Question

\\y = -\frac{1}{4}\sqrt3{x + 2} - 5\\

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Identify the parent function

The parent function is the cube root function:

$$ y = \sqrt[3]{x} $$

Step 2: Identify horizontal translation

The term inside the radical is \( x + 2 \).

$$ x \to x + 2 $$

This shifts the graph left by \( 2 \) units.

Step 3: Identify vertical and horizontal reflections and scaling

The factor multiplying the radical is \( -\frac{1}{4} \).

• The negative sign reflects the graph vertically across the \( x \)-axis.
• The factor of \( \frac{1}{4} \) vertically compresses the graph by a factor of \( \frac{1}{4} \).

Step 4: Identify vertical translation

The constant term outside the radical is \( -5 \).
This shifts the graph down by \( 5 \) units.

Answer:

The function \( y = -\frac{1}{4}\sqrt[3]{x + 2} - 5 \) is obtained by transforming the parent function \( y = \sqrt[3]{x} \) through the following steps:

1. Horizontal Shift: Left by \( 2 \) units.
2. Reflection: Vertically across the \( x \)-axis.
3. Vertical Compression: By a factor of \( \frac{1}{4} \).
4. Vertical Shift: Down by \( 5 \) units.