Chapter 2.5, Problem 39AYU

In Problems 39-62, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Stan with the graph of the basic function (for example, $y\text{ = }{x}^2$) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.

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

To determine

To graph: The function $f\left(x\right)=x^2-1$, 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.

Answer to Problem 39AYU

Domain of the function $f\left(x\right)=x^2-1$ is $\left(-\infty ,\infty \right)$.

Range of the function $f\left(x\right)=x^2-1$ is $[-1,\infty )$.

Explanation of Solution

Given:

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

Graph:

Now use the following steps to obtain the graph of $f\left(x\right)$.

Step 1: The function $f\left(x\right)=x^2$ is the square function.

$f\left(x\right)=x^2$     square function

Step 2: To obtain the graph of $f\left(x\right)=x^2-1$, subtract 1 from each $y\text{-coordinate}$ on the graph of $f\left(x\right)=x^2$, that it is shifted down 1 unit.

$f\left(x\right)=x^2-1$     Subtract 1: vertical shift down 1 unit

Interpretation:

Domain of the function $f\left(x\right)=x^2-1$ is $\left(-\infty ,\infty \right)$.

Range of the function $f\left(x\right)=x^2-1$ is $[-1,\infty )$.