Sure, let's work through the process of converting \(g(x) = -16x + x^2\) into vertex form step by step.
1. Write the function in standard form:
The given function is \(g(x) = -16x + x^2\). By rearranging the terms, we can write it as:
[tex]\[
g(x) = x^2 - 16x
\][/tex]
2. Form a perfect square trinomial by adding and subtracting \(\left(\frac{b}{2}\right)^2\):
Identify the coefficient \(b\) in the quadratic expression. Here, \(b\) is \(-16\). Now, calculate \(\left(\frac{b}{2}\right)^2\):
[tex]\[
b = -16, \quad \left(\frac{-16}{2}\right)^2 = 64
\][/tex]
Add and subtract this value inside the function to form a perfect square trinomial:
[tex]\[
g(x) = x^2 - 16x + 64 - 64
\][/tex]
3. Write the trinomial as a binomial squared:
The expression inside the parentheses is now a perfect square trinomial:
[tex]\[
x^2 - 16x + 64
\][/tex]
This can be written as the square of a binomial:
[tex]\[
(x - 8)^2
\][/tex]
Thus, the function can be written as:
[tex]\[
g(x) = (x - 8)^2 - 64
\][/tex]
So, the function in vertex form is:
[tex]\[
g(x) = (x - 8)^2 - 64
\][/tex]
The vertex form of [tex]\(g(x) = -16x + x^2\)[/tex] is [tex]\((x - 8)^2 - 64\)[/tex].
