Absolute value graph

Solved Examples

Example 1

 

 

Frank took three positive \(x\) values to plot the graph of the function given by \(f_{1}(x)=|x+2|\)

The values are given as follows:

\(x\)	0	1	2
\(f_{1}(x)=|x+2|\)	2	3	4

He plotted these three points on a graph and joined the points.

Did he plot the graph of the function \(f_{1}(x)\) correctly?

Solution

No, Frank did not correctly plot the graph because the absolute value function takes into account the absolute values of numbers.

The function returns the non-negative values for every negative input.

Let's a few negative integers and evaluate the value of the function at those points.

	\(x<0\)			\(x\geq0\)
\(x\)	-5	-3	-1	0	1	2	3
\(f(x)=|x+2|\)	3	1	1	2	3	4	5

Now, plot these points on the graph and join them to obtain the required graph.

So, the correct graph of the function \(f_{1}(x)\) is shown.

Example 2

 

 

Can you help James plot the graph of \(f_{2}(x)=-|x+2|\) using the help of \(f_{1}(x)=|x+2|\)?

Solution

The function \(f_{2}(x)\) is the negative of the function \(f_{1}(x)=|x+2|\).

So, let's take the same points we used for the previous example and change the sign of each number in the last row.

	\(x<0\)			\(x\geq0\)
\(x\)	-5	-3	-1	0	1	2	3
\(f_{2}(x)=|x+2|\)	-3	-1	-1	-2	-3	-4	-5

Now, plot these points on the graph and join them to obtain the required graph.

So, we obtained the graph of \(f_{2}\).

Example 3

 

 

Plot the graph of \(f_{3}(x)=2|x-3|+4\)

Explain the shifts in the graph.

Solution

According to the definition of the absolute value function, if \(x-3\geq0\), then \(|x-3|=x-3\) and if \(x-3<0\), then \(|x-3|=-(x-3)\)

So, the function \(f_{3}(x)\) can be expressed as

\[2|x-3|+4=\left\{\begin{array}{r}
2x-2 \quad \text { if } x \geq 3 \\
10-x \quad \text { if } x<3
\end{array}\right.\]

The graph is shown below.

So, we obtained the graph of \(f_{3}\).

Example 4

 

 

Here is a function that uses the absolute value.

It is given as \(f_{4}(x)=\left\{\begin{matrix}\frac{|x|}{x},&\;x\neq0\\0,&\;x=0\end{matrix}\right.\)

This is the signum function.

Can you plot the graph of this function?

Solution

We already know that the absolute value function returns non-negative values for all real numbers.

When \(x> 0\), then we have \(f_{4}(x)=\frac{|x|}{x}=\frac{x}{x}=1\)

When \(x < 0\), then we have \(f_{4}(x)=\frac{|x|}{x}=-\frac{x}{x}=-1\)

So, the signum function can also be given as:

\[f_{4}(x)=\left\{\begin{matrix}1,&\;x>0\\0,&\;x=0\\-1,&\;x<0\end{matrix}\right.\]

Now, we can easily plot the graph.

Hence, we have a graph of the signum function.