d. 79. Answers will vary. 80. −8 = + − = + + 2 3 ; 2x 1 ; y x2 x y x2 (0, = + 2 y x2 x 78. a. − − −1 = = 2 = x b (1) 2a 2 (−1) (−1)2 (−1) = = + − 2 8 y f = −9 (−1, Each member of this family will be a parabola −9) The vertex is . with the following characteristics: ( ) (i) opens upwards since a > 0; b. = 0 f x + 2 − 8 = x2 x 2 (ii) vertex occurs at = − =− = −1 ; x 0 ( ( + − 4) 2) x x (1) 2a 2 = 0 = 2 x + 4 = 0 or x − = 2 x −4 = x (iii) There is at least one since x-intercept 0 − 4ac ≥ 0 . b2 −4 The of are and 2. Likewise, the f x = − 4x + 1 ; of are −4 and 2. y x2 x-intercepts f ( + = ( ) 2 c. = −8 f x 0 x = + 1 ; y x2 x + 2 − 8 = −8 x2 x + = 81. = + 4 + 1 y x2 x 0 = 0 or x + 2 = 0 x = −2 x The solutions ( ) = −8 are −2 and 0. Thus, f x (−2, (0, the points −8) and −8) are on the graph of f. Each member of this family will be a parabola with the following characteristics: d. (i) opens upwards since > 0 a (ii) occurs at (0, 1). y-intercept 82. The graph of the quadratic function (−4, 0) (2, 0) ( ) = + + will not have any f x ax2 bx c x- − < intercepts whenever 4ac 0 . b2
d. 79. Answers will vary. −8 (0, 80. y = x2 + 2 x − 3 ; y = x2 + 2x + 1 ; y = x2 + 2 x 78. a. x = − b = − 2 = −1 2a 2 (1) y = f (−1) = (−1)2 + 2 (−1) − 8 = −9 The vertex is (−1, −9) . b. f ( x ) = 0 Each member of this family will be a parabola with the following characteristics: (i) opens upwards since a > 0; x2 + 2 x − 8 = 0 (ii) vertex occurs at 2 x = − =− = −1 ; ( x + 4) ( x − 2) = 0 2a 2 (1) x + 4 = 0 or x = −4 x − 2 = 0 x = 2 (iii) There is at least one x-intercept since b2 − 4ac ≥ 0 . The of f are −4 and 2. Likewise, the x-intercepts of f are −4 and 2. c. f ( x ) = −8 x2 + 2 x − 8 = −8 + 2 x = 0 x ( x + = 0 81. y = x2 − 4x + 1 ; y = x2 + 1 ; y = x2 + 4 x + 1 x = 0 or x + 2 = 0 x = −2 The solutions f ( x ) = −8 are −2 and 0. Thus, the points (−2, −8) and (0, −8) are on the graph of f. d. (−4, 0) (2, 0) Each member of this family will be a parabola with the following characteristics: (i) opens upwards since a > 0 (ii) y-intercept occurs at (0, 1). 82. The graph of the quadratic function f ( x ) = ax2 + bx + c will not have any x- intercepts whenever b2 − 4ac < 0 .