Graphs of absolute value functions
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- [Instructor] 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 negative two, the absolute value of that is going to be two. Negative one, absolute value is one. Zero, absolute value is zero. One, absolute value is one. So on and so forth. What I wanna do in this video is think about how the equation will change if we were to shift this graph. So in particular, we're gonna first think about what would be the equation of this graph if we shift, 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? Alright, 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 gonna shift three to the right, it would look like, it would look like this. So that's what we first wanna figure out the equation for and so how would we think about it? Well, one way to think about it is, well, something interesting is happening right over here at x equals three. Before, that interesting thing was happening at x equals zero. Now, it's happening at x equals three. And 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 and that's why we see a switch in direction here of this line and so 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 and this is where it's going to switch signs and 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, 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 minus three, well, let's try it out. Let's see if it makes sense. So when x is equal to three, three minus three is zero, absolute value of that is zero. That works out. When x is equal to four, four minus three is one, absolute value of one is indeed, is indeed one. And if x is equal to two, well, two minus three is negative one 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 an 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 an x value but 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 numbers and think about what's happening here. At this 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 equals three. So whatever is inside the absolute value sign has to be equal to zero at x equals three and this, pause this video and really think about this if it isn't making sense and 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 that why that works. But now, let's start from here and shift four up and shifting four up is in some ways a lot more intuitive. So let me do that. Let me shift four up. Alright, so let me move. I'm gonna go 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, now we are shifting four up. So before, y equaled zero here but now, y needs to be equal to four. So whatever this was evaluating, do 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 minus three 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 'cause 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 minus three plus four.