the graph of g(x) is a translation of the function f(x) = x². the vertex of g(x) is located 5 units above…

Explanation:

Step1: Recall vertex - form of 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. The vertex of $f(x)=x^{2}$ is $(0,0)$.

Step2: Determine the new vertex

The vertex of $g(x)$ is 5 units above and 7 units to the right of $(0,0)$. So the new vertex $(h,k)=(7,5)$.

Step3: Write the equation of $g(x)$

Substitute $h = 7$ and $k = 5$ into the vertex - form $y=a(x - h)^2+k$. Since $a = 1$ (no vertical stretch or compression as it is just a translation of $y=x^{2}$), we get $g(x)=(x - 7)^2+5$.

Answer:

$g(x)=(x - 7)^2+5$ (corresponding to the second option in the multiple - choice list)