Chapter 3, Problem 8CLT

a .

To determine

To compare: the graphs of the functions $y=\sqrt[3]{x}$ and $r\left(x\right)=\frac{1}{2}\sqrt[3]{x}$ .

Answer to Problem 8CLT

The graph of $r\left(x\right)$ is the graph of $y\left(x\right)$ which is vertically shrinked by a factor of $\frac{1}{2}$ .

Explanation of Solution

Concept used:

Vertical transformation:

Relative to the graph of $f\left(x\right)$ , the transformation of graph of $g\left(x\right)=cf\left(x\right)$ is a vertical shrink by a factor of $c$ when $0<c<1$ and is a vertical stretch by a factor of $c$ when $c>1$ .

It can be observed that $r\left(x\right)=\frac{1}{2}y\left(x\right)$ which is of the form $g\left(x\right)=cf\left(x\right)$ , so it can be said that the graph of r is a vertical shrink of the graph of y by a factor of $\frac{1}{2}$ .

b.

To determine

To compare: the graphs of the functions $y=\sqrt[3]{x}$ and $h\left(x\right)=\sqrt[3]{x}+2$ .

Answer to Problem 8CLT

The graph of $h\left(x\right)$ is the graph of $y\left(x\right)$ which is vertically shifted 2 units upward.

Explanation of Solution

Concept used:

Vertical translation:

The graph of $g\left(x\right)=f\left(x\right)+k$ is a vertical translation of the graph of f.

The vertical shift is k units up if $k>0$ and k units down if $k<0$ .

It can be observed that $h\left(x\right)=y\left(x\right)+2$ which is of the form $g\left(x\right)=f\left(x\right)+k$ , so it can be said that relative the graph of y, the graph of h is a vertical shift 2 units upwards.

c.

To determine

To compare: the graphs of the functions $y=\sqrt[3]{x}$ and $g\left(x\right)=\sqrt[3]{x+2}$ .

Answer to Problem 8CLT

The graph of $g\left(x\right)$ is the graph of $y\left(x\right)$ which is horizontally shifted 2 units towards left.

Explanation of Solution

Concept used:

Horizontal translation:

The graph of $g\left(x\right)=f\left(x+h\right)$ is a horizontal translation of the graph of f.

The horizontal shift is h units’ left if $h>0$ and h units’ right if $h<0$ .

It can be observed that $g\left(x\right)=y\left(x+2\right)$ which is of the form $g\left(x\right)=f\left(x+h\right)$ , so it can be said that relative the graph of y, the graph of g is a horizontal shift 2 units left.