Chapter 2.5, Problem 54AYU

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.

54. $g\left(x\right)\text{ = 3}\left|x\text{ + 1}\right|\text{ - 3}$

To determine

To graph: The function $f\left(x\right)=3\left|x+1\right|-3$, 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 54AYU

Domain of the function $f\left(x\right)=3\left|x+1\right|-3$ is $\left(-\infty ,\infty \right)$.

Range of the function $f\left(x\right)=3\left|x+1\right|-3$ is $\left[-3,\infty \right)$.

Explanation of Solution

Given:

$f\left(x\right)=3\left|x+1\right|-3$

Graph:

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

Step 1: The function $f\left(x\right)=\left|x\right|$ is the absolute value function.

$f\left(x\right)=\left|x\right|$     Absolute value function

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

$f\left(x\right)=\left|x+1\right|$     replace $x$ by $x+1$; Horizontal shift left 1 unit.

Step 3: To obtain the graph of $f\left(x\right)=3\left|x+1\right|$, multiply each $y\text{-coordinate}$ of the graph of $f\left(x\right)=\left|x+1\right|$, by 3, that it is vertically stretched by the factor of 3.

$f\left(x\right)=3\left|x+1\right|$     Multiply by 3, vertically stretched by a factor of 3

Step 4: To obtain the graph of $f\left(x\right)=3\left|x+1\right|-3$, subtract 3 from each $y\text{-coordinate}$ on the graph of $f\left(x\right)=3\left|x+1\right|$, that it is shifted down 3 units.

$f\left(x\right)=3\left|x+1\right|-3$     subtract 3, vertically shift down 3 units.

Interpretation:

Domain of the function $f\left(x\right)=3\left|x+1\right|-3$ is $\left(-\infty ,\infty \right)$.

Range of the function $f\left(x\right)=3\left|x+1\right|-3$ is $\left[-3,\infty \right)$.