Question

compared with the graph of the parent function, which equation shows only a vertical compression by a factor of \\(\frac{1}{3}\\) and a shift downward of 4 units? \\(y = \frac{1}{3}\sqrt3{x + 4}\\) \\(y = -\frac{1}{3}\sqrt3{x} - 4\\) \\(y = \frac{1}{3}\sqrt3{x} - 4\\) \\(y = -\frac{1}{3}\sqrt3{x} - 4\\)

Explanation:

Step1: Recall Transformations

For a cube root function \( y = \sqrt[3]{x} \), vertical compression by factor \( \frac{1}{3} \) is \( y=\frac{1}{3}\sqrt[3]{x} \), downward shift 4 units is subtracting 4: \( y=\frac{1}{3}\sqrt[3]{x}-4 \). No reflection (so no negative sign) and no horizontal shift (so no \( x\pm4 \) inside the root).

Step2: Analyze Options

• \( y=\frac{1}{3}\sqrt[3]{x + 4} \): Horizontal shift (left 4), not desired.
• \( y=-\frac{1}{3}\sqrt[3]{x}-4 \): Reflection (negative sign), not desired.
• \( y=\frac{1}{3}\sqrt[3]{x}-4 \): Matches compression and downward shift.
• \( y=-\frac{1}{3}\sqrt[3]{x}-4 \): Reflection, not desired.

Answer:

\( y = \frac{1}{3}\sqrt[3]{x}-4 \)