which translation maps the vertex of the graph of the function $f(x)=x^{2}$ onto the vertex of the function…

which translation maps the vertex of the graph of the function $f(x)=x^{2}$ onto the vertex of the function $g(x)=x^{2}-10x + 2$?\nright 5, down 23\nleft 5, down 23\nright 5, up 27\nleft 5, up 27

which translation maps the vertex of the graph of the function $f(x)=x^{2}$ onto the vertex of the function $g(x)=x^{2}-10x + 2$?\nright 5, down 23\nleft 5, down 23\nright 5, up 27\nleft 5, up 27

Answer

Explanation:

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

The vertex - form of a quadratic function is $y = a(x - h)^{2}+k$, where $(h,k)$ is the vertex. For $f(x)=x^{2}=(x - 0)^{2}+0$, the vertex is $(0,0)$.

Step2: Rewrite $g(x)=x^{2}-10x + 2$ in vertex - form

Complete the square for $g(x)$. [ \begin{align*} g(x)&=x^{2}-10x + 2\ &=(x^{2}-10x)+2\ &=(x^{2}-10x + 25-25)+2\ &=(x - 5)^{2}-25 + 2\ &=(x - 5)^{2}-23 \end{align*} ] The vertex of $g(x)$ is $(5,-23)$.

Step3: Determine the translation

To get from $(0,0)$ to $(5,-23)$, we move 5 units to the right (because the $x$ - coordinate changes from 0 to 5) and 23 units down (because the $y$ - coordinate changes from 0 to - 23).

Answer:

right 5, down 23