Write \( g(x) = -16x + x^2 \) in vertex form.

1. Write the function in standard form.

[tex]\[
g(x) = x^2 - 16x
\][/tex]

2. Form a perfect square trinomial by adding and subtracting \(\left(\frac{b}{2}\right)^2\).

[tex]\[
b = -16 \text{, so } \left(\frac{-16}{2}\right)^2 = 64
\][/tex]

[tex]\[
g(x) = \left(x^2 - 16x + 64\right) - 64
\][/tex]

3. Write the trinomial as a binomial squared.

[tex]\[
g(x) = (x - 8)^2 - 64
\][/tex]

4. The function in vertex form is:

[tex]\[
g(x) = (x - 8)^2 - 64
\][/tex]



Answer :

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]