Question

the vertex form of a function is $g(x) = (x - 3)^2 + 9$. how does the graph of $g(x)$ compare to the graph of the function $f(x) = x^2$?
\bigcirc $g(x)$ is shifted 3 units left and 9 units up.
\bigcirc $g(x)$ is shifted 3 units right and 9 units up.
\bigcirc $g(x)$ is shifted 9 units left and 3 units down.
\bigcirc $g(x)$ is shifted 9 units right and 3 units down.

Explanation:

Step1: Recall the vertex form shift rules

For a function in vertex form \( y = a(x - h)^2 + k \), compared to \( y = x^2 \), the graph shifts \( h \) units horizontally (right if \( h>0 \), left if \( h<0 \)) and \( k \) units vertically (up if \( k>0 \), down if \( k<0 \)).

Step2: Identify h and k in \( g(x) \)

In \( g(x)=(x - 3)^2 + 9 \), \( h = 3 \) (so shift 3 units right) and \( k = 9 \) (so shift 9 units up).

Answer:

B. \( g(x) \) is shifted 3 units right and 9 units up.