Question

which translation maps the vertex of the graph of the function $f(x) = x^2$ onto the vertex of the function $g(x) = x^2 - 10x + 2$? left 5, up 27; right 5, down 23; right 5, up 27; left 5, down 23

Explanation:

Step1: Find vertex of $f(x)=x^2$

The vertex of $f(x)=x^2$ is $(0,0)$.

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

Complete the square for $g(x)=x^2-10x+2$:

$$\begin{align*} g(x)&=x^2-10x+2\\ &=(x^2-10x+25)-25+2\\ &=(x-5)^2-23 \end{align*}$$

Step3: Identify vertex of $g(x)$

From $g(x)=(x-5)^2-23$, the vertex is $(5,-23)$.

Step4: Determine translation from $(0,0)$ to $(5,-23)$

To go from $x=0$ to $x=5$: shift right 5 units. To go from $y=0$ to $y=-23$: shift down 23 units.

Answer:

right 5, down 23