Question

Functions f and g are both defined for x ER and are given by

f(x)=x²-2x+5.
g(x)=x² + 4x + 13.
(a) By first expressing each of f(x) and g(x) in completed square form, express g(x) in the form
f(x+p) + q, where p and q are constants.
(4)

Answer 1

To express g(x) in the form f(x+p) + q, we complete the square for both f(x) and g(x). We find that f(x+p) + q is f(x+3)+9, with p = 3 and q = 9.

Step-by-step explanation:

The goal is to express g(x) in the form f(x+p) + q, where p and q are constants. We start by writing the given functions f(x) and g(x) in completed square form.
The function f(x) is already a perfect square trinomial:
f(x) = (x - 1)^2 + 4.

Next, we complete the square for g(x):
g(x) = x^2 + 4x + 13 = (x^2 + 4x + 4) + 9 = (x + 2)^2 + 9.

We can now express g(x) in the form f(x+p) + q:
g(x) = f(x + 3) + 9,
where p = 3 and q = 9.