Question

complete the square to re - write the quadratic function in vertex form: $y = x^2 - 8x - 8$

Explanation:

Step1: Group the x terms

We start with the quadratic function \( y = x^2 - 8x - 8 \). First, we group the terms involving \( x \): \( y=(x^2 - 8x)-8 \).

Step2: Complete the square for the x terms

To complete the square for the expression \( x^2 - 8x \), we take half of the coefficient of \( x \), which is \( \frac{-8}{2}=-4 \), and then square it: \( (-4)^2 = 16 \). We add and subtract this value inside the parentheses:
\( y=(x^2 - 8x + 16 - 16)-8 \)

Step3: Rewrite the perfect square trinomial

Now, we can rewrite \( x^2 - 8x + 16 \) as a perfect square: \( (x - 4)^2 \). So the expression becomes:
\( y=((x - 4)^2 - 16)-8 \)

Step4: Simplify the constant terms

Next, we simplify the constant terms: \( -16 - 8=-24 \). So the vertex form of the quadratic function is:
\( y=(x - 4)^2 - 24 \)

Answer:

\( y=(x - 4)^2 - 24 \)