Scaling functions introduction | Transformations of functions | Algebra 2 | Khan Academy

So this is a screenshot of Desmos. It's an online graphing calculator. What we're going to do is use it to understand how we can go about scaling functions, and I encourage you to go to Desmos and try it on your own, either during this video or after.

Let's start with a nice interesting function. Let's say ( f(x) ) is equal to the absolute value of ( x ). So that's pretty straightforward. Now, let's try to create a scaled version of ( f(x) ).

We could say ( g(x) ) is equal to... well, I'll start with just the absolute value of ( x ), so it's the same as ( f(x) ). So it just traced ( g(x) ) right on top of ( f ). But now, let's multiply it by some constant. Let's multiply it by two.

Notice the difference between ( g(x) ) and ( f(x) ). You can see that ( g(x) ) is just ( 2 \times f(x) ). In fact, we can write it this way: we can write ( g(x) = 2 \times f(x) ). We get to the exact same place, but you can see that as our ( x ) increases, ( g(x) ) increases twice as fast, at least for positive ( x )'s on the right-hand side.

Actually, as ( x ) decreases, ( g(x) ) also increases twice as fast. So is that just a coincidence that we have a two here and it increased twice as fast? Well, let's put a three here. Well, now it looks like it's increasing three times as fast, and it does that in both directions.

Now, what if we were to put a 0.5 here? 0.5. Well, now it looks like it's increasing half as fast, and that makes sense because we are just multiplying. We're scaling how much our ( f(x) ) is. So before, when ( x ) equals 1, we got to 1. But now, when ( x ) equals 1, we only get to 0.5.

Before, when ( x ) equals 5, we got to 5. Now, when we get to ( x = 5 ), we only get to 2.5. So we're increasing half as fast, or we have half the slope.

Now, an interesting question to think about is what would happen if instead of it just being an absolute value of ( x ), let's say we were to have a non-zero y-intercept. So let's say, I don't know, plus 6.

So notice, then, when we change this constant out front, it not only changes the slope, but it changes the y-intercept because we're multiplying this entire expression by 0.5. So if you multiply it by 1, we're back to where we got before. Now, if we multiply it by 2, this should increase the y-intercept because remember, we're multiplying both of these terms by 2.

We see that it not only doubles the slope, but it also increases the y-intercept. If we go to 0.5, not only did it decrease the slope by a factor of one-half, or I guess you could say multiply the slope by one-half, but it also made our y-intercept be half of what it was before.

We can see this more generally if we just put a general constant here, and we can add a slider. Actually, let me make the constant go from 0 to 10 with a step of, I don't know, 0.05. That's just how much does it increase every time you change the slider.

Notice when we increase our constant, not only are we getting narrower because the slope, the magnitude of the slope is being scaled, but our y-intercept increases. Then, as ( k ) decreases, our y-intercept is being scaled down, and our slope is being scaled down.

Now, that's one way that we could go about scaling. But what if, instead of multiplying our entire function by some constant, we instead just replace the ( x ) with a constant times ( x )?

So instead of ( k \times f(x) ), what if we did ( f(k \times x) )? Another way to think about it is ( g(x) ) is now equal to the absolute value of ( kx + 6 ). What do you think is going to happen? Pause this video and think about it.

Well, now when we increase ( k ), notice it has no impact on our y-intercept because it's not scaling the y-intercept, but it does have an impact on slope. When ( k ) goes from 1 to 2, once again, we are now increasing twice as fast.

Then, when ( k ) goes from 1 to 0.5, we're now increasing half as fast. Now, this is with an absolute value function. What if we did it with a different type of function? Let's say we did it with a quadratic, so ( 2 - x^2 ).

Let me scroll down a little bit, and so you can see when ( k = 1 ), these are the same. Now if we increase our ( k ), let's say we increase our ( k ) to 2, notice our parabola is, in this case, decreasing as we get further and further from zero at a faster and faster rate. That's because what you would have seen at ( x = 2 ), you're now seeing at ( x = 1 , because you are multiplying 2 times that.

If we go between 0 and 1, notice on either side of zero, our parabola is decreasing at a lower rate. It's a changing rate, but it's a lower changing rate, I guess you could put it that way.

We could also try just to see what happens with our parabola here if, instead of doing ( kx ), we once again put the ( k ) out front. What is that going to do? Notice that is changing not only how fast the curve changes at different points, but it's now also changing the y-intercept because we are now scaling that y-intercept.

So I'll leave you there. This is just the beginning of thinking about scaling. I really want you to build an intuitive sense of what is going on here. Really think about mathematically why it makes sense, and go on to Desmos and play around with it yourself. Also, try other types of functions and see what happens.