Questions: Completing the Square Pre-Test Active 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? g(x) is shifted 3 units left and 9 units up. g(x) is shifted 3 units right and 9 units up. g(x) is shifted 9 units left and 3 units down. g(x) is shifted 9 units right and 3 units down.

Solution Steps

To compare the graph of \( g(x) = (x-3)^2 + 9 \) with \( f(x) = x^2 \), we need to identify the transformations applied to \( f(x) \) to obtain \( g(x) \). The expression \( (x-3)^2 \) indicates a horizontal shift, and the \( +9 \) indicates a vertical shift. Specifically, \( (x-3)^2 \) means the graph is shifted 3 units to the right, and \( +9 \) means it is shifted 9 units up.

We have two functions:

• The standard quadratic function \( f(x) = x^2 \).
• The transformed function \( g(x) = (x-3)^2 + 9 \).

To determine how \( g(x) \) compares to \( f(x) \), we analyze the transformations:

• The term \( (x-3)^2 \) indicates a horizontal shift of \( 3 \) units to the right.
• The \( +9 \) indicates a vertical shift of \( 9 \) units up.

Thus, the graph of \( g(x) \) is obtained by shifting the graph of \( f(x) \) \( 3 \) units to the right and \( 9 \) units up.

Final Answer