Question

a manufacturer produces items at a cost of c(x)=2x^2-16x+40 dollars. the items are later sold for a reasonable profit. rewrite the function in the form C(x)=a(x-h)^2+k. what transformations would have to be applied to f(x)=x^2 for it to become c(x) above? find the y intercept and what it means. find any x intercept and what it means and find the vertex

Answer 1

C(x) = 2(x-4)²+8; the graph would be stretched by a factor of 2, shifted up 8 and right 4; the y-intercept is 40; there are no x-intercepts; and the vertex is at (4, 8).

To write this in vertex form, we first factor 2 out of the right side:
C(x) = 2(x²-8x+20)

Divide both sides by 2:
C(x)/2 = x²-8x+20

To complete the square, we will half the value of b and square it:
(-8/2)² = (-4)² = 16
This is the value we add (and subtract) to the right hand side:

C(x)/2 = x²-8x+16+20-16
C(x)/2 = (x-4)²+4

Multiply both sides by 2:
C(x) = 2(x-4)²+8

The y-intercept is given by the value of c in the original function:
C(x) = 2x²-16x+40; c = 40.

Since the y-intercept is at 40 and this opens upward, there are no x-intercepts.

From vertex form, we can see that the original function was multiplied by 2; this is a stretch of 2. 8 was added to the end; this is a shift up 8. 4 was subtracted inside; this is a shift right 4.

In vertex form, f(x)=a(x-h)²+k, the vertex is (h, k):
C(x) = 2(x-4)²+8; the vertex is at (4, 8).²