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

Explanation:

Step1: Identify the vertex form of a quadratic function

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

Step2: Determine the vertex of $g(x)$

From the graph, the vertex of the parabola $g(x)$ is at the point $(- 2,0)$. So, $h=-2$ and $k = 0$.

Step3: Determine the value of $a$

Since the shape of the parabola of $g(x)$ is the same as that of $f(x)=x^{2}$, the value of $a$ remains $1$ (no vertical stretch or compression).

Step4: Write the function rule for $g(x)$

Substitute $a = 1$, $h=-2$, and $k = 0$ into the vertex - form $y=a(x - h)^2 + k$. We get $g(x)=1\times(x-(-2))^{2}+0=(x + 2)^{2}$.

Answer:

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