Graphing a shifted and stretched absolute value function

So we're asked to graph ( f(x) = 2 \times |x + 3| + 2 ).

And what they've already graphed for us, this right over here, is the graph of ( y = |x| ).

Let's do this through a series of transformations. So the next thing I want to graph, let's see if we can graph ( y = |x + 3| ). Now, in previous videos, we have talked about it: if you replace your ( x ) with ( x + 3 ), this is going to shift your graph to the left by 3. You could view this as the same thing as ( y = |x - (-3)| ). Whatever you are 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. Then we're just gonna confirm that it matches up, that it makes sense.

So let's first graph that and get my ruler tool here. If we shift 3 to the left, it's going to look something like this.

On that, whenever we have inside the absolute value sign is positive, we're going to get essentially this slope of 1. Whenever we have inside the absolute value sign is negative, we're going to have a slope of essentially negative 1. So it's going to look like that.

Let's confirm whether this actually makes sense below ( x = -3 ). For ( x ) values less than -3, what we're going to have here is this inside of the absolute value sign is going to be negative, and so then we're going to take its opposite value. So that makes sense; that's why you have this downward line right over here.

Now, for ( x ) greater than -3, when you add 3 to it, you're going to get a positive value, and so that's why you have this upward sloping line right over here. At ( x = -3 ), 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 = 0 ).

So hopefully that makes a little bit more intuitive sense of why if you replace ( x ) with ( x + 3 )—and this isn't just true of absolute value functions; this is true of any function—if you replace ( x ) with ( x + 3 ), you're going to shift 3 to the left.

Alright, now let's keep building. Now, let's see if we can graph ( y = 2 \times |x + 3| ). So what this is essentially going to do is multiply the slopes by a factor of 2. It's going to stretch it vertically by a factor of 2.

So for ( x ) values greater than -3, instead of having a slope of 1, you're gonna have a slope of 2. Let me get my ruler out again and see if I can draw that.

Let me put that there, and then—so, here, instead of a slope of 1, I'm going to have a slope of 2. Let me draw that; it's going to look like that right over there.

Then, instead of having a slope of negative 1 for values less than ( x = -3 ), I'm going to have a slope of negative 2. Let me draw that right over there.

So that is the graph of ( y = 2 \times |x + 3| ).

And now, to get to the ( f(x) ) that we care about, we now need to add this 2. So now I want to graph ( y = 2 \times |x + 3| + 2 ). Well, whatever ( y ) value I was getting for this orange function, I now want to add 2 to it; this is just going to shift it up vertically by 2.

So instead of—this is going to be shifted up by 2. This is going to be shifted; every point is going to be shifted up by 2. Or you can think about shifting up the entire graph by 2. Here in the orange function, whatever ( y ) value I got for the black function, I'm going to have to get 2 more than that.

And so it's going to look like this. Let me see; I'm shifting it up by 2. So for ( x ) less than -3, it'll look like that, and for ( x ) greater than -3, it is going to look like that.

And there you have it! This is the graph of ( y ) or ( f(x) = 2 \times |x + 3| + 2 ).

You could have done it in different ways. You could have shifted up 2 first, then you could have multiplied by a factor of 2, and then you could have shifted. So you could have moved up two first, then you could have multiplied by a factor of 2, then you could have shifted left by 3.

Or you could have multiplied by a factor of 2 first, shifted up 2, and then shifted over. So there are multiple—there are three transformations going up here.

This is a—let me color them all.

So this right over here tells me: shift left by 3. This tells us: stretch vertically by 2, or essentially multiply the slope by 2; stretch vertical by 2.

And then that last piece says: shift up by 2. Shift up by 2, which gives us our final result for ( f(x) ).