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 $g(x)$?

$\bigcirc\\ (4, -7)$
$\bigcirc\\ (-4, -7)$
$\bigcirc\\ (4, 7)$
$\bigcirc\\ (-4, 7)$

Explanation:

Step1: Recall translation rules

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

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

To find the corresponding point on \( g(x) \), we shift the \( x \)-coordinate of \((0,0)\) by \( h = 4 \) (right) and \( y \)-coordinate by \( k=-7 \) (down). So new \( x = 0+4 = 4 \), new \( y = 0+(-7)=-7 \).

Answer:

A. \((4, -7)\)