Question

use transformations of the standard cubic function,
$f(x)=x^{3}$, to graph the function $g(x)=x^{3}-6$
what transformation is needed to graph the function
$g(x)=x^{3}-6$? choose the correct answer below.
a. the graph of $f(x)=x^{3}$ should be shifted 6 units to
the right.
b. the graph of $f(x)=x^{3}$ should be shifted 6 units to
the left.
c. the graph of $f(x)=x^{3}$ should be shifted 6 units up.
d. the graph of $f(x)=x^{3}$ should be shifted 6 units
down.
use the graphing tool to graph the function.

Explanation:

To determine the transformation from \( f(x) = x^3 \) to \( g(x) = x^3 - 6 \), we use the rules of vertical shifts. For a function \( y = f(x) + k \), if \( k>0 \), the graph shifts up \( k \) units; if \( k<0 \), it shifts down \( |k| \) units. Here, \( g(x)=f(x)-6 \) (since \( k = -6 \)), so the graph of \( f(x) \) is shifted 6 units down. Option A is a horizontal shift right (for \( y = f(x - h), h>0 \)), Option B is horizontal shift left (for \( y = f(x + h), h>0 \)), and Option C is vertical shift up, which do not match.

Answer:

D. The graph of \( f(x)=x^3 \) should be shifted 6 units down.