are not equivalent because they do not have 99. a. = 1 x the same solution set. In the first equation −1 2x = 2 we can also have 0 = 9 x2 = −3 . x or + 4 = 0 2x = 1 x and = 3 x = −4 x = b. x (−4, 9 and = 3 are equivalent because x ⎟ (0, , and 0) . The intercepts are −4) , ⎛ 9 = 3 . ⎝ 2 ⎠ ( ( ( −1) − = − − = c. 2) 1)2 and 2 x x x x x (−5)⎤ −1 are not equivalent because they do not = ⎡ 25 3. 1 2 4 have the same solution set. The first equation has the {1} solution set the second equation 4. right; 4 has no solutions. 5. parabola 100. Answers may vary. Methods discussed in this 6. axis (or axis of symmetry) section include factoring, the square root method, completing the square, and the quadratic 7. − b formula. 2a 101. Answers will vary. Knowing the discriminant 8. True; = 2 > 0 . a allows us to know how many real solutions the 4 b equation will have. =− = True; − 9. 2 2a (−1) 102. Answers will vary. One possibility: 2 ( ) − = 14 + 49 f x x2 x Two distinct: ( ) = − 3x − f x x2 One repeated: 10. True 18 11. C No real: 12. E ( ) = + + f x x2 x 13. F 4 103. Answers will vary. 14. A 104. Two quadratic functions can intersect 0, 1, or 2 15. G 16. B Section 2.4 17. H 18. D = 1. To find the let 0 : y-intercept, x 1 = 02 − 9 = −9 . x2 y , compress vertically 4 ( To find the let f x-intercept(s), = 0 : y Using the graph of x) − 9 = 0 x2 = y x2 by a factor of 1 . = = 9 19. x2 4 =± 9 = ±3 x The intercepts are (0, −9), (−3, 0), and (3, 0) .
99. a. x2 = 9 and x = 3 are not equivalent because they do not have the same solution set. In the first equation we can also have x = −3 . 2x −1 = 0 2x = 1 x = 1 2 or x + 4 = 0 x = −4 b. x = 9 and x = 3 are equivalent because The intercepts are (0, −4) , ⎛ ⎟ , and (−4, 0) . 9 = 3 . c. ( x −1) ( x − 2) = ( x − 1)2 and x − 2 = x −1 are not equivalent because they do not have the same solution set. The first equation has the 3. ⎡ 1 (−5)⎤ 2 = 25 4 ⎝ 2 ⎠ solution set {1} the second equation has no solutions. 100. Answers may vary. Methods discussed in this section include factoring, the square root method, completing the square, and the quadratic formula. 101. Answers will vary. Knowing the discriminant allows us to know how many real solutions the 4. right; 4 5. parabola 6. axis (or axis of symmetry) 7. − b 2a 8. True; a = 2 > 0 . b 4 equation will have. 102. Answers will vary. One possibility: 9. True; − 2a =− = 2 2 (−1) Two distinct: One repeated: f ( x ) = x2 − 3x − 18 f ( x ) = x2 − 14 x + 49 10. True 11. C No real: f ( x ) = x2 + x + 4 12. E 13. F 103. Answers will vary. 104. Two quadratic functions can intersect 0, 1, or 2 Section 2.4 1. To find the y-intercept, let x = 0 : 14. A 15. G 16. B 17. H 18. D y = 02 − 9 = −9 . To find the x-intercept(s), let x2 − 9 = 0 x2 = 9 x =± 9 = ±3 y = 0 : 19. f ( x) = 1 x2 4 Using the graph of by a factor of 1 . 4 y = x2 , compress vertically The intercepts are (0, −9), (−3, 0), and (3, 0) .