Absolute Value

In this lab you will learn about absolute values of numbers, and study graphs of functions involving absolute value. You will also learn how to solve equations involving absolute value.

Question 1. As you know, every real number except zero has an opposite, and two numbers that are opposites of each other are the same distance from zero. For example, −3 and 3 are opposites, and each is 3 units away from zero.

The absolute value of a number is the distance between that number and zero. For example, the absolute value of 3 is 3 and the absolute value of −3 is 3. The absolute value of 3 is written $|3|$, so ${|3|}=3$ and ${|-3|}=3$.

Use the number line above to find the absolute value of each number in this table.

number	absolute value
1
-1
2
-2

Question 2. What numbers are 7 units away from zero?

What numbers have an absolute value of 7?

If ${|x|}=7$, what are the possible values of $x$?

If ${|x|}=10$, what are the possible values of $x$?

The graph of $y={|x|}$

Question 3. Complete this table for the equation $y={|x|}$.

$x$	$y$
5
3
0
-3
-5

Click on to see the graph of $y={|x|}$. Are there any values that $y$ cannot have? Explain.

Question 4. Look at the graphs of $y=x$ and $y=-x$ shown below. Compare the graph of $y={|x|}$ shown on the grid to the left to these graphs.

If you only look at x-values that are greater than zero, does $y={|x|}$ look like $y=x$ or $y=-x$?

If you only look at x-values that are less than zero, does $y={|x|}$ look like $y=x$ or $y=-x$?

Question 5. Look at the graph of $y={|x|}$. On this graph, there are two points that have a y-value of 2: the points $(2,2)$ and $(-2,2)$. This means that if ${|x|}=2$, then $x$ can be either 2 or −2.

If ${|x|}=4$, what can $x$ be?

$x=$

If ${|x|}=0$, what can $x$ be?

$x=$

Can ${|x|}=-2$? Explain.

Question 6. Describe the shape and properties of the graph of $y={|x|}$ to someone who doesn't know about absolute values.

Solving equations involving absolute value

You have seen that if ${|x|}=3$, $x$ can be either 3 or −3 because both 3 and −3 are 3 units away from zero. In this section you will learn how to solve equations such as ${|x-2|}=3$. First, let's see what the graph of $y={|x-2|}$ looks like.

Question 7. Click on . Complete this table for $y={|x-2|}$.

$x$	$y$
5
4
2
0
-1

What do you notice about the numbers in the table? Describe any pattern(s) you see. Are there any value(s) that $y$ cannot have?

Click on to look at the graph of $y={|x-2|}$. What does the graph of $y={|x-2|}$ look like?

Question 8. On the graph of $y={|x-2|}$ there are two points that have a y-value of 3: the points $(-1,3)$ and $(5,3)$. This means that if ${|x-2|}=3$, then $x$ can be either −1 or 5. Use the graph to answer the following questions.

If ${|x-2|}=1$, what can $x$ be?

$x=$

If ${|x-2|}=0$, what can $x$ be?

$x=$

Can ${|x-2|}=-1$? Explain.

Question 9. Click on . The graph of $y={|x-2|}$ is shown in green and the graph of $w={|x|}$ is shown in red. Describe any similarities and differences between the two graphs.

You can also solve equations involving absolute value algebraically. You know that if ${|x|}=3$, then $x=3$ or $x=-3$. Similarly, if ${|x-2|}=3$, then $x-2$ must equal 3 or −3.

$$\table \colspan 3 {|x-2|}=3; x-2=3, \text"OR", x-2=-3; x=5, \text"OR", x=-1$$

Always check your solutions by substituting them back in the original equation:

Test $x=5$: ${|5-2|}={|3|}=3$. True.

Test $x=-1$: ${|-1-2|}={|-3|}=3$. True.

Question 10. Solve the equation ${|x+3|}=2$ algebraically as shown above. Test your solutions by substituting them in the original equation.

$x=$ or

More graphs involving absolute value

Question 11. Click on . Compare the graph of $y=-\,{|x|}$ (in blue) to the graph of $w={|x|}$ (in red).

Question 12. Click on . The purple line is the graph of $y=a\,{|x-h|}+k$. Use the sliders to find an equation for the graph shown below. The equation for your purple graph is shown below the grid.

When does $y$ have the value −2?

$x=$ or

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