Name: Class: Date: A. 70 B. 62 C. 38 D. 46
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1 Class: Date: Test 2 REVIEW Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Divide: (4x 2 49y 2 ) (2x 7y) A. 2x 7y B. 2x 7y C. 2x 7y D. 2x 7y 2. What is the remainder when x x 3x 2 is divided by 6 x? A. 70 B. 62 C. 38 D The volume of a shipping box with the shapeof a rectangular prism can be expressed as the polynomial 2x 3 11x 2 17x 6. Each dimension of the box can be expressed as a binomial. Which binomial could represent one dimension of the box? A. 2x 1 B. x 1 C. 2x 3 D. x 6 4. Which statements are always true for the graph of a cubic function? i) When the graph has exactly 1 x-intercept, the graph has no hills and no valleys. ii) When the graph has 2 or 3 x-intercepts, the graph has 1 hill and 1 valley. iii) When the x 3 -term is positive, the graph falls to the left and rises to the right. iv) When the x 3 -term is negative, the graph rises to the left and falls to the right. A. i, ii, iv B. i, iii, iv C. i, ii, iii D. ii, iii, iv 1
2 5. For the graph of y f(x) shown below, which graph best represents y f(x)? A. C. B. D. 2
3 6. For the graph of y f(x) shown below, what are the domain and range of y f(x)? A. domain: x 3orx 1; range: 0 y 1 B. domain: x 3orx 1; range: y 0 C. domain: 3 x 1; range: y 0 D. domain: 3 x 1; range: 0 y 1 3
4 7. For the graph of y f(x) shown below, which graph best represents y f(x)? A. C. B. D. 4
5 8. The graph of which function below has a hole? A. y x 2 x 2 2 B. y x 2 9 x 3 C. y x 2 x 4 D. y x 2 3 x The graph of which function below has a horizontal asymptote? A. y x 2 7x 12 x 7 B. y x 2 3 x 7 C. y x 2 3 x 2 2 D. y x 2 x What is the non-permissible value of x for this function? y x 2 4 x 2 A. x 4 C. x 2 B. x 2 D. x For the graph of this rational function, identify the equation of any asymptote. y 6x 8 x 2 4 A. The graph has an oblique asymptote at y 6x 8. B. The graph has a vertical asymptote at x 2. C. The graph has a horizontal asymptote at y 0. D. The graph has no vertical or horizontal asymptotes. 12. For the graph of this rational function, identify the equations of any asymptotes and the coordinates of any hole. x 4 y x 2 8x 15 A. The graph has a vertical asymptote at x 5, a hole at (3,7), and a horizontal asymptote at y 0. B. The graph has vertical asymptotes at x 5 and x 3, and a horizontal asymptote at y 0. C. The graph has a vertical asymptote at x 0, a hole at (3,7), and a horizontal asymptote at y 5. D. The graph has a vertical asymptote at x 0, and a horizontal asymptote at y 5. 5
6 13. What is the solution of this radical equation, to the nearest tenth if necessary? x 2 x 5 x 4 4 x A. x 4 or x 1 C. x Ö 5.2 or x Ö 2.3 or x Ö 0.5 B. x Ö 5.2 or x Ö 2.3 D. x Which function below describes this graph? A. y x 2 x 6 x 3 B. y x 2 x 6 x 3 C. y x 2 6x 1 x 3 x 3 D. y x 2 x 6 6
7 15. Which graph represents the function y x 3 x 1? A. C. B. D. 7
8 16. Which graph represents the function y 4x 2? x A. C. B. D. 8
9 17. Which graph represents the function y x 3 x 2 2x 8? A. C. B. D. 18. Compared to the graph of the base function f(x) x, the graph of the function g(x) 5x is A. compressed by a factor of 1 and not reflected 5 B. stretched by a factor of 5 and reflected in the y-axis C. compressed by a factor of 1 and reflected in the y-axis 5 D. stretched by a factor of 5 and not reflected 19. Compared to the graph of the base function f(x) x, the graph of the function g(x) x 8 4 is translated A. 4 units to the right and 8 units up C. 8 units to the left and 4 units down B. 4 units to the left and 8 units down D. 8 units to the right and 4 units up 9
