To rewrite the function \( g(x) = -16x + x^2 \) in vertex form, let's go through the process step-by-step:
### Step 1: Write the function in standard form
We start with the given function:
[tex]\[ g(x) = -16x + x^2 \][/tex]
In standard quadratic form, it should be written as:
[tex]\[ g(x) = x^2 - 16x \][/tex]
### Step 2: Form a perfect square trinomial by adding and subtracting \(\left(\frac{b}{2}\right)^2\)
Identify the \(b\) term:
[tex]\[ b = -16 \][/tex]
Calculate \(\left(\frac{b}{2}\right)^2\):
[tex]\[ \left(\frac{-16}{2}\right)^2 = 64 \][/tex]
Add and subtract \(\left(\frac{b}{2}\right)^2\) inside the function:
[tex]\[ g(x) = x^2 - 16x + 64 - 64 \][/tex]
[tex]\[ g(x) = \left(x^2 - 16x + 64\right) - 64 \][/tex]
### Step 3: Write the trinomial as a binomial squared
Rewrite the perfect square trinomial as a binomial squared:
[tex]\[ g(x) = (x - 8)^2 - 64 \][/tex]
### Step 4: Write the function in vertex form, if needed
The vertex form of a quadratic function is
[tex]\[ g(x) = a(x - h)^2 + k \][/tex]
where \((h, k)\) is the vertex of the parabola.
Thus, the function in vertex form is:
[tex]\[ g(x) = (x - 8)^2 - 64 \][/tex]
In conclusion, the function \( g(x) = -16x + x^2 \) can be written in vertex form as:
[tex]\[ g(x) = (x - 8)^2 - 64 \][/tex]
