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Shifting functions introduction

The graph of y=f(x)+k (where k is a real number) is the same as the graph of y=f(x) only it's shifted up (when k>0) or down (when k<0). Similarly, the graph of y=f(x-h) (where h is a real number) is the same as the graph of y=f(x) only it's shifted to the right (when h>0) or to the left (when h<0).

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Video transcript

- [Instructor] So I am here at desmos.com, which is an online graphing calculator, and 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? And 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 can 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 can 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 zero, zero. Now it is at zero, negative three, so it shifted it down. And we can set up a slider here to make that a little bit clearer, so if I just replace this with, 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, and notice, we have shifted up, and if we increase the value of k, notice how it shifts the graph up, and as we decrease the value of k, if k is zero, we're back where our vertex is right at the origin, and as we decrease the value of k, it shifts our graph down. And that's pretty intuitive, 'cause 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? So 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. So I'm gonna 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 zero, zero. Now our vertex is at one, zero. 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, 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. And to see how this can be generalized, let's put another variable here and let's add a slider for h. And then we can see that when h is zero and k is zero, our function is really then just x squared, and then if h increases, we are replacing our x with x minus a larger value. That's shifting to the right and that is, 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. And of course, we can shift both of them together, like this. So here, we're shifting it up, and then we are, we could get back to our neutral horizontal shift and then we can shift it to the right like that. And 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 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 gonna 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 negative five, negative five, which is right over there. So what you would do is you would replace your x with x plus five, or you would make this h variable to negative five right over here, 'cause notice, if you replace your h with a negative five, inside the absolute value, you would have an x plus five, and then if you want to shift it down, you just reduce the value of k, and 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, go to desmos.com. Try this out for yourself, and really play around with these functions to give yourself an intuition of how things and 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.