, the input values are different. Every point (x, y) on the graph of y = x2 is transformed to (x + 5, y). How do the input changes relate to the translation direction?8 MHR Chapter 1 Therefore, the graph of y - 2 = x2 is the graph of y = x2 translated vertically 2 units up. Each point (x, y) on the graph of y = x2 is transformed to become the point (x + 5, y) on the graph of y = (x - 5)2. In mapping notation, (x, y) (x + 5, y). Therefore, the graph of y = (x - 5)2 is the graph of y = x2 translated horizontally 5 units to the TurnHow do the graphs of y + 1 = x2 and y = (x + 3)2 compare to the graph of y = x2? Justify your and Vertical TranslationsSketch the graph of y = |x - 4| + y = |x - 4| + 3, h = 4 and k = -3. yx246-22460y = |x|y = |x - 4| Start with a sketch of the graph of the base function y = |x|, using key the horizontal translation of 4 units to the right to obtain the graph of y = |x - 4|.