Question

use transformations of the cube root function, $f(x)=\sqrt3{x}$, to graph the function $h(x)=\frac{1}{3}\sqrt3{x - 5}$. choose the correct graph below. \\(\bigcirc\\) a. \\(\bigcirc\\) b. \\(\bigcirc\\) c. \\(\bigcirc\\) d.

Explanation:

Step1: Analyze the transformation of the function

The parent function is \(y = \sqrt[3]{x}\). For the function \(h(x)=\frac{1}{3}\sqrt[3]{x - 5}\), there is a horizontal shift and a vertical compression.
The transformation \(y=\sqrt[3]{x}\to y=\sqrt[3]{x - 5}\) is a horizontal shift of 5 units to the right (using the rule \(y = f(x - c)\) is a shift of \(c\) units to the right for \(c>0\)).
The transformation \(y=\sqrt[3]{x-5}\to y=\frac{1}{3}\sqrt[3]{x - 5}\) is a vertical compression by a factor of \(\frac{1}{3}\) (using the rule \(y=af(x)\) with \(0 < a<1\) is a vertical compression).

Step2: Check key points

For the parent function \(y = \sqrt[3]{x}\), when \(x = 0\), \(y = 0\); when \(x = 1\), \(y = 1\); when \(x=-1\), \(y=-1\).
For the function \(h(x)=\frac{1}{3}\sqrt[3]{x - 5}\), when \(x = 5\), \(h(5)=\frac{1}{3}\sqrt[3]{5 - 5}=0\); when \(x=6\), \(h(6)=\frac{1}{3}\sqrt[3]{6 - 5}=\frac{1}{3}\); when \(x = 4\), \(h(4)=\frac{1}{3}\sqrt[3]{4 - 5}=-\frac{1}{3}\).

Answer:

The graph of \(h(x)=\frac{1}{3}\sqrt[3]{x - 5}\) is the graph of \(y = \sqrt[3]{x}\) shifted 5 units to the right and vertically compressed by a factor of \(\frac{1}{3}\). Without seeing the exact details of the graphs A - D, but based on the transformation rules:

• Option D (assuming it has the graph shifted 5 units to the right and vertically compressed) is the correct answer.