- In this lesson, you're going to learn how to graph absolute value functions with transformations. We're going to go through three examples. So let's dive in. The first thing you want to know is the basic shape of the parent function, this y equals absolute value of x graph. And the way you would get points for this graph is by picking a few negative, 0, and a few positives. So let's go ahead and do that. 

So negative 3, when we put it into between these absolute value bars, that tells us to make the quantity positive. It's like the distance from 0. And so this is going to be 3. If we put negative 2 in, we get the absolute value. That gives us positive 2. Negative 1 is 1. 0 is 0. 1 is 1. It's always going to be positive. Remember, it's that distance from 0 and so on. So if we plot these points, we're going to get negative 3, 3. Negative 2, 2. negative 1, 1. 0, 0. 1, 1. 2, 2. 3, 3. 

And it keeps going like that. And notice how we're getting this real sharp V, this real sharp corner here, as opposed to a quadratic, where it's got that U shape, that parabola shape. So this point here where the graph bends, this is referred to as the vertex. And we're going to be looking at that closer when we graph absolute value graphs that are in this form, y equals a absolute value of x minus h plus k. 

The h and the k is involved with shifting the graph left and right, up and down, and the a is involved with stretching the graph. So let's talk a little bit more about that. So you see this h here that's grouped with the x? This is going to shift it in the x-axis direction, like left and right. But you want to remember that this quantity here has actually the opposite effect on the graph. 

What I mean by that is if you had x minus 1, it looks like it's-- minus 1, you'd think it would go left 1. It actually goes positive 1. It goes the opposite way. Whereas over here, this k value, this is what's involved with shifting the graph up and down. If this is plus 2, it would actually go up 2. If it's minus 2, it would go down 2. Say, for example, this was absolute value of x plus 3. See, plus 3 you would think would be right 3, right? But it's actually the opposite. It goes to the left 3. 

Now, the a value, if it's greater than 1, it's going to be a vertical stretch, meaning that the graph is going to be narrower. It's like you're pulling it in a vertical direction. If a is between 0 and 1, we call it a vertical shrink or a vertical compress. It's going to make it wider, like that. And if it's negative, it's going to reflect it over the x-axis, meaning it's going to open down like that. So let's jump into the three examples, and let's practice. So for number 1, we have y equals the absolute value of x plus 1 minus 3. 

So the first thing I like to do is I like to find that vertex, that point where the graph bends. And this is going to shift left 1 and down 3. So left 1, down 3-- so you're going to be right there. That's your vertex. And then the a value here, you don't actually see an a value, which means that this is really like 1, because 1 times anything is itself. So we usually don't write the 1. But you can think of it as a 1 here in front. 

And 1 is like 1/1. Anything divided by 1 is itself. And the reason I do that is because you can think of this a value like the slope of a line. So if the slope is 1, that means from this point, we're going to go rise 1, run 1. And because this absolute value graph is symmetric about this axis of symmetry, if you fold it, it's going to map to itself. Because we went up 1 and right 1, we can also go up 1 and left 1. It's like I'm folding it over the vertex, like that. 

And we can continue. We can go up 1, over 1. And same thing here-- up 1, over 1, and up 1, over 1, and up 1, over 1, et cetera. And so you can see you're going to get that real nice V-shape absolute value graph. Now, if you're interested in finding the domain and range, the domain is what the x values can be. And you can see that this graph's going to the left and the right forever and ever. So the domain would be all real numbers. 

But the range, those are what the y values can be. And here, you can see that the lowest y can be is negative 3 or greater. So for the range, we'd say y is greater than or equal to negative 3. OK. Let's do another example. See if you can do this one. Number 2, it says y equals negative 2 times the absolute value of x minus 3 plus 4. 

So where is that vertex? Well, remember, this one here in the parentheses-- again, the absolute value bars, I should say-- has the opposite effect. The minus 3 is actually going to shift it right 3. And the plus 4 is going to have the same effect as that positive sign. It's going to go up positive 4. So we're going to go right 3, up 4-- 1, 2, 3, 4-- right about there. That's our vertex. 

But this a value is negative. So we know that the graph is going to be opening down, something like that. And the negative 2, we can think of like the slope. That's like negative 2/1. So from here, I'm going to go down 2-- 1, 2. And right 1. And I'm going to reflect it. So I could go down 2 and left 1. And I'll repeat that again-- down 2, left 1. From here, down 2, right 1. 

So you see how that slope-- but I'm then reflecting it over this axis of symmetry, like that. So here, on this one, again, the domain is all real numbers. The range is going to be y is less than or equal to 4. See, 4 is the highest or below. So we'll just write that down, positive 4. OK, one more example. If you're enjoying this video so far, be sure to check out my algebra 1 and my algebra 2 video courses, as well as my huge ACT Math review video course and my huge SAT math review video course. I'll put a link in the description below. 

But number 3, what we have is y equals 1/3 absolute value of x plus 1 minus 2. So what do you think for this one? Where is the vertex? Or you can think about it as, what's the transformation? Where's the graph being shifted or stretched? Well, we can see, this one that's grouped with the x-- we talked about how that shifts it in the x direction, but it has the opposite effect. So plus 1 is actually shifting it left 1. 

This one here has the same effect. It's going to go minus 2. That's down 2. That's going to put us right there. That's our vertex. And we can treat the a value like the slope, like the slope of a line, rise over run. We know it's positive, so the graph's going to open up like that. The 1/3, though, is between 0 and 1. So this is going to be like a vertical shrink or a vertical compression. It's going to make the graph wider. 

But let's go ahead and graph it. So from the vertex, I'm going to go up 1-- that's the rise-- run 3. 1, 2, 3. I can repeat that, up 1 and then 3 to the right. I could also go the other direction, rise 1 and go left 3. Same thing again-- rise 1, left 3. And that's because it's symmetric about the axis of symmetry. So the graph's going to look something like that. The domain is all real numbers. And the range is going to be y is greater than or equal to negative 2. And you got it. If you want to see another example, follow me over to that video right there, and I'll show you some more practice problems.