Flipping and compressing a graph

The graph of y is equal to the absolute value of x is reflected across the x-axis and then compressed vertically by a factor of 8/3. What is the equation of the new graph?

All right, so let's think about this step by step. If I start, and I'm just going to draw some quick hand-drawn sketches here, so that's my x-axis; that is my y-axis. If we're talking about the graph of y is equal to the absolute value of x, that looks like this. In the first quadrant, it looks like y equals x, and in the second quadrant, it looks like y = |x| because the absolute value of a negative number is its opposite.

So let me make it look a little bit more symmetric than that, so it looks something like that. That's what the graph of y is equal to the absolute value of x looks like. Now, they're asking us to do what amounts to two different transformations. The first one is they want to reflect it; they want to reflect across the x-axis, so they want us to flip it across the x-axis like this.

So instead, it looks like this. So that graph, that equation that describes this graph, well, this is going to be the opposite. Whatever y you were getting on this orange graph, you're going to get the negative of that. So you're going to get y is equal to the negative of the absolute value of x.

And if this doesn't make intuitive sense to you, try it out in the orange graph. When x is equal to, let's say, 2, well, the absolute value of two is two; the absolute value of -2 is two. But now we want to take the absolute value, but then take the negative of it. This thing stays nonpositive the entire time, so the absolute value of 2 is two, but we want the opposite of that; we want -2. The absolute value of -2 is two, but we want the opposite of that.

But they didn't ask us to just reflect across the x-axis; they then want us to compress vertically by a factor of 8/3. So let's think about this a little bit. Compress vertically by a factor of 8/3. So if they said stretch vertically by a factor of 8/3, then I would just multiply this by 8/3.

Sometimes it's helpful to think of this in terms of a mixed number: 8/3 is the same thing as 2 and 2/3. So, if you were to stretch by two and 2/3, you would get taller; you would look something like this. But if you are compressing, then it's going to look something like this. If you're compressing vertically, you can think about it as being stretched horizontally.

So, if you're compressing by a factor, you should multiply by the reciprocal of that factor. Think about it: if you were compressing by three, you would multiply by 1/3. So, if you are compressing vertically by 8/3, well, that means whatever y you would have gotten, you multiply that times the reciprocal of 8/3.

Or you could—another way you could think about it—you could divide that by 8/3. So, if we want to get this right over here, this is going to be y is equal to the negative absolute value of x. Since we're compressing by 8/3, we would divide that by 8/3, or another way to think about it: dividing by 8/3 is the same thing as multiplying by the reciprocal, so 3/8 times the absolute value of x.

And when we look at our choices, we see that it is that choice right over there. And I really want to stress this point because I think it can get a little bit confusing. The reason why I multiplied that by the reciprocal is we're saying compressing vertically. If we said stretching vertically, we would just multiply by 8/3, but since we're compressing, we would divide by 8/3 or multiply by its reciprocal.