10 20. The two functions in the graph shown are reflections of each other. Select the type of reflection(s). A. a reflection in the y-axis C. a reflection in the line y = x B. a reflection in the x-axis and the y-axis D. a reflection in the x-axis 21. What are the coordinates of the invariant point(s) when the function y A. (9, 3) C. (0, 3) B. ( 3, 0) and (9, 0) D. (0, 9) x 3 is reflected in the y-axis? 22. Which radical equation can be solved using the graph shown below? A. 4 x x 2 C. x 2 4 x B. 4 x x 2 D. 4 x x What is the solution to the radical equation 0 x 9 3? A. 18 C. 18 B. 36 D. 0 10
11 24. Which graph represents an even-degree polynomial function with a y-intercept of 9? A. C. B. D. 11
12 25. Based on the graph of fx ( ) x 4 2x 3 24x 2 8x 96, what are the real roots of x 4 2x 3 24x 2 8x 96 0? A. 6, 2, 2, 4 C. there are no real roots B. 6, 2, 2, 4 D. impossible to determine 12
13 26. Which of the following graphs of polynomial functions corresponds to a cubic polynomial equation with roots 4, 1, and 3? A. C. B. D. 27. Which of the reciprocal functions has a vertical asymptote with equation x 9 2? 1 9 A. f(x) C. f(x) 2x 9 x B. f(x) D. f(x) 2x 9 x What is the x-intercept of f(x) 2x 4? A. There is no x-intercept. C. 2 B. 1 2 D What is the value of k in the function f(x) 2 k 5x k if its graph passes through the point (3, 2 19 )? A. 19 C B. 4 D. No such k exists 13
14 30. Which function has a point of discontinuity at x 3? A. x 3 x 3 f(x) C. f(x) 2x 2 2x 12 x 2 6x 12 B. x 3 x 3 f(x) D. f(x) x 2 6x 12 x 2 6x Which function has a y-intercept of 8 27? A. 8 f(x) x 2 12x 27 B. 8 f(x) ( 8x 3)(x 9) C. f(x) 8 x 2 12x 27 D. all of the above 32. Which function has an x-intercept of 1 3? A. f(x) 6x 2 5x 3 B. f(x) 5x 3 6x 2 C. f(x) 6x 2 5x 3 D. f(x) 5x 2 6x What is the equation for the horizontal asymptote of the graph of the function shown? A. y 2 C. y 3 B. x 5 D. x 6 14
15 34. Given the functions fx ( ) x 2 and g( x) x 2 2x 7, what is the domain of the combined function kx ( ) fx ( ) gx ( )? A. cannot be determined C. {x x 2, x R} B. {x x 7, x R} D. {x x 9, x R} 35. Given the functions fx ( ) x and g( x) 1 ( 3 x 5 ), which of the following is most likely the graph of y f(g(x))? A. C. B. D. 15
16 Matching Match the functions to their corresponding graphs. A. f(x) 4 2(x 7) 6 D. f(x) 4 2(x 6) 7 B. f(x) 2 4(x 7) 6 E. f(x) 4 2(x 7) 11 C. f(x) 2 4(x 6) 7 F. f(x) 2 4(x 6)
17 Short Answer 1. Determine the quotient and remainder when the polynomial 3x 3 3x 2 30x 15 is divided by x Factor: x 3 2x 2 19x Determine the remainder when 20x 3 33x 2 45x 30 is divided by 5x Use a table of values to sketch the graph of the polynomial function f(x) x 4 6x 3 8x 2 6x 9. x f(x) 17
18 5. The zeros of the polynomial function f(x) x 4 3x 3 7x 2 15x 18 are 1, 2, and 3. The zeros 1 and 2 have multiplicity 1. The zero 3 has multiplicity 2. Sketch a graph of the function. 6. Write an equation in standard form for a cubic function with zeros 1, 2, and Sketch the graph of this function. y x 3 x 1 18
