Graphing Absolute Value Equations 06:25 minutes

Video Transcript

Transcript Graphing Absolute Value Equations

Mom is a real pool shark. She wants to teach her young son, Jimmy, everything she knows about the game. Let's take a look at how learning to graph absolute value functions can help you hit the balls into the pockets! First, let’s review a couple of concepts. We already know the absolute value of a number is the distance of the number from zero.

Also for the positive AND negative values of a number, the absolute value is the same. For example, the absolute value of negative one and positive one is the same: one. This number line shows the solution set for the |x| = 1. 'x' = ±1. And we know how to graph linear equations. Here's the graph of y = 3x + 2.

First absolute value function

But, when we graph the absolute value of a function, we get something totally different. Let’s start at the beginning with the simplest absolute value function, y = |x|. This is also known as the parent function for absolute value. Let's call this function "Mom". For 'x' is equal to 1, 2, 3, and so on, you get something that looks like a linear equation, but hold on, that's only one part of the graph.

Vertex and the axis of symmetry

Look what happens when 'x' = -1, -2, -3, and so on. The graph looks like a 'V'. All absolute value function graphs have a v-shape. That makes them easy to recognize, right? For absolute value graphs, the vertex is the lowest point of the graph and the axis of symmetry is the invisible vertical line that passes through the vertex.

Change the function

Let's change the Mom function and see what happens to the graph. We'll add one so that now, y = |x + 1|. Cool, the vertex moved one unit to the left, but otherwise, the graph stays the same. What do you think happens if you add 2 to 'x'? Did you guess correctly? Compared to the original "Mom" graph, the vertex moves two units to the left.


Let’s get a little crazy here and subtract two from 'x'… this time, the vertex moves two units to the right. D'you get the picture here? There's a pattern. Inside the absolute value bars, the opposite of the constant is the 'x' coordinate of the vertex.


Let’s change it up a bit. This time, add 1 outside the absolute value bars. How does the Mom graph change? The vertex moves up, while the rest of the graph stays the same. And, when you add two to the original "Mom" graph, the graph moves up another unit. What happens when you subtract two from the original graph? The graph moves down two units from the position of the original graph.

One last operation. Look what happens when you multiply the absolute value by two? The vertex stays the same, but the slope changes. The slope was one, and now it's two. What happens when you multiply by three? Oh look, the larger the multiplier, the more narrow the graph and the steeper the slope.


What about if we multiply by 1/2? Now the graph gets wider, and the slope gets flatter. Can you guess what happens when we multiply the "Mom" function by -2? Check out the graph. The v-shape flips itself over! Now that you understand how to graph absolute value functions, let’s get back to the pool table.

Check out the trajectory of the ball. The shape looks familiar. Right? It's a v-shape, just like the graph of an absolute value function. Let’s write an equation for the path of the ball. To do this, let's start with the Mom function, 'y' is equal to the absolute value of 'x', then add or subtract inside the absolute value bars, add or subtract outside the bars, or multiply the absolute value by a constant.

The path of the ball positions the vertex at point (6, 0). The vertex is 6 units to the right of the "Mom" graph, so inside the absolute value bars, we subtract 6 from 'x'. The 'y' coordinate indicates the graph has not moved up or down, so outside the absolute value bars, we don't need to add or subtract.

The v-shape

Now, take a look at the graph's v-shape. The v-shape is more narrow but open to the top, so we need to multiply the binomial inside the absolute value bars by a positive number. What's the slope? Check the rise over the run. It's up 2 and over 1, so the multiplier is 2. I’m not so sure if Jimmy was able to follow this explanation... He seems preoccupied with the cue chalk.