Chapter 3.2, Problem 7E

(a)

To determine

To sketch: The graph of the function $P\left(x\right)=x^3-8$ by transforming the graph of an appropriate function of the form $y=x^n$ and indicate all $x$ and $y$ intercepts on graph.

Explanation of Solution

Given information:

The given functions is $P\left(x\right)=x^3-8$ .

Graph:

The graph of the functions $P\left(x\right)=x^3-8$ is as follows.

  

Interpretation:

The graph of $P\left(x\right)=x^3-8$ is the graph of $y=x^3$ shifted downward 8 units.

(b)

To determine

To sketch: The graph of the function $Q\left(x\right)=-x^3+27$ by transforming the graph of an appropriate function of the form $y=x^n$ and indicate all $x$ and $y$ intercepts on graph.

Explanation of Solution

Given information:

The given functions is $Q\left(x\right)=-x^3+27$ .

Graph:

The graph of the functions $Q\left(x\right)=-x^3+27$ is as follows.

  

Interpretation:

The graph of $Q\left(x\right)=-x^3+27$ is the reflection of the graph of $y=-x^3$ in the $x$ -axis, which is shifted to the upward 27 units.

(c)

To determine

To sketch: The graph of the function $R\left(x\right)=-{\left(x+2\right)}^3$ by transforming the graph of an appropriate function of the form $y=x^n$ and indicate all $x$ and $y$ intercepts on graph.

Explanation of Solution

Given information:

The given functions is $R\left(x\right)=-{\left(x+2\right)}^3$ .

Graph:

The graph of the functions $R\left(x\right)=-{\left(x+2\right)}^3$ is as follows.

  

Interpretation:

The graph of $R\left(x\right)=-{\left(x+2\right)}^3$ is the reflection of the graph of $y=x^3$ in the x -axis, which is shifted to the left 2 units.

(d)

To determine

To sketch: The graph of the function $S\left(x\right)=\frac{1}{2}{\left(x-1\right)}^3+4$ by transforming the graph of an appropriate function of the form $y=x^n$ and indicate all $x$ and $y$ intercepts on graph.

Explanation of Solution

Given information:

The given functions is $S\left(x\right)=\frac{1}{2}{\left(x-1\right)}^3+4$ .

Graph:

The graph of the functions $S\left(x\right)=\frac{1}{2}{\left(x-1\right)}^3+4$ is as follows.

  

Interpretation:

The graph of $S\left(x\right)=\frac{1}{2}{\left(x-1\right)}^3+4$ is the graph of $y=x^3$ shifted to the right 1 unit and 4 units upward direction.