19 1 8. Consider the function f(x) x 2 2x 8. a) Determine the key features of the function: i) domain and range ii) intercepts iii) equations of any asymptotes b) Sketch the graph of the function. Problem 1. Here is a student s solution for dividing 3x 4 2x 2 x by x 2 1. a) Identify the errors in the solution. Write a correct solution. b) Verify your answer. 3x 2 5 x 2 1 3x 4 0x 3 2x 2 x 0 3x 4 3x 2 5x 2 x 5x 2 5 x 5 The quotient is 3x 2 5 and the remainder is x 5. 19
20 2. Is x 2 a factor of 4x 4 2x 3 6x 2 7x 10? How do you know? 3. A cubic function has zeros 3, 1, and 2. The y-intercept of its graph is 12. Sketch the graph, then determine an equation of the function in standard form. Explain your work. 4. Sketch the graph of this polynomial function: f(x) (x 1) 2 (x 1) 3 5. Kara and 3 friends each have birthdays on January 9. Leon is 4 years older than Kara, Max is 3 years younger than Kara, and Norma is 5 years younger than Kara. On January 9, 2011, the product of their ages was greater than the sum of their ages. How old was Kara and each friend on that day? Show your work. 20
21 6. Sketch the graph of this function, and state the domain and range. Show your work. y x 2 7x 12 x 3 7. Sketch the graph of this function, and state the domain and range. Show your work. 3x y x
22 8. Write an equation for the graph of the rational function shown. Explain your reasoning. 22
23 9. Given the functions fx ( ) x 2 3x 10 and g( x) x 2 5x, graph the function h( x) fx ( ) gx ( ) intercepts and identify any asymptotes and/or points of discontinuity.. Label all 23
24 Test 2 REVIEW Answer Section MULTIPLE CHOICE 1. ANS: C PTS: 0 DIF: Easy REF: 1.1 Dividing a Polynomial by a Binomial LOC: 12.RF11 TOP: Relations and Functions KEY: Conceptual Understanding Procedural Knowledge 2. ANS: C PTS: 0 DIF: Moderate REF: 1.1 Dividing a Polynomial by a Binomial LOC: 12.RF11 TOP: Relations and Functions KEY: Conceptual Understanding Procedural Knowledge 3. ANS: A PTS: 0 DIF: Difficult REF: 1.2 Factoring Polynomials LOC: 12.RF11 TOP: Relations and Functions KEY: Procedural Knowledge Problem-Solving Skills 4. ANS: D PTS: 0 DIF: Moderate REF: 1.3 Graphing Polynomial Functions LOC: 12.RF12 TOP: Relations and Functions KEY: Conceptual Understanding 5. ANS: B PTS: 1 DIF: Easy REF: 2.1 Properties of Radical Functions LOC: 12.RF13 TOP: Relations and Functions KEY: Conceptual Understanding 6. ANS: B PTS: 1 DIF: Moderate REF: 2.1 Properties of Radical Functions LOC: 12.RF13 TOP: Relations and Functions KEY: Procedural Knowledge Conceptual Understanding 7. ANS: A PTS: 1 DIF: Moderate REF: 2.1 Properties of Radical Functions LOC: 12.RF13 TOP: Relations and Functions KEY: Conceptual Understanding 8. ANS: B PTS: 1 DIF: Easy REF: 2.2 Math Lab: Graphing Rational Functions LOC: 12.RF14 TOP: Relations and Functions KEY: Conceptual Understanding 9. ANS: C PTS: 1 DIF: Easy REF: 2.2 Math Lab: Graphing Rational Functions LOC: 12.RF14 TOP: Relations and Functions KEY: Conceptual Understanding 10. ANS: C PTS: 1 DIF: Easy REF: 2.2 Math Lab: Graphing Rational Functions LOC: 12.RF14 TOP: Relations and Functions KEY: Conceptual Understanding 11. ANS: C PTS: 1 DIF: Moderate REF: 2.3 Analyzing Rational Functions LOC: 12.RF14 TOP: Relations and Functions KEY: Conceptual Understanding Procedural Knowledge 12. ANS: B PTS: 1 DIF: Moderate REF: 2.3 Analyzing Rational Functions LOC: 12.RF14 TOP: Relations and Functions KEY: Conceptual Understanding Procedural Knowledge 13. ANS: C PTS: 1 DIF: Moderate REF: 2.3 Analyzing Rational Functions LOC: 12.RF14 TOP: Relations and Functions KEY: Procedural Knowledge 14. ANS: B PTS: 1 DIF: Moderate REF: 2.4 Sketching Graphs of Rational Functions LOC: 12.RF14 TOP: Relations and Functions KEY: Conceptual Understanding 15. ANS: B PTS: 1 DIF: Moderate REF: 2.4 Sketching Graphs of Rational Functions LOC: 12.RF14 TOP: Radical and Rational Functions KEY: Conceptual Understanding 1
