Chapter 2.5, Problem 21E

To determine

To graph: the function $f\left(x\right)=x^2-1$ using the graph of a standard function and then applying transformations.

Explanation of Solution

Given information:

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

Graph:

Start with the standard function graph $f(x)=x^2$ .

To plot the graph of $f(x)=x^2$ ,use the plotting points method, substitute different value of $x$ on $f(x)=x^2$ to determine corresponding value of $f(x)$ .

The table is thus obtained as:

x	-3	-2	-1	1	2	3
y	9	4	1	1	4	9

now plot each point of $x$ along the horizontal axis and the corresponding value of $f(x)=y$ along vertical axis and connect each point to form a smooth curve.

The graph of $f(x)=x^2$ is obtained as:

  

Now, the graph of $f\left(x\right)=x^2-1$ is the graph of $f(x)=x^2$ shifted $1$ unit downwards.

The graphof $f\left(x\right)=x^2-1$ is obtained as:

  

Interpretation:

From the above graph it can be obtained that the graph of $f\left(x\right)=x^2-1$ is obtained by shifting the graph of $f(x)=x^2$

  $1$ unit downwards.