By the end of this video, you're going to easily be able to graph piecewise functions, and in about 30 seconds, here's how this video is going to help you do that. We'll be graphing four piecewise functions in this video, and they'll be getting harder as we go. So in the first problem, we're just going to be graphing two lines, but in the second problem, now I'm going to throw in a quadratic, and in the third problem, we'll have to graph three different functions.
Finally, with the fourth problem, we'll have three functions to graph, yeah, but they'll be harder to graph because we're going to have things like quadratics in the mix. So we'll go through all of that and then after we do that I'll give you a problem to try and answer in the comments and by that point it should honestly be breezy And if you're looking for the notes for this video because I mean they are looking pretty good I have a printable version of these notes linked right in the description and also in the description I have an extra video where we go through and we graph eight more piecewise functions And we also go through four problems where we're evaluating piecewise function So that extra video is also going to be linked in the description So we're going to start off with this first piecewise function here. And this piecewise function has two little functions inside here.
So let's just graph them one at a time. So the first one is just 5. So f of x, that's y. So what this is really saying is that y is equal to 5 when x is less than or equal to negative 3. And y equals 5, well, we can graph that. y equals 5 is 1, 2, 3, 4, 5 up on the y-axis. So it's going to be all of that.
this right here. So that's the full graph of y equals 5, but that's not what we're graphing here. We're graphing this piecewise function here where y equals 5 is only a part of that. It's only a part of that when x is less than or equal to negative 3. And so where is that? Well, negative 3, that's 3 over in the x direction to the left.
And so that means that this graph needs to stop right here. So let's scale that back real quick. Now at the point where this line ends, we need to either have a closed circle or we need to have an open circle. It depends whether there's actually a point there or not. And well, that's going to depend on the inequality.
If this inequality is less than or equal to or greater than or equal to, we'll be using a closed circle. But if that inequality is a greater than or a less than and it doesn't have that or equal to bar underneath it, then we'll be using an open circle. Now, in this case, we're using a closed circle because it's less than or equal to.
And so that does it for the first part of our graph. Let's look at the second part now. And that is negative 2x minus 3. And that line's what we're going to use for x's greater than negative 3. So let's look at this line here. We know it has a slope of negative 2. And it's got a y-intercept of negative 3. So let's start at that y-intercept of negative 3. And let's just use that slope, negative 2. And remember, slope is rise over run. And you can write that slope as being negative 2 over 1. So the rise is negative 2 and the run is 1. That means for every 2 we go down, we go over by 1. And we keep doing that.
And we can do that in the other direction as well. So now we can go through connect these dots and graph our line, but we need to stop it right here at x equals negative three because remember that line doesn't go past that. So let's drop that line now.
So I went ahead and I graphed that line. And now the only thing that we need to worry about before we call it quits on this problem is, well, is there going to be a closed circle or an open circle here? And well, this is just x is greater than negative three. It's not greater than or equal.
And so yeah, that's our first piecewise function already graphed. Now let's graph another one here. Here we have another piecewise function where we have a line and we have a quadratic.
And you can see it's getting cut off at x equals 0. So for this line here, y equals 2x plus 1, we'll graph that first. That's got a slope of 2 and a y-intercept of 1. So let's start at that y-intercept of 1. And let's just graph this line out. The slope says that we can write that as 2 over 1, that for every 2 we go up, we're going to go over by 1. And so we go up by 2, over by 1, up by 2, over by 1, and we can do that in the other direction as well. And now we can graph our line. So I went through and I graphed our line, but that's not the graph that we're going to use here.
That's just the full graph of y equals 2x plus 1. We're graphing this piecewise function here where 2x plus 1 only makes up a part of it, the part where x is less than 0. And so we're actually cutting it off here and only using what's to the left of this. So let's skip. this graph back to x equals 0. And now that we've done that, we need to see is there going to be an open circle here or is there going to be a closed circle? And that depends on this inequality. Here x is less than 0, but x can't equal 0. There's no line under here.
