Q. 45 - Precalculus Enhanced with Graphing Utilities

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. 

$f\left(x\right)={\left(x-1\right)}^3+2$

Step-by-Step Solution

Answer

The graph of the function is given as: 

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

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 - 1 3

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

Similarly, for $h\left(x\right)={(x-1)}^3$, the graph $g\left(x\right)=x^3$ is shifted to the right  by $1$ unit.

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 upwards by $k$ units.

Similarly, for $f\left(x\right)={(x-1)}^3+2$, the graph $h\left(x\right)={(x-1)}^3+2$ is shifted upwards by $2$ units.

The graph of the given function is given as:

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.