Q. 39 - 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. 

$f\left(x\right)=x^2-1$

Step-by-Step Solution

Answer

The graph of the function $f\left(x\right)=x^2-1$is given as:

The domain of the function is $(-\infty ,\infty )$ and the range is $[-1,\infty )$

Step 1. given data

The given function is $f\left(x\right)=x^2-1$

Step 2. Graph the parent function.

The parent function of the given function is $g\left(x\right)=x^2$

Plot the graph of the function$g\left(x\right)=x^2$

Step 3. Graph of the function

Graph of function $f\left(x\right)=x^2-1$can be obtained by shifting the graph of the function$g\left(x\right)=x^2$ towards down by one unit.

The graph of the function is given as: 

Step 4. domain and range of the function

From the graph, we infer that 

The domain of the function is $(-\infty ,\infty )$

The range of the function is  $[-1,\infty )$

Step 5. Verification

plot the graph of the function$f\left(x\right)=x^2-1$ using a graphing utility 

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