Question

the graph of the parent function $f(x) = x^3$ is translated to form the graph of $g(x) = (x - 4)^3 - 7$. the point $(0, 0)$ on the graph of $f(x)$ corresponds to which point on the graph of $g(x)$?
○ $(4, -7)$
○ $(-4, -7)$
○ $(4, 7)$
○ $(-4, 7)$

Explanation:

Step1: Recall function translation rules

For a function \( y = f(x - h)+k \), the graph is translated \( h \) units right (if \( h>0 \)) and \( k \) units up (if \( k>0 \)). Here, \( g(x)=(x - 4)^{3}-7=f(x - 4)-7 \), so \( h = 4 \) (right translation) and \( k=-7 \) (down translation).

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

For the \( x \)-coordinate: \( 0+4 = 4 \) (since we translate 4 units right).
For the \( y \)-coordinate: \( 0-7=-7 \) (since we translate 7 units down).

Answer:

\( (4, -7) \)