Chapter 2, Problem 10CT

Graph each function using the techniques of shifting, compressing or stretching, and reflecting. Start with the graph of the basic function and show all the steps.

$h(x)=-2{(x+1)}^3+3$

$g(x)=|x+4|+2$

To determine

To graph: The function $h(x)=-2{(x+1)}^3+3$ using techniques of shifting, compressing or stretching, and reflecting.

Explanation of Solution

Given Information:

The function is $h(x)=-2{(x+1)}^3+3$.

Graph:

To get the graph of $h(x)=-2{(x+1)}^3+3$, perform the transformations as follows:

Step 1: Begin with the graph of the base function $h(x)=x^3$.

The graph will look like:

Step 2: Shift the graph horizontally to the left by 1 units. It gives $h(x)={(x+1)}^3$.

For this the graph is:

Step 3: Reflect the graph over $x-axis$ to get $h(x)=-{(x+1)}^3$.

For this step the graph is:

Step 4: Stretch the graph vertically by 2 units, which gives the graph of $h(x)=-2{(x+1)}^3$.

The graph will be:

Step 5:Shift the graph vertically by 3 units upward. It will give the graph of $h(x)=-2{(x+1)}^3+3$.

The graph of the function $h(x)=-2{(x+1)}^3+3$ is as follows:

Interpretation:

The graph represents the function $h(x)=-2{(x+1)}^3+3$.

To determine

To graph: The function $g(x)=|x+4|+2$ using techniques of shifting, compressing or stretching, and reflecting.

Explanation of Solution

Given Information:

The function is $g(x)=|x+4|+2$.

Graph:

To get the graph of $g(x)=|x+4|+2$, perform the transformations as follows:

Step 1: Begin with the graph of the base function $g(x)=\left|x\right|$.

Step 2: Shift the graph horizontally to the left by 1 unit. It gives $g(x)=|x+4|$.

Step 3: Shift the graph vertically by 2 units upward. It will give the graph of $g(x)=|x+4|+2$.

The graph of the function $g(x)=|x+4|+2$ is as follows:

Interpretation:

The graph represents the function $g(x)=|x+4|+2$ .