Q. 63 - 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.

$f\left(x\right)=-{(x+1)}^3-1$

Step-by-Step Solution

Answer

The graph of the function $f\left(x\right)=-{(x+1)}^3-1$ is:

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

And the range of the function is the set of all real numbers.

Step 1. Given

The function $f\left(x\right)=-{(x+1)}^3-1$

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

Step 2. Graph the basic function

Graph the basic function $f\left(x\right)=x^3$

Step 3. Replace x 3 with ( x + 1 ) 3

Replace $y=x^3$ by

              $y={(x+1)}^3$ so that the graph shifted horizontally left to $1$  unit.

Step 4. Multiply the right side by - 1

Multiply the graph $y={(x+1)}^3$ by $-1$, so that the graph reflected about $x-$axis.

Step 5. subtract the right side by 1

Replace $y=-{(x+1)}^3$ by

               $y=-{(x+1)}^3-1$  so that the graph shifted vertically downward by the factor $1$.

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 the set of all real numbers.

Step 7. Verify the graph

Verify the graph by using graphing utility.

The graph of the function using graphing utility is: