( = 4 − 4 , = 4, = −4, = 0. f x) x2 x a b c occurs at = −b = −12 = −12 = x Since = 4 > 0, the graph opens up, so the a 3. vertex is a minimum point. The minimum 2a 2(−2) −4 occurs at = −b = −(− 4) = 4 = 1 . x The maximum value is The (3) = −2(3)2 + 12(3) = −18 + 36 = f minimum value is 2a 18 . 2(4) 8 2 61. For ⎛ 1 ⎞ ⎛ 1 ⎞2 ( = 2 + 12 − = 2, = f x) x2 x a b − ⎛ ⎞ − −1 4 1 = 1 2 = . ⎜ ⎟ = 4 ⎜ ⎟ f 3, 12, = c −3. ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ Since = 2 > 0, the graph opens up, so 2 2 2 a the vertex is a minimum point. The minimum = − + . f x a x h k 2 occurs at = −b = −12 = −12 = x The ( ) ( ) −3. 67. Use the form 2a 2(2) 4 The vertex is (0, 2) , so = 0 and = 2. h k = − + = + 0 2 2 . f x a x ax2 minimum value is 2 (−3) = 2(−3)2 + 12(−3) − 3 = 18 − 36 − 3 f ( ) ( ) = −21 . (1) = 8 . f Since the graph passes through (1, 8) , 62. For ( = 4 − 8x + = 4, = = 3. f x) x2 a b c ( ) = + 2 f x ax2 3, −8, Since = 4 > 0, the graph opens up, so the a (1)2 vertex is a minimum point. The minimum = + 8 2 a 8 = + 2 a occurs at = −b = −(−8) = 8 x The = 1. 2a 6 = a 2(4) 8 minimum value is ( ) = 6 + 2 . f x x2 (1) = 4(1)2 − 8(1) + 3 = 4 − 8 + 3 f = 6, = 0, = 2 a b c = −1 .
f ( x) = 4 x2 − 4 x , a = 4, b = −4, c = 0. occurs at x = −b = −12 = −12 = 3. 2a 2(−2) −4 The maximum value is Since a = 4 > 0, the graph opens up, so the vertex is a minimum point. The minimum occurs at x = −b = −(− 4) = 4 = 1 . The f (3) = −2(3)2 + 12(3) = −18 + 36 = 18 . 2a minimum value is 2(4) 8 2 61. For f ( x) = 2 x2 + 12 x − 3, a = 2, b = 12, ⎛ 1 ⎞ ⎛ 1 ⎞2 f ⎜ ⎟ = 4 ⎜ ⎟ − 4 ⎛ 1 ⎞ = 1 − 2 = −1 . c = −3. Since a = 2 > 0, the graph opens up, so ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ the vertex is a minimum point. The minimum f x = a x − h 2 + k . occurs at x = −b = −12 = −12 = −3. The 67. Use the form ( ) ( ) 2a 2(2) 4 The vertex is (0, 2) , so h = 0 and k = 2. minimum value is f x = a x − 0 2 + 2 = ax2 + 2 . f (−3) = 2(−3)2 + 12(−3) − 3 = 18 − 36 − 3 = −21 . ( ) ( ) Since the graph passes through (1, 8) , f (1) = 8 . 62. For f ( x) = 4 x2 − 8x + 3, a = 4, b = −8, c = 3. f ( x ) = ax2 + 2 Since a = 4 > 0, the graph opens up, so the vertex is a minimum point. The minimum 8 = a (1)2 + 2 8 = a + 2 occurs at x = −b = −(−8) = 8 = 1. The 2a minimum value is 2(4) 8 6 = a f ( x ) = 6 x2 + 2 . f (1) = 4(1)2 − 8(1) + 3 = 4 − 8 + 3 = −1 . a = 6, b = 0, c = 2
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