Question

write the quadratic function in standard form.
$g(x) = x^2 - 8x$
$g(x) = $
sketch its graph.

Explanation:

Step1: Recall the standard form of a quadratic function

The standard form of a quadratic function is \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. To convert \( g(x)=x^{2}-8x \) to standard form, we complete the square.

For the quadratic expression \( x^{2}-8x \), the coefficient of \( x \) is \(- 8\). To complete the square, we take half of the coefficient of \( x \), square it, and add and subtract it (or just add it by using the formula for completing the square).

Half of \(-8\) is \(\frac{-8}{2}=-4\), and squaring it gives \((-4)^{2} = 16\).

Step2: Complete the square

We rewrite \( x^{2}-8x\) as \(x^{2}-8x + 16-16\). This is because we added \( 16\) to complete the square and then subtracted \( 16\) to keep the expression equivalent.

So, \(x^{2}-8x=(x^{2}-8x + 16)-16\).

And \(x^{2}-8x + 16=(x - 4)^{2}\) (by the formula \((a - b)^{2}=a^{2}-2ab + b^{2}\), here \(a = x\) and \(b = 4\)).

Therefore, \(g(x)=x^{2}-8x=(x - 4)^{2}-16\)

Answer:

\(g(x)=(x - 4)^{2}-16\)