 In this video, I'm going to talk about parent functions. Now, parent functions, they're basically your bare bones graphs that you're going to see in your algebra or your algebra two level of courses. So we're going to talk about what those parent functions are today. So it looks like there's a bunch of them. We have constants. We have linear. We have quadratic. We have cubic. And we have square root. So these are the 1, 2, 3, 4, 5. These are the five that we're going to talk about today. There are many, many more that we can have. But these are just the basic ones that we're going to talk about today. Now, the rule, so those are the different families, the constant, linear, quadratic, cubic, and square root. The rule then is in function notation. So it's the function is equal to just a constant. Now, you see for constant, but we could have functions say f of x is equal to 5 or something to that effect. Now, that's basically what your function will look like, is that you'll have a 5 right here for x. It'll be a constant. No x is, no nothing like that, just a constant number. It could be positive. It could be negative. It could be decimals, fractions. It could be just about anything. Then you have your linears. Your linears are your lines. Are your lines. So you're going to have just an x function. Now, this could be a 3x. It could be a negative 1 half x. I mean, it could be a bunch of different things. But mainly, you're just going to have an x to the first power. And as we see when we go to quadratics, this is simply just going to be an x to the second power. Now, of course, there's going to be numbers in front here. There might be parentheses around it or something to that effect. But anything that's a quadratic is going to be x to the second power. Cubic then is anything x to the third power, x cubed. And then the square root is going to be anything that is the square root of x. Notice that these x is underneath the square root. So the variable is being square rooted. So those are the different rules. Now, when they say rule, they just mean, OK, this is the function. This is the notation that you're going to see. These are all the equations that you're going to see. Now, what do these graphs look like? What do they look like? Well, a constant graph is going to be a horizontal line somewhere. Now, if this was x equals 5, this point on the y-axis would be a y-intercept of 5. So that's what a constant graph looks like. Linear, those are your lines. Now, don't mistake them for your lines, for your constant graphs. Linears are going to be diagonal. They're going to be diagonal. Now, they could be going uphill. This is a positive slope. They could be going uphill. Or they could be going downhill. They could be going the other way. Just kind of depends on what the function looks like. Like up here, this example, 3x, that's going to go uphill. This negative 1 half x, that's going to go downhill. It just depends on what the function is. And again, these are parent functions. These are just functions just in general of what they're going to look like. This isn't hard fact. Well, I guess these are facts. But not every graph is going to look like this. They will look something like this. Anyway, moving on to quadratics, which is x squared. These are what we call parabolas. A little bit of vocabulary there. Parabolas. These are your curves. This is your first introduction to curves as a parabola. So it looks like a cup of some sort. Looks like a bowl of some sort, a smiley face. A lot of different students will think of it in a couple of different ways. But that's a parabola. That's a quadratic graph. A cubic, on the other hand, a cubic, looks a little bit different. From the origin, it does curve up when it goes to the right. But on the other side, on the other side, on this left side, it's actually going to curve down. So it looks a lot like your quadratic, except for this left half over here is actually now pointing down. So this is what a cubic graph looks like. Some students will say it looks like an S, or sometimes students will think a cubic looks like a snake of some sort. There's a couple of different ways to describe. But in general, that's what a cubic graph is going to look like. And last but not least, square root. When you have a square root over the x, it's going to look something like this. From the origin, it's going to curve. Now, it doesn't curve up. It actually curves to the right. It goes that way. And it kind of tape, it curves to the right. It kind of tapers off. And it doesn't rise very fast. It doesn't rise very fast. OK, now that's in general what these parent functions would look like. So constants, this is what constants look like. Linear functions, this is what they look like. Quadratic functions, they look like parabolas. They look like this bowl effect. Cubics, they look like snakes or Ss. However you want to look at this. Square roots, they look like arcs going off to the right side. They don't increase as fast as these other two, like your parabolas and your cubics. But they go off to the right. They curve off to the right. Now let's get down to some of the technical stuff. So now let's go down here to domain. Now from previous lessons, you've learned what domain is. Domain is your input. Domain is what your input is. So these are the numbers. What can we plug in? That's basically what it is. What can we plug in? Now if you remember this little symbol, this funky-looking r, this is all real numbers. All real numbers. That's what this little funky r means, which means what numbers can I plug in? What are the domains? Well, you can plug in all numbers. So let's look back up here to the rule. Let's look back up here to function. I can plug in whatever number I want to here. It doesn't really matter. I can plug in a 5. I can plug in a 1 half. I can