Question

Graph the function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function 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.

$f\left(x\right)={\left(x+2\right)}^3-3$

Short answer

The graph of the function is given as:

The domain and the range of the given function are all real numbers.

Step-by-step solution

Step 1. Graph the parent function.  

The parent function of the given function is $g\left(x\right)=x^3$.

So, the graph of $g\left(x\right)=x^3$ is given as:

Step 2. Graph the function h(x)=x+23

We know that for $y=f(x+k)$, the original graph is shifted to the left by $k$units.

Similarly, for $h\left(x\right)={\left(x+2\right)}^2$, the graph localid="1645807375470" $g\left(x\right)=x^3$is shifted to the left by localid="1645807379159" $2$ unis.

The graph is given as:

Step 3. Graph the given function 

We know that for,$y=f\left(x\right)-k$, the original graph is shifted downwards by $k$units.

Similarly, for $f\left(x\right)={\left(x+2\right)}^3-3$, the graph is shifted downwards by $3$units.

The graph of the given function is:

Step 4. Find the domain and range  

From the graph, we infer that

The domain and the range of the given function are all real numbers.

Step 5. Verify the graph 

The graph of the given function using graphing utility is given as:

The graph is the same as we have drawn. Hence, it is correct.