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

So I am here at desmos.com, which is an online graphing calculator. The goal of this video is to explore how shifts in functions happen.

How do things shift to the right or left? Or how do they shift up and down? What we're going to start off doing is just graph a plain vanilla function: f of x is equal to x squared. That looks as we would expect it to look.

But now let's think about how we could shift it up or down. Well, one thought is, well, to shift it up, we just have to make the value of f of x higher. So we could add a value, and that does look like it shifted it up by one. Whatever f of x was before, we're now adding one to it, so it shifts the graph up by one. That's pretty intuitive.

If we subtract one, or actually, let's subtract three. Notice it shifted it down. The vertex was right over here at (0, 0), now it is at (0, -3). So it shifted it down.

We can set up a slider here to make that a little bit clearer. If I just replace this with the variable k, then let me delete this little thing here, that little subscript thing that happened. Then we can add a slider k here, and this is just allowing us to set what k is equal to. So here k is equal to one, so this is x squared plus one. Notice we have shifted up.

If we increase the value of k, notice how it shifts the graph up. As we decrease the value of k, if k is zero, we're back where our vertex is right at the origin. Then, as we decrease the value of k, it shifts our graph down. That's pretty intuitive because we're adding or subtracting that amount to x squared. So it changes, we could say, the y value. It shifts it up or down.

But how do we shift to the left or to the right? What's interesting here is to shift to the left or the right, we can replace our x with an x minus something. So, let's see how that might work.

I'm going to replace our x with an x minus, let's replace it with an x minus one. What do you think is going to happen? Do you think that's going to shift it one to the right or one to the left? So let's just put the one in.

Well, that's interesting! Before, our vertex was at (0, 0), now our vertex is at (1, 0). So by replacing our x with an x minus one, we actually shifted one to the right. Now, why does that make sense?

Well, one way to think about it is before we put this x, before we replaced our x with an x minus one, the vertex was when we were squaring zero. Now, in order to square zero, squaring zero happens when x is equal to one. When x is equal to one, you do one minus one, you get zero, and then that's when you are squaring zero.

So it makes sense that you have a similar behavior of the graph at the vertex now when x equals one as before you had when x equals zero. To see how this can be generalized, let's put another variable here and let's add a slider for h.

Then we can see that when h is 0 and k is 0, our function is really then just x squared. If h increases, we are replacing our x with x minus a larger value, that's shifting to the right. Then as h decreases, as it becomes negative, that shifts to the left.

Now, right here, h is equal to negative five. You typically won't see x minus negative five; you would see that written as x plus five. So if you replace your x's with an x plus five, that actually shifts everything five units to the left.

Of course, we can shift both of them together like this. So here we're shifting it up, and then we could get back to our neutral horizontal shift and then we can shift it to the right like that.

Everything we did just now is with the x squared function as our core function, but you could do it with all sorts of functions. You could do it with an absolute value function. Let's do it! Let's do absolute value! That's always a fun one.

So instead of squaring all this business, let's have an absolute value here. So I'm going to put an absolute—whoops, absolute value! And there you have it. You can start at—let me make both of these variables equal to zero—so that would just be the graph of f of x is equal to the absolute value of x.

But let's say you wanted to shift it so that this point right over here that's at the origin is at the point (-5, -5), which is right over there. So what you would do is you would replace your x with x plus 5, or you would make this h variable to -5 right over here.

Because notice if you replace your h with a -5 inside the absolute value, you would have an x plus 5. Then, if you want to shift it down, you just reduce the value of k. If you want to shift it down by five, you reduce it by five, and you could get something like that.

So I encourage you to go to desmos.com, try this out for yourself, and really play around with these functions to give yourself an intuition of how things—why things shift up or down when you add a constant, and why things shift to the left or the right when you replace your x's with an x minus—in this case, an x minus h, but it really could be x minus some type of a constant.