Lesson 2-6: Graphs of Absolute Value Equations

Where we’re headed today…

Today we’re going to take the next graphing step…we’ll learn how to graph absolute value equations. Here are the three things you are going to need to be able to do:

Match an absolute value equation with its graph.
Find the value of its vertex.
Sketch the graph of an absolute value equation.

First things first though…we need to understand what the graph of an absolute value equation looks like. It is very distinctive!

What do these things look like?

The simplest absolute value equation in two variables is. In order to understand its graph, please do the following:

Graph the equation. Compare your graph with the one at this link (or go to page 5).
Now, graph the absolute value equation. To do so, fill out a T-chart; I’d suggest using the following values for x: -2, -1, 0, 1, & 2. Compare your graph with the one at this link (or go to page 6).
Next, consider these two questions:
How does the graph fordiffer from the graph of?
How are the two graphs similar?

How are the graphs different?

Well, for negative values of x, the absolute value graph looks like it bends back up at the origin; on the left of the y-axis, the line comes back up. It almost looks like we took the graph and bent it at the origin into a big V.

How are the graphs similar?

On the right side of the y-axis (positive values of x) the graphs look identical.

The vertex

The bend in the absolute value equation graph is at what we call the vertex. The vertex is a point. For the equationwhat is the vertex? It is (0, 0). In a bit, I’ll tell you how you can figure out where the vertex is just by looking at the equation.

Will an absolute value equation always look like that?

You may wonder if all absolute value equations are a V opening up like the one for.

The simple answer is absolute value equation graphs always are a V but they may be:

an up-side-down V
shifted up or down
shifted left or right
wider or narrower

The cool thing is, when you learn a few tricks, you can simply look at the equation and basically predict how it will graph out! Let me show you.

Graph

Build a T-chart using the same x values from before: -2, -1, 0, 1, & 2. To see the graph, follow this link or go to page 7.

How does it differ from the graph for? It is flipped up-side-down.

What is different about the equation? The coefficient before the absolute value is a negative.

So, a negative before the absolute value flips the graph up-side-down.

Graph

Again, build a T-chart using the same x values from before: -2, -1, 0, 1, & 2. To see the graph, follow this link or go to page 8.

How does it differ from the graph of? It is shifted up one.

What is different about the equation? We’re adding one to the absolute value.

What do you suppose will happen if you subtract one from the absolute value? It will shift down by one.

So, add a number outside the absolute value to shift up that amount. Subtract to shift down.

Graph

Again, build a T-chart using the same x values from before: -2, -1, 0, 1, & 2. To see the graph, follow this link or go to page 9.

How does it differ from the graph of? It is shifted left one.

What is different about the equation? We are adding one inside the absolute value.

What do you suppose will happen if you subtract one inside the absolute value? It will shift right by one.

So, add a number inside the absolute value to shift left that amount. Subtract to shift right.

How to tell where the vertex will be

To figure out where the vertex will be, take everything inside the absolute value, set it equal to zero and solve for x. This will give you the x coordinate of the vertex. Here is an example:

… solve for x. You get x = 2. The vertex is at x = 2.

To find the y-coordinate, plug that value of x back into the equation and solve for y.

Problems 9-19 in the homework assignment will practice finding the vertex.

Cookbook: How to quick graph an absolute value equation

Find the vertex
Determine if the graph will be:
Flipped (up-side-down if negative before the absolute value)
Shifted up or down (number added or subtracted outside the absolute value)
Shifted left or right (number added or subtracted inside the absolute value)
Build a T-chart of a few values each side of the vertex’s x-coordinate value.

Problems 5-8 and 21-24 will practice matching absolute value equations to their graphs.

Lesson 2-6: Graphs of Absolute Value Equations