x f x - 1 10 0 14 1 20 For the quadratic function f , the table shows three

Question

x f x - 1 10 0 14 1 20 For the quadratic function f , the table shows three values of x and their corresponding values of f x . Which equation defines f ?

Accepted Answer

Choice D is correct. The equation of a quadratic function can be written in the form f x = a x - h 2 + k , where a , h , and k are constants. It&rsquo;s given in the table that when x = - 1 , the corresponding value of f x is 10 . Substituting - 1 for x and 10 for f x in the equation f x = a x - h 2 + k gives 10 = a - 1 - h 2 + k , which is equivalent to 10 = a 1 + 2 h + h 2 + k , or 10 = a + 2 a h + a h 2 + k . It&rsquo;s given in the table that when x = 0 , the corresponding value of f x is 14 . Substituting 0 for x and 14 for f x in the equation f x = a x - h 2 + k gives 14 = a 0 - h 2 + k , or 14 = a h 2 + k . It&rsquo;s given in the table that when x = 1 , the corresponding value of f x is 20 . Substituting 1 for x and 20 for f x in the equation f x = a x - h 2 + k gives 20 = a 1 - h 2 + k , which is equivalent to 20 = a 1 - 2 h + h 2 + k , or 20 = a - 2 a h + a h 2 + k . Adding 20 = a - 2 a h + a h 2 + k to the equation 10 = a + 2 a h + a h 2 + k gives 30 = 2 a + 2 a h 2 + 2 k . Dividing both sides of this equation by 2 gives 15 = a + a h 2 + k . Since 14 = a h 2 + k , substituting 14 for a h 2 + k into the equation 15 = a + a h 2 + k gives 15 = a + 14 . Subtracting 14 from both sides of this equation gives a = 1 . Substituting 1 for a in the equations 14 = a h 2 + k and 20 = a h 2 - 2 a h + a + k gives 14 = h 2 + k and 20 = 1 - 2 h + h 2 + k , respectively. Since 14 = h 2 + k , substituting 14 for h 2 + k in the equation 20 = 1 - 2 h + h 2 + k gives 20 = 1 - 2 h + 14 , or 20 = 15 - 2 h . Subtracting 15 from both sides of this equation gives 5 = - 2 h . Dividing both sides of this equation by - 2 gives - 5 2 = h . Substituting - 5 2 for h into the equation 14 = h 2 + k gives 14 = - 5 2 2 + k , or 14 = 25 4 + k . Subtracting 25 4 from both sides of this equation gives 31 4 = k . Substituting 1 for a , - 5 2 for h , and 31 4 for k in the equation f x = a x - h 2 + k gives f x = x + 5 2 2 + 31 4 , which is equivalent to f x = x 2 + 5 x + 25 4 + 31 4 , or f x = x 2 + 5 x + 14 . Therefore, f x = x 2 + 5 x + 14 defines f . Choice A is incorrect. If f x = 3 x 2 + 3 x + 14 , then when x = - 1 , the corresponding value of f x is 14 , not 10 . Choice B is incorrect. If f x = 5 x 2 + x + 14 , then when x = - 1 , the corresponding value of f x is 18 , not 10 . Choice C is incorrect. If f x = 9 x 2 - x + 14 , then when x = - 1 , the corresponding value of f x is 24 , not 10 , and when x = 1 , the corresponding value of f x is 22 , not 20 .

$x$	$f (x)$
$negative 1$	$10$
$0$	$14$
$1$	$20$

Question x-f-x-