Question

completing the square
which translation maps the vertex of the graph of the function $f(x) = x^2$ onto the vertex of the function $g(x) = -8x + x^2 + 7$?
right 4, down 9
left 4, down 9
right 4, up 23
left 4, up 23

Explanation:

Step1: Find vertex of $f(x)$

For $f(x)=x^2$, vertex form is $f(x)=(x-0)^2+0$, so vertex is $(0,0)$.

Step2: Rewrite $g(x)$ in vertex form

Start with $g(x)=x^2-8x+7$. Complete the square:

$$\begin{align*} g(x)&=x^2-8x+16-16+7\\ &=(x-4)^2-9 \end{align*}$$

Step3: Identify vertex of $g(x)$

From $g(x)=(x-4)^2-9$, vertex is $(4,-9)$.

Step4: Determine translation from $(0,0)$ to $(4,-9)$

Move right 4 units (from $x=0$ to $x=4$) and down 9 units (from $y=0$ to $y=-9$).

Answer:

right 4, down 9