25 16. ANS: C PTS: 1 DIF: Moderate REF: 2.4 Sketching Graphs of Rational Functions LOC: 12.RF14 TOP: Radical and Rational Functions KEY: Conceptual Understanding 17. ANS: A PTS: 1 DIF: Moderate REF: 2.4 Sketching Graphs of Rational Functions LOC: 12.RF14 TOP: Relations and Functions KEY: Conceptual Understanding 18. ANS: A PTS: 1 DIF: Average OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: horizontal stretch 19. ANS: C PTS: 1 DIF: Average OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: horizontal translation vertical translation 20. ANS: A PTS: 1 DIF: Average OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: reflection 21. ANS: C PTS: 1 DIF: Difficult OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: invariant points 22. ANS: B PTS: 1 DIF: Easy OBJ: Section 2.3 NAT: RF13 TOP: Solving Radical Equations Graphically KEY: graphical solution 23. ANS: D PTS: 1 DIF: Average OBJ: Section 2.3 NAT: RF13 TOP: Solving Radical Equations Graphically KEY: algebraic solution 24. ANS: B PTS: 1 DIF: Average OBJ: Section 3.1 NAT: RF12 TOP: Characteristics of Polynomial Functions KEY: even-degree x-intercepts y-intercept 25. ANS: B PTS: 1 DIF: Easy OBJ: Section 3.4 NAT: RF12 TOP: Equations and Graphs of Polynomial Functions KEY: polynomial equation roots 26. ANS: B PTS: 1 DIF: Average OBJ: Section 3.4 NAT: RF12 TOP: Equations and Graphs of Polynomial Functions KEY: polynomial equation roots graph 27. ANS: B PTS: 1 DIF: Easy OBJ: Section 9.1 NAT: RF14 TOP: Exploring Rational Functions Using Transformations KEY: reciprocal of linear function vertical asymptote 28. ANS: A PTS: 1 DIF: Easy OBJ: Section 9.1 NAT: RF14 TOP: Exploring Rational Functions Using Transformations KEY: reciprocal of linear function x-intercept 29. ANS: B PTS: 1 DIF: Difficult OBJ: Section 9.1 NAT: RF14 TOP: Exploring Rational Functions Using Transformations KEY: reciprocal of linear function 30. ANS: A PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF14 TOP: Analysing Rational Functions KEY: reciprocal of quadratic function vertical asymptote 31. ANS: C PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF14 TOP: Analysing Rational Functions KEY: reciprocal of quadratic function y-intercept 2
26 32. ANS: C PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF14 TOP: Analysing Rational Functions KEY: linear expressions in numerator and denominator x-intercept 33. ANS: C PTS: 1 DIF: Easy OBJ: Section 9.2 NAT: RF14 TOP: Analysing Rational Functions KEY: quadratic denominator horizontal asymptote 34. ANS: C PTS: 1 DIF: Easy OBJ: Section 10.1 NAT: RF1 TOP: Sums and Differences of Functions KEY: subtract functions domain 35. ANS: D PTS: 1 DIF: Average OBJ: Section 10.3 NAT: RF1 TOP: Composite Functions KEY: composite functions transformations graph MATCHING 1. ANS: D PTS: 1 DIF: Average OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: graph transformations translations vertical translation horizontal translation 2. ANS: C PTS: 1 DIF: Average OBJ: Section 2.1 NAT: RF13 TOP: Radical Functions