Question

the graph of the parent function (y = x^3) is horizontally stretched by a factor of (\frac{1}{5}) and reflected over the (y)-axis. what is the equation of the transformed function?

(y = left(\frac{1}{5}x
ight)^3)

(y = left(-\frac{1}{5}x
ight)^3)

(y = (5x)^3)

(y = (-5x)^3)

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Apply the horizontal stretch

A horizontal stretch or compression of a function \( f(x) \) is represented by replacing \( x \) with \( b \cdot x \), where the horizontal stretch factor is \( \frac{1}{b} \).

Given a horizontal stretch by a factor of \( \frac{1}{5} \):

$$ \frac{1}{b} = \frac{1}{5} \implies b = 5 $$

Replacing \( x \) with \( 5x \) in the parent function \( y = x^3 \):

$$ y = (5x)^3 $$

Step 2: Apply the reflection over the y-axis

A reflection over the \( y \)-axis is represented by replacing \( x \) with \( -x \).

Replacing \( 5x \) with \( -5x \):

$$ y = (-5x)^3 $$

Answer:

\( y = (-5x)^3 \)