which translation maps the graph of the function f(x) = x² onto the function g(x) = x² + 2x + 6?\nleft 1…

Explanation:

Step1: Rewrite $g(x)$ in vertex - form

Complete the square for $g(x)=x^{2}+2x + 6$. [ \begin{align*} g(x)&=x^{2}+2x+6\ &=(x^{2}+2x + 1)-1 + 6\ &=(x + 1)^{2}+5 \end{align*} ]

Step2: Recall the rules of function translation

For a quadratic function $y = f(x)=x^{2}$, the transformation to $y=(x - h)^{2}+k$ is a horizontal translation of $h$ units (left if $h>0$, right if $h < 0$) and a vertical translation of $k$ units (up if $k>0$, down if $k < 0$). For $g(x)=(x + 1)^{2}+5$, comparing with $y=(x - h)^{2}+k$, we have $h=-1$ and $k = 5$. So the graph of $f(x)=x^{2}$ is translated 1 unit to the left and 5 units up to get the graph of $g(x)$.

Answer:

left 1 unit, up 5 units