To find the equation of the transformed function, we need to apply two transformations to the parent function [tex]\( y = x^3 \)[/tex]. These transformations are:
1. A horizontal stretch by a factor of [tex]\(\frac{1}{5}\)[/tex].
2. A reflection over the [tex]\( y \)[/tex]-axis.
### Step 1: Horizontal Stretch
A horizontal stretch by a factor of [tex]\(\frac{1}{5}\)[/tex] means we replace [tex]\( x \)[/tex] with [tex]\( 5x \)[/tex] in the equation. Thus, the function becomes:
[tex]\[ y = (5x)^3 \][/tex]
### Step 2: Reflection Over the [tex]\( y \)[/tex]-Axis
A reflection over the [tex]\( y \)[/tex]-axis means we replace [tex]\( x \)[/tex] with [tex]\(-x\)[/tex]. Applying this to our already horizontally stretched function, we replace [tex]\( x \)[/tex] in [tex]\( y = (5x)^3 \)[/tex] with [tex]\(-5x\)[/tex]. Thus, the function becomes:
[tex]\[ y = (-5x)^3 \][/tex]
The correct equation of the transformed function is therefore:
[tex]\[ y = (-5x)^3 \][/tex]
Hence, the correct choice is:
[tex]\[ y = (-5x)^3 \][/tex]
