x y 21 - 8 23 8 25 - 8 The table shows three values of x and their corresponding

Question

x y 21 - 8 23 8 25 - 8 The table shows three values of x and their corresponding values of y , where y = f x + 4 and f is a quadratic function. What is the y -coordinate of the y -intercept of the graph of y = f x in the xy -plane?

Accepted Answer

The correct answer is - 2,112 . It's given that f is a quadratic function. It follows that f can be defined by an equation of the form f x = a x - h 2 + k , where a , h , and k are constants. It's also given that the table shows three values of x and their corresponding values of y , where y = f x + 4 . Substituting a x - h 2 + k for f x in this equation yields y = a x - h 2 + k + 4 . This equation represents a quadratic relationship between x and y , where k + 4 is either the maximum or the minimum value of y , which occurs when x = h . For quadratic relationships between x and y , the maximum or minimum value of y occurs at the value of x halfway between any two values of x that have the same corresponding value of y . The table shows that x -values of 21 and 25 correspond to the same y -value, - 8 . Since 23 is halfway between 21 and 25 , the maximum or minimum value of y occurs at an x -value of 23 . The table shows that when x = 23 , y = 8 . It follows that h = 23 and k + 4 = 8 . Subtracting 4 from both sides of the equation k + 4 = 8 yields k = 4 . Substituting 23 for h and 4 for k in the equation y = a x - h 2 + k + 4 yields y = a x - 23 2 + 4 + 4 , or y = a x - 23 2 + 8 . The value of a can be found by substituting any x -value and its corresponding y -value for x and y , respectively, in this equation. Substituting 25 for x and - 8 for y in this equation yields - 8 = a 25 - 23 2 + 8 , or - 8 = a 2 2 + 8 . Subtracting 8 from both sides of this equation yields - 16 = a 2 2 , or - 16 = 4 a . Dividing both sides of this equation by 4 yields - 4 = a . Substituting - 4 for a , 23 for h , and 4 for k in the equation f x = a x - h 2 + k yields f x = - 4 x - 23 2 + 4 . The y -intercept of the graph of y = f x in the xy -plane is the point on the graph where x = 0 . Substituting 0 for x in the equation f x = - 4 x - 23 2 + 4 yields f 0 = - 4 0 - 23 2 + 4 , or f 0 = - 4 - 23 2 + 4 . This is equivalent to f 0 = - 2,112 , so the y -intercept of the graph of y = f x in the xy -plane is 0 , - 2,112 . Thus, the y -coordinate of the y -intercept of the graph of y = f x in the xy -plane is - 2,112 .

$x$	$y$
$21$	$negative 8$
$23$	$8$
$25$	$negative 8$

Question x-y-21