Main content

# Graphing absolute value functions

## Video transcript

- [Instructor] So we're asked to graph f of x is equal to two times the absolute value of x plus three, plus two. And what they've already graphed for us, this right over here, this is the graph of y is equal to the absolute value of x. So let's do this through a series of transformations. So the next thing I wanna graph, let's see if we can graph y. Y is equal is to the absolute value of x plus three. Now in previous videos we have talked about it. If you replace your x, with an x plus three, this is going to shift your graph to the left by three. You could view this as the same thing as y is equal to the absolute value of x minus negative three. And whatever you're subtracting from this x, that is how much you are shifting it. So you're going to shift it three to the left. And we're gonna do that right now and then we're gonna just gonna confirm that it matches up. That it makes sense. So let's first graph that. Get my ruler tool here. So if we shift three to the left, it's gonna look something like... It's gonna look something like this. So on that... When whatever we have inside the absolute value sign is positive, we're going to get essentially, this slope of one. And whenever we have inside the absolute value sign is negative, we're gonna have a slope of essentially negative one. So it's going to look... It's going to look like that. And let's confirm whether this actually makes sense. Below x equals negative three, for x values less than negative three, what we're gonna have here, is this inside of the absolute value sign, is going to be negative and so then we're gonna take it's opposite value and so that makes sense. That's why you have this downward line right over here. Now for x is greater than negative three, when you add three to it, you're gonna get a positive value and so that's why you have this upward sloping line right over here. And at x equals negative three, you have zero inside the absolute value sign, just as if you didn't shift it, you would have had zero inside the absolute value sign at x equals zero. So hopefully that makes a little bit more intuitive sense of why if you replace x, if you replace x with x plus three, and this isn't just true of absolute value functions, this is true of any function, if you replace x with x plus three, you are going to shift three to the left. All right, now let's keep building. Now let's see if we can graph y is equal to two times the absolute value of x plus three. So what this is essentially going to do is multiple the slopes by a factor of two. It's going to stretch it vertically by a factor of two. So for x values greater than negative three, instead of having a slope of one, you're gonna have a slope of two. So let me get my ruler out again and see if I can draw that. So let me put that there. And then, so here instead of a slope of one, I'm gonna have a slope of two. Let me draw that. It's gonna look like that, right over there. And then instead of having a slope of negative one for values less than x equals negative three, I'm gonna have a slope of negative two. Let me draw that right over there. So that is the graph of y is equal to two times the absolute value of x plus three. And now to get to the f of x that we care about, we now need to add this two. So now I wanna graph, and we're in the home stretch, I wanna graph, y is equal to two times the absolute value of x plus three, plus two. Well whatever y value I was getting for this orange function, I now wanna add two to it. So this is just gonna shift it up vertically by two. So instead of... So this is gonna be shifted up by two. This is going to be shifted. Every point is going to shifted up by two or you can think about shifting up the entire graph by two. Here, in the orange function, whatever y value I got for the black function, I'm gonna have to get two more than that. And so it's going to look... It's going to look like this. So let me see, I'm shifting it up by two. So for x less than negative three, it'll look like that. And for x greater than negative three, it is going to look like... It is going to look like that. And there you have it. This is the graph of y or f of x is equal to two times the absolute value of x plus three, plus two. And you could've done it in different ways. You could have shifted up two first, then you could have multiplied by a factor of two, and then you could have shifted, and then, so you could have moved up two first, then you coulda multiplied by a factor of two, then you could've shifted left by three. Or could have multiplied by a factor of two first, shifted up two and then shifted over. So there's multiple, there's three transformations going up here. This is an... This is a, let me color them all. So this right over here tells me... This over here says hey, shift left. Shift left by three. This told us, stretch vertically by two. Or essentially multiply the slope by two. Stretch vert by two. And then that last piece, says shift up by two. Shift up by two, which gives us our final result for f of x.