Q. 60 - Precalculus Enhanced with Graphing Utilities

Question

In Problems $39-68$, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, $y=x^2$) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.Verify your results using a graphing utility.

$g\left(x\right)=3\left|x+1\right|-3$

Step-by-Step Solution

Answer

The graph of the function $g\left(x\right)=3\left|x+1\right|-3$ is:

The domain of the function is the set of all real numbers and the range is $\{y\in \mathrm{ℝ}:y\ge -3\}$

Step 1. Given

The function $g\left(x\right)=3\left|x+1\right|-3$

To graph the function and to find its domain and range.

Step 2. Graph the basic function

Graph the basic function $y=\left|x\right|$

Step 3. Replace x with x + 1

Replace $y=\left|x\right|$ by

              $y=\left|x+1\right|$ so that the graph shifted horizontally left to one unit.

Step 4. Multiply the right side by 3

Multiply the graph $y=\left|x+1\right|$ by $3$, so that the graph stretched vertically by the factor $3$.

Step 5. Subtract the right side by 3

Replace $y=3\left|x+1\right|$ by

              $y=3\left|x+1\right|-3$, so the graph shifted vertically downward by $3$

Step 6. Find domain and range

The domain of the function is the set of all real numbers.

And the range of the function is $\{y\in \mathrm{ℝ}:y\ge -3\}$

Step 7. Verify the graph

Verify the graph by using graphing utility.

The graph of the function using graphing utility is: