Question

use transformations of the cube root function, ( f(x)=sqrt3{x} ), to graph the function ( h(x)=\frac{1}{3} sqrt3{x + 1} ).

Explanation:

Step1: Horizontal shift

The function \(h(x)=\frac{1}{3}\sqrt[3]{x + 1}\) is obtained from \(y=\sqrt[3]{x}\) by a horizontal shift. For the transformation \(y = f(x + c)\) (where \(c>0\)), the graph of \(y = f(x)\) is shifted \(c\) units to the left. Here \(c = 1\), so the graph of \(y=\sqrt[3]{x}\) is shifted 1 unit to the left to get \(y=\sqrt[3]{x+1}\).

Step2: Vertical compression

For the transformation \(y=kf(x)\) (where \(0

Answer:

First, shift the graph of \(y = \sqrt[3]{x}\) 1 unit to the left to get \(y=\sqrt[3]{x + 1}\). Then, vertically compress the resulting graph by a factor of \(\frac{1}{3}\) to obtain the graph of \(h(x)=\frac{1}{3}\sqrt[3]{x+1}\).