the graph shows g(x), which is a translation of f(x) = x². write the function rule for g(x).

Explanation:

Step1: Recall translation rule

The general form of a horizontal translation of $y = f(x)$ is $y=f(x - h)$ and vertical translation is $y = f(x)+k$, where $(h,k)$ is the translation vector. The vertex of $f(x)=x^{2}$ is $(0,0)$. The vertex of $g(x)$ from the graph is $(- 4,0)$.

Step2: Determine the translation

Since the vertex of $f(x)=x^{2}$ moves from $(0,0)$ to $(-4,0)$, it is a horizontal - translation of 4 units to the left. For a horizontal translation of $y = f(x)$ to the left by $a$ units, the new function is $y=f(x + a)$. Here $a = 4$.

Step3: Write the function rule

Substitute $x$ with $x + 4$ in $f(x)=x^{2}$. So $g(x)=(x + 4)^{2}$.

Answer:

$g(x)=(x + 4)^{2}$