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The graph of the parent function [tex]$y=x^3$[/tex] is horizontally stretched by a factor of [tex]$\frac{1}{5}$[/tex] and reflected over the [tex]y[/tex]-axis. What is the equation of the transformed function?

A. [tex]y=(5 x)^3[/tex]
B. [tex]y=(-5 x)^3[/tex]
C. [tex]y=\left(\frac{1}{5} x\right)^3[/tex]
D. [tex]y=\left(-\frac{1}{5} x\right)^3[/tex]

Sagot :

GinnyAnsweridn
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]
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