PreCalc 12 Chapter 2 Review Pack v2 Answer Section

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1 PreCalc 12 Chapter 2 Review Pack v2 Answer Section MULTIPLE CHOICE 1. ANS: D PTS: 1 DIF: Moderate REF: 2.1 Properties of Radical Functions LOC: 12.RF13 KEY: Procedural Knowledge 2. ANS: B PTS: 1 DIF: Easy REF: 2.2 Math Lab: Graphing Rational Functions 3. ANS: B PTS: 1 DIF: Easy REF: 2.3 Analyzing Rational Functions KEY: Procedural Knowledge 4. ANS: B PTS: 1 DIF: Easy REF: 2.2 Math Lab: Graphing Rational Functions 5. ANS: A PTS: 1 DIF: Moderate REF: 2.2 Math Lab: Graphing Rational Functions 6. ANS: C PTS: 1 DIF: Easy REF: 2.3 Analyzing Rational Functions KEY: Procedural Knowledge 7. ANS: C PTS: 1 DIF: Moderate REF: 2.3 Analyzing Rational Functions KEY: Procedural Knowledge 8. ANS: B PTS: 1 DIF: Moderate REF: 2.3 Analyzing Rational Functions Procedural Knowledge 9. ANS: B PTS: 1 DIF: Moderate REF: 2.3 Analyzing Rational Functions Procedural Knowledge 10. ANS: C PTS: 1 DIF: Moderate REF: 2.3 Analyzing Rational Functions Procedural Knowledge 11. ANS: A PTS: 1 DIF: Easy REF: 2.4 Sketching Graphs of Rational Functions 12. ANS: D PTS: 1 DIF: Moderate REF: 2.4 Sketching Graphs of Rational Functions 1

2 SHORT ANSWER 13. ANS: PTS: 1 DIF: Moderate REF: 2.1 Properties of Radical Functions LOC: 12.RF13 Procedural Knowledge 14. ANS: The graph has a horizontal asymptote with equation: y 2. PTS: 1 DIF: Moderate REF: 2.3 Analyzing Rational Functions Procedural Knowledge 15. ANS: Vertical Asymptote(s) Horizontal Asymptote Non-permissible values of x x 4, x 2 y 0 4, 2 PTS: 1 DIF: Easy REF: 2.2 Math Lab: Graphing Rational Functions Procedural Knowledge 2

3 PROBLEM 16. ANS: The equation has 1 binomial factor in the denominator that is not a factor of the numerator. Students equations may vary. For example, the graphs of these functions have a vertical asymptote at x 2: y x 5 x 2 or y 7 x 2 PTS: 1 DIF: Moderate REF: 2.2 Math Lab: Graphing Rational Functions Problem-Solving Skills 17. ANS: From the graph: f(x) 0 for 2 x 6, so the domain of y f(x) is 2 x 6. Mark the points where y 0 or y 1. Where 0 f(x) 1, the graph of y f(x) lies above the graph of y f(x). Where f(x) 1, the graph of y f(x) lies below the graph of y f(x). The vertex of the graph of y f(x) has coordinates (2,4), so the vertex of the graph of y f(x) has coordinates (2, 4), or (2,2). Join the points on each side of the y-axis with a smooth curve for the graph of y f(x). PTS: 1 DIF: Moderate REF: 2.1 Properties of Radical Functions LOC: 12.RF13 Procedural Knowledge Problem-Solving Skills 3

4 18. ANS: a) The denominator of the function is, so the graph has a non-permissible value of x at x 3. Check whether the equation can be simplified: y x 2 8x 15 y ()(x 5) y x 5, x 3 So, the non-permissible value represents a hole in the graph of y x 5 at x 3. b) Substitute x 3 in y x 5 to determine the y-coordinate of the hole: y The hole has coordinates ( 3,2). PTS: 1 DIF: Moderate REF: 2.3 Analyzing Rational Functions Procedural Knowledge Communication 4

5 19. ANS: The graph may have an oblique asymptote because the degree of the numerator is 1 more than the degree of the denominator. Check for common factors. Factor the numerator. y x 2 x 6 y ( x 2) ( ) The denominator is not a factor of the numerator, so the equation does not simplify, and the graph has an oblique asymptote. Determine: (x 2 x 6) () The quotient is x 5 and the remainder is 14. The function can be written as: y x As x, 0, and y x 5 0 So, the oblique asymptote has equation y x 5. PTS: 1 DIF: Moderate REF: 2.3 Analyzing Rational Functions Procedural Knowledge Communication 5

6 20. ANS: Mark points on the graph of y f(x) with y-coordinates that are perfect cubes. The perfect cubes between 10 and 10 are: 0, 1, and 8 Record the coordinates of the points in a table. 3 Calculate the y-coordinates of corresponding points on the graph of y f(x). x y f(x) 3 y f(x) Join the points with a smooth curve for the graph of y 3 f(x). The invariant points are the points that lie on both graphs. They are: (1,1), (2,0), and (3, 1) PTS: 1 DIF: Difficult REF: 2.1 Properties of Radical Functions LOC: 12.RF13 6

7 KEY: Procedural Knowledge Conceptual Understanding Problem-Solving Skills 21. ANS: The graph of the rational function has a vertical asymptote with equation x 3, so is a factor of the denominator. Since the oblique asymptote has equation y x 2, the numerator divided by the denominator is x 2 with a non-zero remainder. So, to find a possible numerator, multiply the denominator by x 2 and then add a non-zero constant. Students answers will vary. For example: (x 2)() ( 5) x 2 5x 6 5 Graph of y x 2 5x 1 x 2 5x 1 : (x 2)() 3 x 2 5x 6 3 x 2 5x 9 Graph of y x 2 5x 9 : So, each of the functions y x 2 5x 1 and y x 2 5x 9 and a vertical asymptote with equation x 3. has an oblique asymptote with equation y x 2, PTS: 1 DIF: Difficult REF: 2.3 Analyzing Rational Functions Communication Problem-Solving Skills 7

8 22. ANS: a) Factor the denominator to determine the non-permissible values of x. y x 2 x 6 y ( ) ( x 2) So, the non-permissible values are x 3 and x 2. Neither factor of the denominator is a factor of the numerator, so both non-permissible values indicate vertical asymptotes. b) The vertical asymptotes have equations x 3 and x 2. PTS: 1 DIF: Moderate REF: 2.3 Analyzing Rational Functions Procedural Knowledge Communication 8