And so that means that there's no actual point here. So we're going to be using an open circle So that was the first part of our piecewise function. Now, let's do the second Let's do that parabola x squared minus 3 for that. I can see that the y-intercept is negative 3 and that actually is the point where x is equal to zero that's where i'm getting cut off and i know that's going to be a closed circle because it's greater than or equal to zero so there is an actual point there and also i know that my y intercept is negative three because y intercepts for quadratics for for parabolas they work in the same way as y intercepts for lines it's still the number without the x on it so my y intercept is one for this line and my y intercept is negative three for that quadratic so that's a little tip there if you didn't already know that because I don't think I did when I was doing this kind of stuff. So here's my xy chart, and I know that when x is 0, y is negative 3. That was the y-intercept part.
And so now let's look at all of these x's that are greater than or equal to 0. Let's just go 1, 2, 3. Now when x is equal to 1, let's plug that in here, we'll get a 1 squared, which is 1, minus 3. And then negative 2. When x is equal to 2, we'll get a 2 squared, that's 4. 4 minus 3 is 1. And then when x is equal to 3, that's a 3 squared, which is 9 minus 3 is 6. And so now we can go through and we can plot all these points. We have one comma negative two, we have two comma one, and we have three comma six. This is four, five, six.
And now we can just graph that parabola. So that is it for our second piecewise function. And moving on to our third piecewise function here, our third problem for this video, we have three functions in here that we're going to have to graph, and the cutoffs are going to be at negative 2 and 2. I always think it's good to note that.
So first off, We need to look at our first little function in here, and that's y equals negative 4. And we know that that is for x is less than or equal to negative 2. That means it's going to be at negative 2 and everything to the left of that. nothing over here. We also know that this is going to be a what closed circle or open circle? Well, it's going to be a closed circle, right?
Because that's less than or equal to. So we're going to have a closed circle wherever we start at x equals negative two. And well, this is y equals negative four, right? So we're going to go down by four.
And that line, that y equals negative 4 line, would be this. But since we're only graphing the part that is to the left of x equals negative 2, then we're going to start here with this point and just go left from there. And again, remember, we have that closed dot because it's less than or equal to. But yeah, that's our first function graphed.
Now we have to graph the other stuff. Let's move on to x minus 2. Now x minus 2. That's a line as well. This is a line in y equals mx plus b form, where the slope is 1, and our y-intercept is negative 2. So let's go through here.
We'll put our y-intercept in. And then we've got that slope of one. So we're just going to go up one over one, up one over one, and the same thing in the other direction.
And we'll actually end up at this closed circle right here. So now I'm going to connect these dots. And that's going to give me my second line. And notice that I stopped at x equals two.
And look, if you're having trouble just remembering where to stop with these lines, because it's a lot of things to think about all at once. I get that. Here's what I recommend you do. Before you even start graphing anything, look at where your cutoff points are.
We know it's... it's x equals negative 2, and we know x equals 2. That's where we're changing over from one function to another function. And so what we can do is we can just put little lines down. Here's x equals negative 2, and here's x equals 2. And I think that is really going to help you see, like, okay, here's where we're graphing y equals negative 4. Here's where we're graphing that line y equals x minus 2. And then here's where we're going to graph that third line, y equals negative 2x plus 4. And at these boundaries, you're going to need to either have closed dots or open dots.
And speaking of which, we need to figure out for this line whether there is a closed circle or an open circle here. So let's look, this is at x equals two. So do we have a point there for this line? Well, no, because there's no bar under here. X can only be less than two, it can't be equal to two.
So that means we're going to be using an open circle. Now, you also might see that x can't be equal to negative 2 either, so you might think that there needs to be an open circle here. But no, there's a closed circle there because there is actually a point there. That point isn't from this function. It's from the first function that we did.
So you're not going to change that and make it an open circle now because there is a point there. It's just not from the function that you're working with. It's from one that we did before. So great, that works for the second.
line and now let's move on to the third. For that one we can see that the slope is negative 2 and the y-intercept is 4. So let's start graphing this by going up 4 to get to our y-intercept and then we're going to go down 2 and over 1. That's what the slope tells us. So we're going to have a point there and then we'd have a point here and actually that's where this line starts right.
