Question

what is the x - intercept of the graph of the function $f(x)=x^{2}-16x + 64$?
$\bigcirc$ $(-8,0)$
$\bigcirc$ $(0,8)$
$\bigcirc$ $(8,0)$
$\bigcirc$ $(0,-8)$

Explanation:

Step1: Recall x - intercept definition

To find the x - intercept of a function \(y = f(x)\), we set \(y = 0\) (i.e., \(f(x)=0\)) and solve for \(x\). The x - intercept is in the form \((x,0)\).

Step2: Set \(f(x)=0\) for the given function

Given \(f(x)=x^{2}-16x + 64\), set \(f(x) = 0\):
\[x^{2}-16x + 64=0\]

Step3: Factor the quadratic equation

We recognize that the quadratic \(x^{2}-16x + 64\) is a perfect square trinomial. The formula for a perfect square trinomial is \(a^{2}-2ab + b^{2}=(a - b)^{2}\). Here, \(a=x\), \(2ab = 16x\) (so \(b = 8\) since \(2\times x\times8=16x\)) and \(b^{2}=64\). So we can factor the quadratic as:
\[(x - 8)^{2}=0\]

Step4: Solve for \(x\)

Taking the square root of both sides of \((x - 8)^{2}=0\), we get \(x-8 = 0\), so \(x=8\).

Since the x - intercept is \((x,0)\) and \(x = 8\), the x - intercept is \((8,0)\).

Answer:

C. (8, 0) (assuming the options are labeled as A. (-8, 0), B. (0, 8), C. (8, 0), D. (0, -8))