and Transformations KEY: graph transformations translations vertical translation horizontal translation SHORT ANSWER 1. ANS: 3x 2 9x 6 R( 9) PTS: 0 DIF: Moderate REF: 1.1 Dividing a Polynomial by a Binomial LOC: 12.RF11 TOP: Relations and Functions KEY: Conceptual Understanding Procedural Knowledge 2. ANS: (x 1)(x 4)(x 5) PTS: 0 DIF: Moderate REF: 1.2 Factoring Polynomials LOC: 12.RF11 TOP: Relations and Functions KEY: Conceptual Understanding Procedural Knowledge 3. ANS: The remainder is 16. PTS: 0 DIF: Difficult REF: 1.2 Factoring Polynomials LOC: 12.RF11 TOP: Relations and Functions KEY: Conceptual Understanding Procedural Knowledge 3
27 4. ANS: x f(x) PTS: 0 DIF: Moderate REF: 1.4 Relating Polynomial Functions and Equations LOC: 12.RF12 TOP: Relations and Functions KEY: Procedural Knowledge Communication 5. ANS: PTS: 0 DIF: Moderate REF: 1.4 Relating Polynomial Functions and Equations LOC: 12.RF12 TOP: Relations and Functions KEY: Procedural Knowledge 4
28 6. ANS: Answers may vary by a common numerical factor. For example: y x 3 3x 2 6x 8 PTS: 0 DIF: Moderate REF: 1.4 Relating Polynomial Functions and Equations LOC: 12.RF12 TOP: Relations and Functions KEY: Procedural Knowledge Communication 7. ANS: y x 3 x 1 PTS: 1 DIF: Moderate REF: 2.4 Sketching Graphs of Rational Functions LOC: 12.RF14 TOP: Relations and Functions KEY: Conceptual Understanding Procedural Knowledge Communication 5
29 8. ANS: a) i) {x x 2, 4, x R}, {y y 0, y R} ii) x-intercept: none, y-intercept: 1 8 iii) x 2, x 4, y 0 b) PTS: 1 DIF: Difficult OBJ: Section 9.2 Section 9.3 NAT: RF14 TOP: Analysing Rational Functions Connecting Graphs and Rational Equations KEY: reciprocal of quadratic function key features graph 6
30 PROBLEM 1. ANS: a) The student should have subtracted 3x 2 from 2x 2 instead of adding 3x 2 to 2x 2. A similar error occurs later: the student should have subtracted 5 from x instead of adding 5 to x. The correct solution is: 3x 2 1 x 2 1 3x 4 0x 3 2x 2 x 0 3x 4 3x 2 x 2 x x 2 1 x 1 The quotient is 3x 2 1 and the remainder is x 1. b) To check, multiply the quotient by the divisor, then add the remainder. (x 2 1)( 3x 2 1) ( x 1) 3x 4 x 2 3x 2 1 x 1 3x 4 2x 2 x Since 3x 4 2x 2 x is the original polynomial, the answer is correct. PTS: 0 DIF: Difficult REF: 1.1 Dividing a Polynomial by a Binomial LOC: 12.RF11 TOP: Relations and Functions KEY: Procedural Knowledge Communication Problem-Solving Skills 2. ANS: Let P(x) 4x 4 2x 3 6x 2 7x 10. x 2 is a factor of P(x) if P( 2) 0. P( 2) 4( 2) 4 2( 2) 3 6( 2) 2 7( 2) 10 0 Since P( 2) is 0, x 2 is a factor of 4x 4 2x 3 6x 2 7x 10. PTS: 0 DIF: Easy REF: 1.2 Factoring Polynomials LOC: 12.RF11 TOP: Relations and Functions KEY: Procedural Knowledge Communication 7
31 3. ANS: Sketch the graph: The zeros of the function are the roots of its related polynomial equation. Let k represent the leading coefficient. y k(x 3)(x 1)(x 2) The constant term in the equation is 12. So, k( 3)( 1)(2) 12 k 2 So, an equation is: y 2(x 3)(x 1)(x 2) y 2(x 2 4x 3)(x 2) y 2x 3 4x 2 10x 12 PTS: 0 DIF: Moderate REF: 1.4 Relating Polynomial Functions and Equations LOC: 12.RF12 TOP: Relations and Functions KEY: Procedural Knowledge Communication 8