It's going to be from x equals 2 onward. So is that going to be a closed circle now? Well Let's look at here.
Yeah, it will be because it's greater than or equal to. So there is a point at x equals two from that function, from the line that we're on right now. So this is going to become a closed circle, and I know that's atrociously big, so I will shrink that real quick. So we're at that point, and now we'll continue with our slope, down two over one, down two over one, and we can connect this and make our line. And now that we've done that, we're finished graphing this piecewise function for problem three.
Now moving on to the last piecewise function for this video. Again, we've got three functions to graph. And let's actually mark the cutoffs this time to make it just easier on ourselves.
We have x equals zero and x equals five. So that's the two places where we're going to be cutting our graph off. So let me grab a dotted line here.
And let's take x equals zero, which is the middle. And we'll look at x equals five. So that's going to be right here. So we'll draw our line there.
And so now you can see what we're going to be graphing our three functions. For our first function here, x squared minus 1, we know that's a parabola. And that parabola has a y-intercept of negative 1. And so we can mark our point here.
And actually, since that's the boundary, is that going to be a closed circle or an open circle? Well, it's going to be a closed circle because x can be equal to 0 there. x is less than or equal to 0. So we make that a closed circle there and now let's make a xy chart to graph the rest of this so we know that at x equals 0 we have a y of negative 1 but we need to figure out all the x's to the left so we'll go negative 1 negative 2 and negative 3. so let's see what happens when we plug these x's into this function if we plug in the x's negative 1 we're going to get a negative 1 squared and that's a positive 1 positive 1 minus 1 is 0. plug in negative 2 for x here and we'll get Negative 2 squared, that's 4, and 4 minus 1 is 3. Plug in negative 3 for x, and we'll get negative 3 squared. That's 9. 9 minus 1 is 8. And now we have the points that we can plot.
So we have negative 1 comma 0. We've got negative 2 comma 3. And we've got negative 3 comma 8. Now we just connect the dots. Great, so that's our first function done, and it only gets easier from here. Now we have a line, y equals 2x minus 1, and in that line we know that the slope is 2, and the y-intercept is negative 1. So we'll start with that y-intercept, negative 1, and actually we've already got that point there from the parabola, and now what we're going to do is we're going to use our slope and go up 2 and over 1, and continue doing that until we get to our boundary at x equals 5. So now let's go through and graph our line. Connecting the dots here and now that we've done that at our boundary is this going to be an open circle or a closed circle?
Well for this function x can be less than or equal to 5 and the less than or equal to sign means we're going to Be using a closed circle. There is a point at x equals 5 from this function now That's great. No, but we do have one function left We have y equals 3 and so for y equals 3 we know that that is what we're using for the last section here Let's find y equals 3. We're going to go up by 3. And at this point here, we can have an open circle or a closed circle.
Well, we'll have a open circle because we don't have greater than or equal to here. x can only be greater than 5. So there's no actual point here from y equals 3. We've got an open circle there. and we're going to go to the right on the line y equals 3 and there you go now we have our final piecewise function graphed for this video so that is how you graph piecewise functions and if you feel pretty comfortable with that at this point then here's a problem for you to try and answer in the comments here we've got a piecewise function with two functions in there so it's not terribly big or it's not one of those three function ones that we were doing and since you can't actually go and graph the thing in the comments here's what i can have you do what i want you to do is there's going to be an open circle on this graph and there's going to be a closed circle on this graph. And so what I want you to do in the comments is give me the points of each.
Give me the coordinates where those two circles are. So give that a shot and let me know what your answer is in the comments. And if you have any questions on anything we talked about in this video, again, let me know in the comments and I'll try to get back to you when I can.
Now remember I do have that extra video where you and I will go through and graph eight more piecewise functions and we'll also do four problems where we are evaluating piecewise functions. So especially if you're studying for a quiz or a test on this kind of stuff and you're looking for that review definitely check out that link in the description down below. And while you're there remember you can also snag the printable notes for this video.
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Anyways, guys, that's going to do for this video and I'll see you soon.