f x = a x 2 + 4 x + c In the given quadratic function, a and c are

Question

f x = a x 2 + 4 x + c In the given quadratic function, a and c are constants. The graph of y = f x in the xy -plane is a parabola that opens upward and has a vertex at the point h , k , where h and k are constants. If k &#60; 0 and f - 9 = f 3 , which of the following must be true? c &#60; 0 a &#8805; 1

Accepted Answer

Choice D is correct. It's given that the graph of y = f x in the xy -plane is a parabola with vertex h , k . If f - 9 = f 3 , then for the graph of y = f x , the point with an x -coordinate of - 9 and the point with an x -coordinate of 3 have the same y -coordinate. In the xy -plane, a parabola is a symmetric graph such that when two points have the same y -coordinate, these points are equidistant from the vertex, and the x -coordinate of the vertex is halfway between the x -coordinates of these two points. Therefore, for the graph of y = f x , the points with x -coordinates - 9 and 3 are equidistant from the vertex, h , k , and h is halfway between - 9 and 3 . The value that is halfway between - 9 and 3 is - 9 + 3 2 , or - 3 . Therefore, h = - 3 . The equation defining f can also be written in vertex form, f x = a x - h 2 + k . Substituting - 3 for h in this equation yields f x = a x - - 3 2 + k , or f x = a x + 3 2 + k . This equation is equivalent to f x = a x 2 + 6 x + 9 + k , or f x = a x 2 + 6 a x + 9 a + k . Since f x = a x 2 + 4 x + c , it follows that 6 a = 4 and 9 a + k = c . Dividing both sides of the equation 6 a = 4 by 6 yields a = 4 6 , or a = 2 3 . Since 2 3 < 1 , it's not true that a e 1 . Therefore, statement Iwe isn't true. Substituting 2 3 for a in the equation 9 a + k = c yields 9 2 3 + k = c , or 6 + k = c . Subtracting 6 from both sides of this equation yields k = c - 6 . If k < 0 , then c - 6 < 0 , or c < 6 . Since c could be any value less than 6 , it's not necessarily true that c < 0 . Therefore, statement we isn't necessarily true. Thus, neither we nor Iwe must be true. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

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