Question

the graph of g(x) is a translation of the function f(x) = x². the vertex of g(x) is located 5 units above and 7 units to the right of the vertex of f(x). which equation represents g(x)?
○ g(x) = (x + 7)² + 5
○ g(x) = (x − 7)² + 5
○ g(x) = (x + 5)² + 7
○ g(x) = (x − 5)² + 7

Explanation:

Step1: Recall vertex form of parabola

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

Step2: Apply horizontal translation

A shift 7 units right changes $h$ to $7$, so the term becomes $(x - 7)^2$.

Step3: Apply vertical translation

A shift 5 units up changes $k$ to $5$, so we add 5 to the horizontal translation term.

Answer:

$g(x)=(x - 7)^2 + 5$ (the second option)