Hey, everyone, and welcome back. So up to this point, we've been talking about transformations of functions. In the last video, we looked at reflections. And in this video, we're specifically going to be taking a look at shifts. Now, shifts can be a bit complicated because rather than just dealing with negative signs like we were with reflections, you're going to be dealing with actual numbers inside and or outside your function. So the notation and graphs can get a little bit complicated but don't worry because in this video we're going to be going over some examples and scenarios that will hopefully make this concept crystal clear. So let's get right into this.

A shift occurs when a function is moved either vertically or horizontally from its original position. Now, something that's important to note is that you will often have situations where the graph is moved both vertically and horizontally because a shift is just moving your graph to a new location or whatever function you originally had. Now the general form for shifts is when you have your original function f(x) that goes to f(x−h +k) after you've shifted the function. In this notation, the h here represents the horizontal shift, whereas the k represents the vertical shift.

Now to understand this concept a bit better, let's take a look at the strictly vertical shift. This means that you're only going to move your graph vertically. So an example of a vertical shift would be if we took this parabola that we see here, this green curve, and we shifted it up here, so it looked something like this. Notice how we went from a position of 0 to a position of 2. The x-values stayed the same, but the y-value changed by 2. So when having the vertical shift we would say that it's the y-values that change, But now let's take a look at the horizontal shift. An example of a horizontal shift is whenever your graph moves strictly horizontally. So we could say that we have our original curve right here at 0, the green curve, and then let's say that this gets shifted over to the right. So our graph went from 0 to this position of 3, and because the x-values changed and the y-value stayed the same we would say that for the horizontal shift it's the x's that change.

Now it's also important to know how the function is going to behave when dealing with the shift transformation. So notice in both cases we have the same general form that we saw at the start here, but for the vertical shift notice that our h value is always going to be 0, so we can just ignore this h value that we see in the equation. So our function starts as f(x), and then this gets transformed tof(x +k), where k is the vertical shift, and since we shifted up by 2 we would say that k is 2. And notice how whenever we have a plus 2 the graph shifts up. So in this situation if you ever see f(x) +k you're always going to shift up, whereas if you see f(x) −k, the graph is going to shift down. Since we had a positive 2, we went up by 2. That's the idea of a vertical shift.

Now for the horizontal shift in this situation it's the k value that's equal to 0. So we can take this k and pretend like it's not even there when dealing with a strictly horizontal shift. So we started with our initial function f(x) just like before, but then we went to f(x−h), and because h represents the horizontal shift, and we ended up at an x-value of 3, we would say we have f(x−3) and that's the horizontal shift. Now whenever you see f(x) −h, this means your graph is going to shift to the right, whereas if you see f(x) +h your graph shifts to the left, and we can see that because f(x−3) caused our graph to shift to the right by 3.

But this seems a bit counterintuitive because isn't the minus sign typically associated with the left side of the graph? Well, here's the reason why this happens. Notice how there's already a negative sign or a minus sign inside the equation that we started with. So if we wanted to shift over to the left, let's say at negative 2, our graph would look something like this, and that means that the h value would be negative 2, so we would end up having our original function f(x) become f(x−−2), but we know that two negative signs are going to cancel each other, so this would actually become f(x) +2). So notice how even though we have a plus sign here, we actually(shift to the left because realistically we actually plugged in a negative 2 here for the h value. So this is just something you want to keep in mind. Pay very close attention to what this sign is in front of the h value.

So now that we've gotten to look at the vertical and the horizontal shifts, let's see if we can apply this to an example. So in this example, we're given the function f(x), and we're asked to sketch the transformation f(x−2 +3). We know that this is the standard equation when dealing with a shift, where h is the horizontal shift, and k is the vertical shift. Looking at the equation that we have I can see that this h value here matches with the 2 that we have right there, and I can see that there's a minus sign in front of both which means we're going to shift to the right. So I can keep this as positive 2 for our h value, I can see also that we have a k value of positive 3, So what this means is since we have positive 2 for our H, and positive 3 for our K, that means we're going to shift 2 units to the right from our original position, and we're going to shift 3 units up. So we started here at 0, and we're going to finish at this point which is at 23. So our new function is going to look something like this. Notice how we have the same overall shape that we had before, but now we've been shifted to this new location based on the transformation we were given. So this is how you can handle situations where you have a shift transformation. Hopefully, you found this video helpful, and let me know if you have any questions.