Question

reflecting the cubic function practice
complete this assessment to review what youve learned. it will not count toward your g
if $f(x) = x^3$, then what is the effect on its graph when $f(x)$ is replaced by $f(-3x)$?
option #1: a horizontal compression and reflection over the $y$-axis
option #2: a horizontal stretch and reflection over the $y$-axis
option #3: a vertical compression and reflection over the $x$-axis
option #4: a vertical stretch and reflection over the $x$-axis
(1 point)

Explanation:

Step1: Recall Transformations

For a function \( y = f(x) \), replacing \( x \) with \( -x \) reflects over the \( y \)-axis. Replacing \( x \) with \( kx \) ( \( |k|>1 \)) horizontally compresses the graph (since horizontal compression/stretch is by factor \( \frac{1}{|k|} \), so \( k = - 3 \), \( |k| = 3>1 \), horizontal compression by \( \frac{1}{3} \)).

Step2: Analyze \( f(-3x) \)

Given \( f(x)=x^{3} \), \( f(-3x)=(-3x)^{3}=-27x^{3} \). But the transformation from \( f(x) \) to \( f(-3x) \) is a horizontal transformation (since the change is in the input \( x \)). The \( - \) in \( -3x \) reflects over the \( y \)-axis, and the \( 3 \) ( \( |3|>1 \)) causes horizontal compression (because \( x \) is multiplied by a number with absolute value greater than 1, so the graph is compressed horizontally by \( \frac{1}{3} \)).

Answer:

Option #1: a horizontal compression and reflection over the \( y \)-axis