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 $g(x)=-8x + x^{2}+7$?\nleft 4, down 9\nleft 4, up 23\nright 4, down 9\nright 4, up 23

which translation maps the vertex of the graph of the function $f(x)=x^{2}$ onto the vertex of the function $g(x)=-8x + x^{2}+7$?\nleft 4, down 9\nleft 4, up 23\nright 4, down 9\nright 4, up 23

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 $g(x)=-8x + x^{2}+7$ in vertex - form

Complete the square for $g(x)=x^{2}-8x + 7$. We know that $x^{2}-8x=(x - 4)^{2}-16$. So $g(x)=(x - 4)^{2}-16 + 7=(x - 4)^{2}-9$. The vertex of $g(x)$ is $(4,-9)$.

Step3: Determine the translation

To get from $(0,0)$ to $(4,-9)$, we move 4 units to the right (because the $x$ - coordinate changes from 0 to 4) and 9 units down (because the $y$ - coordinate changes from 0 to - 9).

Answer:

right 4, down 9