Question

the graph of $f(x)=x^2$ is translated to form $g(x)=(x - 5)^2 + 1$. which graph represents $g(x)$?

Explanation:

Step1: Identify parent function vertex

The parent function $f(x)=x^2$ has vertex at $(0,0)$.

Step2: Apply horizontal translation

For $g(x)=(x-5)^2+1$, the $(x-5)$ term shifts the vertex 5 units right: new x-coordinate is $0+5=5$.

Step3: Apply vertical translation

The $+1$ term shifts the vertex 1 unit up: new y-coordinate is $0+1=1$.

Step4: Confirm new vertex

The vertex of $g(x)$ is $(5,1)$, and the parabola opens upward (same as parent function).

Answer:

The graph with vertex at $(5,1)$ opening upward (the option shown in the lower grid with vertex at the point 5 units right and 1 unit up from the origin) is the correct representation of $g(x)$.