A manufacturer produces items at a cost of Cx 2x2 16x 40 d

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 interpret what it means in the context of the problem. Find any x-intercepts and interpret what they mean in the context of the problem. Find the vertex and interpret the coordinates in the context of the problem.(vertex is max/min)

Solution

a)C(x) = 2x^2 -16x +40

C(x) = 2(x^2 - 8x +20)

= 2(x -4)^2 +16

Comparing it with C(x) = a(x-h)^2 +k

b)   Parent function f(x) = x^2

Transformations applied:

i) Horizontal shift of 4 units to right (x-4)^2

ii) Vertical scaling of 2 units 2(x-4)^2

iii) Vertical shift of 16 units : 2(x-4)^2 +16

c) y intercept when x=0

C(0) = 2*4^2 +16 = 48

This is the fixed cost of manfucaturing

d) x intercept : when C(x) =0

0= 2x^2 -16x +40

x^2 - 8x +20 =0

solve for x : x = imaginary roots

So,no x intercepts exist

e) C(x) = 2(x-4)^2 +16

Vertex : ( 4 , 16)

Minimum value of C(x) is at vertex