32 4. ANS: To determine the zeros, solve f(x) 0. 0 (x 1) 2 (x 1) 3 The roots of the equation are x 1 and x 1. So, the zeros of the function are 1 and 1. The zero 1 has multiplicity 2. The zero 1 has multiplicity 3. So, the graph just touches the x-axis at x 1 and crosses the x-axis at x 1. The equation has degree 5, so it is an odd-degree polynomial function. The leading coefficient is positive, so as x, the graph falls and as x, the graph rises. The y-intercept is: (1) 2 ( 1) 3 1 Plot points at the intercepts, then draw a smooth curve that falls to the left and rises to the right. f(x) (x 1) 2 (x 1) 3 PTS: 0 DIF: Moderate REF: 1.4 Relating Polynomial Functions and Equations LOC: 12.RF12 TOP: Relations and Functions KEY: Procedural Knowledge Communication 9
33 5. ANS: Let Kara s age in years be x. Then Leon s age in years is x 4, Max s age in years is x 3, and Norma s age in years is x 5. The sum of their ages is: x (x 4) (x 3) (x 5) 4x 4 Product of ages sum of ages = So, x(x 4)(x 3)(x 5) (4x 4) x(x 4)(x 3)(x 5) (4x 4) x(x 4)(x 3)(x 5) 4x Use a graphing calculator to graph the equation y x(x 4)(x 3)(x 5) 4x Age cannot be negative, so the positive x-intercept, which is 17, represents Kara s age. So, Kara was 17 years old on that day. The ages of the other friends, in years, were: Leon: Max: Norma: On January 9, 2011, Leon was 21, Max was 14, Norma was 12, and Kara was 17. PTS: 0 DIF: Moderate REF: 1.5 Modelling and Solving Problems with Polynomial Functions LOC: 12.RF12 TOP: Relations and Functions KEY: Procedural Knowledge Communication Problem-Solving Skills 10
34 6. ANS: y x 2 7x 12 x 3 The denominator is x 3, so the graph has a non-permissible value of x at x 3. Factor: y (x 4)(x 3) x 3 Simplify: y x 4, x 3 So, the graph of the rational function is the line y x 4, with a hole at x 3. The y-coordinate of the hole is: y ( 3) 4, or 1. So, the hole has coordinates ( 3, 1). Draw the line y x 4 with an open circle at ( 3, 1). The function has domain x 3 and range y 1. PTS: 1 DIF: Moderate REF: 2.4 Sketching Graphs of Rational Functions LOC: 12.RF14 TOP: Relations and Functions KEY: Conceptual Understanding Procedural Knowledge Communication 11
35 7. ANS: 3x y x x Factor: y (x 4)(x 4) The denominator has two binomial factors, x 4 and x 4, so the non-permissible values of x are x 4 and x 4. Since neither factor of the denominator is a factor of the numerator, the graph has vertical asymptotes at x 4 and x 4 The degree of the denominator is greater than the degree of the numerator, so there is a horizontal asymptote at y 0. Draw broken lines for the asymptotes. Determine the approximate coordinates of some points on the graph: ( 6, 0.9), ( 5, 1.67), ( 3, 1.29), (0, 0), (3, 1.29), (5, 1.67), (6, 0.9) Determine the behaviour of the graph close to the asymptotes. x y Join the points to form 3 smooth curves. y 3x x 2 16 The function has domain x 4 and range y ò. PTS: 1 DIF: Moderate REF: 2.4 Sketching Graphs of Rational Functions LOC: 12.RF14 TOP: Relations and Functions KEY: Conceptual Understanding Procedural Knowledge Communication 12
36 8. ANS: Answers may vary. Sample answer: 1 f(x) x 2 4 k Any function of the form f(x), k 0, is a reasonable candidate since it is difficult to tell from the x 2 4 graph how stretched the function is. PTS: 1 DIF: Average OBJ: Section 9.2 NAT: RF14 TOP: Analysing Rational Functions KEY: reciprocal of quadratic function 9. ANS: PTS: 1 DIF: Difficult OBJ: Section 10.2 NAT: RF1 TOP: Products and Quotients of Functions KEY: divide functions vertical asymptotes hole 13
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