Question

use transformations of the cube root function, ( f(x) = sqrt3{x} ), to graph the function ( h(x) = \frac{1}{2}sqrt3{x - 1} ). choose the correct graph below. \\( \bigcirc \\) a. \\( \bigcirc \\) b. \\( \bigcirc \\) c. \\( \bigcirc \\) d.

Explanation:

Step1: Analyze the transformation of the cube - root function

The parent function is \(y = \sqrt[3]{x}\). For the function \(h(x)=\frac{1}{2}\sqrt[3]{x - 1}\), compared to \(y=\sqrt[3]{x}\), there is a horizontal shift and a vertical compression.
The transformation rule for \(y = f(x - a)\) is a horizontal shift of \(y = f(x)\) to the right by \(a\) units. Here \(a = 1\), so the graph of \(y=\sqrt[3]{x}\) is shifted to the right by 1 unit.
The transformation rule for \(y=cf(x)\) (\(0\lt c\lt1\)) is a vertical compression of \(y = f(x)\) by a factor of \(c\). Here \(c=\frac{1}{2}\), so the graph of \(y=\sqrt[3]{x-1}\) is vertically compressed by a factor of \(\frac{1}{2}\).

Step2: Check the key point of the transformed function

For the parent function \(y = \sqrt[3]{x}\), when \(x = 0\), \(y = 0\).
For the function \(h(x)=\frac{1}{2}\sqrt[3]{x - 1}\), when \(x=1\), \(h(1)=\frac{1}{2}\sqrt[3]{1 - 1}=0\).

Answer:

C.