Question

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

Explanation:

Step1: Recall Translation Rules

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

Step2: Apply Translation to (0,0)

Original point: \( (0,0) \). Horizontal translation: \( x \)-coordinate: \( 0-3=-3 \) (since left 3 units, subtract 3). Vertical translation: \( y \)-coordinate: \( 0-4=-4 \) (since down 4 units, subtract 4). So the new point is \( (-3,-4) \).

Answer:

D. \((-3, -4)\) (assuming the last option is D, as the options are \((3, 4)\), \((3, -4)\), \((-3, 4)\), \((-3, -4)\))