plug in a million. I can plug in whatever I want to. There's no real rule. There's no real number I can't plug in. So same thing over here. For the domain, the numbers I can plug in for linear functions, I can plug in whatever I want to. I can take any number times 3. There's really no rule against that. For most of our functions, this is going to be the case. I can plug in whatever number I want to. Same thing with quadratics. I can plug in whatever number I want to. I can put a 5. I can put a negative 2. I can put a 12. I can put whatever I want to in here. So all real numbers are going to work. Same thing with cubics. I can plug in whatever number I want to here. But when we get over here, when we get to square roots, now there's one thing we've got to remember. With a square root, we cannot take the square root of a negative number. We can't do it. So the square root of a negative x can't do it. Can't do it. So this is why in the domain, this is why we have a little bit of stipulation here. Instead of all the real numbers, like all the rest of them, we're saying here x has to be greater than or equal to 0. x has to be bigger than 0. We can't plug a negative number into here. So that limits our domain. Notice here, all of the positive x's are represented by this curve, but I got no picture on this side. No negative x's. I can't plug any of those in because it simply will not work. I get what's called imaginary numbers. Imaginary numbers, I'll save that for another video. We do deal with imaginary numbers a little bit, but for right now, we're just gonna stick with the old rule that you just cannot take the square root of a negative number. We'll just stay with that. Okay, now that was the domain, which is the inputs, now about down to range, which is your output. Your output. So when I plug in all these numbers, what am I gonna get out of it? Okay, so my range in this case, so whenever I plug in a number into my rule, I'm just gonna get that number out there. So whatever real number I plug in, I'm simply just gonna get that constant number out of it. That's really all I'm gonna get. Nothing really changes, okay? I'm always gonna get this number. Whatever number I plug in, I'm always gonna get this constant. This X actually really does not affect this number over here. So it doesn't matter what number I plug in here, I'm always gonna get that constant number, okay? Now my output, my range for my linear function, again, is gonna be your all real numbers. So whatever number I plug in, I can plug in whatever number I want to and then I'll just get a real number out of it. I mean, I'm gonna multiply, I'm gonna divide, I'm gonna add a subtract, but anything that I do, I'm still gonna get a real number, okay? Now, when I go over to the range for my quadratic, now it gets a little bit tricky. Notice here that whatever X I plug in, I'm always gonna square it. So if I plug in a three, I'm gonna get nine. If I plug in a negative three, I'm gonna get nine again. That's kinda odd. So plug in three, I get nine, okay? Plug in three, negative three over here, I'm gonna get nine. Now, that causes a little bit of confusion because what happens is I get no negative Y numbers down here, look at this graph. This graph is just going up, up. It's not going down at all. And so what happens is that these Ys down here, we don't get any graphs down here. So that's why we have a little bit of different range. My outputs, my Ys are gonna be greater than or equal to zero. My Ys are always gonna be bigger than zero. So look up here, no graph down here. My Ys are always gonna be bigger than zero. Now again, this is a general case for quadratics. We do have some quadratics that will have negative Ys, but we'll look at those at a later time, okay? So in general, my Ys for the most part are gonna be bigger than zero. My outputs are gonna be bigger than zero. Okay, and then same thing over here. Now this one's a little bit different. If I plug in that same example I did for quadratics, but I do that same example for cubics. If I take three to the third power, I get 27. But if I plug in negative three to the third power, I get negative 27. So I actually do get negative numbers with this one. See, the line comes down here, we do get negative Ys. So we got negative Ys, negative Ys, then we get to zero, then we get positive Ys, positive Ys. So I can plug in all my Xs, I can plug in negative Xs, I can plug in positive Xs, I get negative Ys, I get positive Ys. So all of these real numbers, I can plug in a bunch of real numbers and I'm always gonna get real numbers out of it. There's really no limitation to what numbers I'm gonna get. Unlike the quadratics where there was a limitation of what I could get. Okay, last but not least, square roots. Square roots, notice here, I got a limitation. Ys cannot be bigger than zero. And we can look at the graph here. We start here at zero, zero, and then we move to the right side. Notice that the graph just kind of stays up. The graph doesn't really go below this axis, this X axis here, it just continues up. And just the curve just continues up and just keeps going and going. But there's no negative Ys that are represented. There's no graph down here. So again, Ys gotta be greater than or equal to zero. My Ys have to be bigger than zero. So whatever X that I plug in, I can only plug in the positive Xs, but when I get the numbers I get out of it, I'm only gonna get the positive Ys. I'm not gonna get any Ys down here. All right, so that's a quick explanation of what parent functions are. So whenever you look at a graph or whenever you look at an equation, you can quickly identify what type of parent function there is. And I'll be here in a minute, I'm gonna be going over. In the next video, I'm gonna be going over a couple of examples of where we see parent functions.