We'll be upgrading our systems on Saturday, Jun 22, 2024 from 9:00 AM to 9:30 AM eastern time. Elephango may be unavailable for up to 3 minutes.
Contributor: Rebekah Brown. Lesson ID: 13430
What happens if you score a safety for the other team instead of a touchdown for your own? You have graphed the absolute value incorrectly! Learn the easy way not to make that mistake!
Nathan was so excited. He was going to score a touchdown!
He felt like he was flying, and no one from the other team could even get close to tackling him.
He was knocked out of his reverie, however, when he finally realized the crowd was not cheering him on but screaming, "Wrong way! Wrong way!"
He had run in the wrong direction and scored points for the other team. How embarrassing!
Football blooper! Player runs wrong way after fumble recovery, scores safety not touchdown from immaginevideo:
A football field is a good representation of the graph of absolute value. Keep reading to see how.
Look at the two types of graphs below. One is a linear function, and the other is an absolute value function:
You should be familiar with linear functions, like y = x.
(If you need to review, check out the video found under Additional Resources in the right-hand sidebar.)
You can think of the graph of an absolute value function as the graph of a straight line with the part below the x-axis broken off and reflected up across the x-axis. This results in a V-shaped graph with the vertex of the V touching the x-axis at (0,0).
You can think of the the vertex of the "V" as the 50-yard line on a football field.
Each side of the graph is pointing up just like each team's goal is to make progress toward their own end zone. The two sides of the graph are equal but opposite like the two end zones are equal distances from the center but face in opposite directions.
First, let's review the concept of absolute value.
Absolute value represents the distance from 0 to a number. For example:
|-4| = 4 because on a number line, -4 is four spaces away from 0.
|4| = 4 is also true because positive 4 is also four spaces away from 0.
In other words, absolute value will always result in the positive version of the number.
Let's look at an absolute value equation:
|x - 3| = 5
This can be written and solved two ways because x - 3 can equal 5 and -5 before you find the absolute value:
x - 3 = 5 | x - 3 = -5 | ||
x - 3 + 3 = 5 + 3 | x - 3 + 3 = -5 + 3 | ||
x = 8 | x = -2 |
So in this case, x can equal 8 or -2. Plug both numbers into the original equation to check!
(If you need more help understanding absolute value, visit our lesson found under Additional Resources in the right-hand sidebar.)
Now, let's look at an algebraic function using absolute value:
y = |x + 2|
From what we reviewed above, we know that x + 2 can equal a positive value or a negative value:
y = x + 2 | -y = x + 2 |
First, let's look at y = x + 2:
x + 2 needs to have a positive value, which would look like this:
x + 2 > 0
x > -2
Now, let's graph it!
We can find the x and y intercepts by using an input/output table:
x | y | |
0 | 2 | |
-2 | 0 |
Since we are dealing with absolute value, we found that we could only use the x values greater than -2. We can get rid of any part of the line that does not fall within that constraint:
Next, let's look at -y = x + 2.
x + 2 needs to have a negative value, which would look like this:
x + 2 < 0
x < -2
Before we graph this line, we need to simplify the equation and put it into standard y = mx + b form. Then, we can use an input/output table.
-y = x + 2
y = -(x+2)
y = -x - 2
x | y | |
0 | -2 | |
-2 | 0 |
Just like before, since we are dealing with absolute value, we have constraints to follow. The x values can only be less than -2.
The vertex is at (-2,0), which is where the two lines intersected. If you plug, (-2,0) into the absolute value function above, it would be true.
Now that you understand what absolute value functions are and how they are graphed, let's go over some other methods to help you interpret these graphs.
The Input/Output Method
We can graph any absolute value function with an input/output table and plotting points as we normally would.
A good choice for the x inputs in y = |x| would be -2, -1, 0, 1, and 2:
The Shift Method
The absolute value function is very easy to graph if we learn a few shortcuts!
Start with the idea of the basic y = |x| graph. Then, using shortcuts, we can shift the graph vertical or horizontal.
Shortcut 1: Numbers added to or subtracted from the x inside the absolute value bars indicate a left (+) or right (-) shift.
For example, look at y = |x - 2|.
A number is being subtracted inside the absolute value brackets signaling a horizontal shift. The minus sign (-) tells us to shift the vertex to the right. 2 tells us how many spaces to shift the vertx along the x-axis.
You can doublecheck the shortcut by plugging the vertex (2,0) or any other point into the function.
The graph for y = |x - 2| is pictured below. Notice the vertex is shifted to the right two units from (0,0):
Shortcut 2: Numbers added to or subtracted from the absolute value itself indicate an up (+) or down (-) shift.
For example, look at y = |x| - 2.
A number is being subtracted outside of the absolute value brackets singaling a vertical shift. The minus sign (-) tells us to shift the vertex down. 2 tells us to shift the vertx down 2 places along the y-axis.
You can doublecheck the shortcut by plugging the vertex (0,-2) or any other point into the function.
The graph for y = |x| - 2 is pictured below. Notice the vertex is shifted down two units from (0,0):
We can also have both kinds of shifts in one function as in y = |x - 1| + 2. Notice the vertex is shifted up two units and to the right one unit from (0,0):
To review the shift method, watch Graphing Absolute Value Functions - 1 from mathman1024:
When you think you are ready, head over to the Got It? section to try it out!