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

Explanation:

Step1: Find vertex of $f(x)$

The vertex - form of a quadratic function is $y = a(x - h)^2+k$, where $(h,k)$ is the vertex. For $f(x)=x^{2}$, $a = 1$, $h = 0$, $k = 0$. So the vertex of $f(x)$ is $(0,0)$.

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

We have $g(x)=x^{2}+2x + 1$. Using the perfect - square formula $(a + b)^2=a^{2}+2ab + b^{2}$, where $a=x$ and $b = 1$, we can rewrite $g(x)$ as $g(x)=(x + 1)^{2}$. For $g(x)=(x + 1)^{2}$, $a = 1$, $h=-1$, $k = 0$. So the vertex of $g(x)$ is $(-1,0)$.

Step3: Determine the translation

To get from $x = 0$ (the $x$ - coordinate of the vertex of $f(x)$) to $x=-1$ (the $x$ - coordinate of the vertex of $g(x)$), we need to move 1 unit to the left.

Answer:

left 1 unit