Shifting absolute value graphs | Mathematics II | High School Math | Khan Academy

This right over here is the graph of y is equal to absolute value of x, which you might be familiar with. If you take x is equal to -2, the absolute value of that is going to be two. Negative -1, absolute value is one. Zero, absolute value is zero. One, absolute value is one, so on and so forth.

What I want to do in this video is think about how the equation will change if we were to shift this graph. So in particular, we're going to first think about what would be the equation of this graph if we shift three to the right and then think about how that will change if not only do we shift three to the right but we also shift four up. Shift four up.

And so once again, pause this video like we always say, and figure out what would the equation be if you shift three to the right and four up. All right, now let's do this together. So let's just first shift three to the right and think about how that might change the equation. So let's just visualize what we're even talking about.

So if we're going to shift three to the right, it would look like this. So that's what we first want to figure out the equation for. And so how would we think about it? Well, one way to think about it is something interesting is happening right over here at x equal 3. Before that, interesting thing was happening at x equal 0.

Now it's happening at x equal 3. The interesting thing that happens here is that you switch signs inside the absolute value. Instead of taking an absolute value of a negative, you're now taking the absolute value as you cross this point of a positive. That's why we see a switch in direction here of this line. You see the same thing happening right over here.

So at this point right over here, we know that our function, we know that our equation needs to evaluate out to zero. This is where it's going to switch signs. So we say, okay, well this looks like an absolute value, so it's going to have the form y is equal to the absolute value of something.

And so you say, okay, how do I, if x is three, how do I make that equal to zero? Well, I can subtract three from it. If I say y is equal to the absolute value of x - 3, well let's try it out. Let's see if it makes sense.

So when x is equal to 3, 3 - 3 is 0, absolute value of that is zero, that works out. When x is equal to 4, 4 - 3 is 1, absolute value of 1 is indeed one. And if x is equal to 2, well 2 minus 3 is -1, but the absolute value of that is one.

So once again, I'm showing you this by really trying out numbers, trying to give you a little bit of intuition because that wasn't obvious to me when I first learned this, that if I'm shifting to the right, which it looks like I'm increasing in x value, that what I would really do is replace my x with an x minus the amount that I'm shifting to the right.

But I encourage you to try the numbers and think about what's happening here at the vertex right over here. Whatever was in the absolute value sign was equaling zero. It's when whatever was in the absolute value sign is switching from negative signs to positive signs.

So once again, if you shift three to the right, that has to happen at x equal 3. So whatever is inside the absolute value sign has to be equal to zero at x equal 3. And you know, pause this video and really think about this if it isn't making sense.

Even as you get more and more familiar with this, I encourage you to try out the numbers. That will give you more instead of just memorizing, "Hey, if I shift to the right, I replace x with x minus the amount that I shift." Always try out the numbers and try to get an intuition for why that works.

But now, let's start from here and shift four up. Shifting four up is in some ways a lot more intuitive. So let me do that. Let me shift four up. All right, so let me move up one, two, three, and four. I think I got that right.

So now I've shifted four up. And just as a reminder of what we've even done, the first part, we shifted three to the right, and now we are shifting four up. So now we are shifting four up. Before, y equal zero here, but now y needs to be equal to four.

So whatever this was evaluating to, we now have to add four to it. So when we just shifted three to the right, our equation was y is equal to the absolute value of x - 3, but now whatever we were getting before, we now have to add four to it.

We're going up in the vertical direction, so we just have to add four. Now this makes a little bit more sense if you're shifting in the vertical direction. If you shift up in the vertical direction, well, you just add a constant by the amount you're shifting. If you shift down in the vertical direction, well, you would subtract.

If we said shift down four, you would subtract four right over here. The less intuitive thing is what we did with x. When you shift to the right, you actually replace your x with x minus the amount that you shifted. But once again, try out numbers until it really makes some intuitive sense for you.

But this is what we would finally get. The equation of this thing right over here is y is equal to the absolute value of x - 3 + 4.