Chapter 3, Problem 6VE

Solution

To describe how the graph of $y=\sqrt[3]{x-4}+5$ is related to the graph of the parent function $y=\sqrt[3]{x}$ .

Given information:

A function and its inverse.

Concept used:

Transformation of functions: For any positive integer a , we have

• $y=f\left(x+a\right)$ is the graph of $y=f\left(x\right)$ shifted a units left.
• $y=f\left(x-a\right)$ is the graph of $y=f\left(x\right)$ shifted a unit right.
• $y=f\left(x\right)+a$ is the graph of $y=f\left(x\right)$ shifted a unit up.
• $y=f\left(x\right)-a$ is the graph of $y=f\left(x\right)$ shifted a unit down.
• $y=af\left(x\right)$ is the graph of $y=f\left(x\right)$ stretches a unit in y direction if $a>1$ and compresses a unit in y direction if $0<a<1$
• $y=f\left(ax\right)$ is the graph of $y=f\left(x\right)$ stretches a unit in x direction if $0<a<1$ and compresses a unit in x direction if $a>1$ .
• $y=-f\left(x\right)$ is the graph of $y=f\left(x\right)$ reflected about x axis.
• $y=f\left(-x\right)$ is the graph of $y=f\left(x\right)$ reflected about y axis.

Calculation:

The graph of the function $y=\sqrt[3]{x-4}$ obtained by shifting the graph of parent function $y=\sqrt[3]{x}$ to the right side by 4 units.

The on shifting the graph of the function $y=\sqrt[3]{x-4}$ in upward direction by 5 units, the resulting graph will be of the function $y=\sqrt[3]{x-4}+5$ .

Thus, the graph of the function $y=\sqrt[3]{x-4}+5$ is obtained by shifting the parent function to the right side by 4 units and then 5 units up.