# Graphing Absolute Value Equations 06:25 minutes

**Video Transcript**

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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**.

### Subtraction

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.

### Addition

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.

### Multiplication

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.

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