Question

the graph of the parent function $f(x) = x^3$ is translated to form $g(x) = (x - 7)^3 + 9$. the point $(0, 0)$ on the graph of $f(x)$ corresponds to which point on the graph of $g(x)$?
○ $(7, 9)$
○ $(-7, 9)$
○ $(9, 7)$
○ $(9, -7)$

Explanation:

Step1: Recall translation rules

For a function \( y = f(x - h)+k \), the graph is translated \( h \) units to the right (if \( h>0 \)) and \( k \) units up (if \( k>0 \)). Here, \( f(x)=x^{3} \), \( g(x)=(x - 7)^{3}+9 \), so \( h = 7 \) (right translation) and \( k=9 \) (up translation).

Step2: Apply translation to the point \((0,0)\)

To find the corresponding point on \( g(x) \), we apply the translation to the \( x \)-coordinate and \( y \)-coordinate of the point \((0,0)\) on \( f(x) \).

• For the \( x \)-coordinate: Since we translate 7 units to the right, we add 7 to the original \( x \)-coordinate. So \( 0+7 = 7 \).
• For the \( y \)-coordinate: Since we translate 9 units up, we add 9 to the original \( y \)-coordinate. So \( 0 + 9=9 \).

Answer:

A. \((7, 9)\)