Question

What is the shifted form of the function f(x) = 4x^2 - 32x + 45?

a) f(x) = 4(x - 4)^2 + 1
b) f(x) = 4(x - 2)^2 + 41
c) f(x) = 4(x - 2)^2 - 41
d) f(x) = 4(x + 2)^2 - 45

Answer 1

The shifted form of the function f(x) = 4x^2 - 32x + 45 is f(x) = 4(x - 2)^2 + 41.

Step-by-step explanation:

The shifted form of the function f(x) = 4x^2 - 32x + 45 is f(x) = 4(x - 2)^2 + 41 (option b).

To determine the shifted form of a quadratic function, we need to complete the square. Rearranging the original function, we have f(x) = 4(x^2 - 8x) + 45. To complete the square, we take half of the coefficient of x (-8) and square it, which is (-8/2)^2 = 16. Adding and subtracting 16 inside the parentheses, we have f(x) = 4(x^2 - 8x + 16 - 16) + 45. Simplifying, we get f(x) = 4((x - 4)^2 - 16) + 45. Finally, distributing the 4 and simplifying further gives us f(x) = 4(x - 2)^2 + 41.