Common Core Algebra II.Unit 2.Lesson 1.Introduction to Functions

Hello and welcome to another common core algebra two lesson by E math instruction. My name is Kirk weiler and today we're going to be doing unit two lesson number one on introduction to functions. There is hardly any concept in math that is more important or more fundamental than that of a function. So we're going to begin our discussion today by reviewing just what a function is. Let's go ahead and do that. All right. A function is any rule. Notice how I put rule in quotes because rule is taken by a lot of different things by a lot of different people. But a function is any rule that it signs exactly one output value for each input value. Then we typically think of the outputs as being Y and the inputs as being X so think of it that way. When I put an X value in, what happens is only one Y value comes out. Now functions come in a lot of different forms, but the three most common are equations. You know, like Y equals 5 X plus two. Graphs like this beautiful one. And tables. Something like that. Now, because functions are all about converting inputs to outputs. We have special names for them. The input variable is often called the independent variable. In other words, X is just X, it can not exactly be anything at once to be, but it's sort of independent. Then again, we call the output, the Y variable, the D dependent variable, because it really doesn't get to choose what it wants to be. When I put X in, then the rule converts that X into a Y and I don't really have any choice about it. The Y variable depends on the value of the X variable. Typically not the other way around. So let's jump right into a real world scenario where we have a function. Exercise one, an Internet music service offers a plan whereby users pay a flat monthly fee of $5. And then can download songs for ten cents each. Sounds like a good deal. I like it. Especially if I'm downloading a lot of songs in a month. What are the independent and dependent variables in this scenario? Well, the independent variable is simply the number of downloads, right? That's independent. All it depends on is how much music you want to download. But the dependent is how much we pay or how much we are charged, right? Obviously, the more songs we download, the more we're going to pay. The less songs we download, the less we're going to pay. Letter B, fill in the table below for a variety of independent values. In other words, how much are we going to have to pay if we don't download anything? How much are we going to have to pay if we download 5 songs or ten songs or 20 songs? Now, you should be able to do this with the information that the problem gave you. So why don't you go ahead and do that? And if you want to use your calculator that's fine, but why don't you fill out that table, okay? All right, let's go through it. Well, there's not much we have to do for zero downloads, but it is important to note that we are charged something. In other words, we're charged that flat monthly fee $5. So if we have a bad month, we don't get on to the music service. We don't order anything. Well, we're still being charged $5. On the other hand, if we do 5 downloads, think about this at ten cents apiece, well, that's going to be 50 cents. We're going to have to pay. But then we're going to have to take that 50 cents. And add it to the $5 because we are always charged that $5. And we're going to get 5 50. We could do the same thing for ten songs, right? We would have ten songs times ten cents. Oh, that just gives us $1. But then we'd have to have that $1 and add $5 to it. To get $6. We would find the same for 20, that would be $7. Okay. Now, the reason I like doing something like letter B, even though it's relatively easy, is it's going to lead us to the formula for this function and keep in mind it really is a function. The definition of a function is that any given input gives us exactly one output. It wasn't like you said, hey, if you downloaded ten songs, you would be charged $6 or 6 25. It was no. You download ten songs. Your charge $6. One input gives you exactly one output. Okay, pause the video now, and then we'll clear out the text. Okay, here we go. All right, we're going to continue to work on this. Let her see says let the number of downloads be represented by the variable X and the amount charged be represented by the variable Y write an equation that models Y as a function of X so we can do mathematical modeling in many different ways, but here we like that formula, right? So think about that for a little bit and look back at B if you need help. All right, well, what did we do? In B, we kept taking ten cents, and multiplying it by the number of songs that we downloaded. But then we kept having to add that to $5 to get our total cost. So there's our equation. Y equals 0.10 times X plus 5. So we took what we did in letter B and we just turned it into an equation. Letter D, based on the equation you found in part C, produce a graph of this function. For all values of X on the interval zero to 40, use a calculator table to generate additional coordinate pairs to the ones you found in part B all right, well, why don't we do it? Let's bring open the TI 84 plus. There it is. There's not a lot of room on the screen. I apologize for that. I'll try to make the calculator screen as big as I can when I need to. But let's generate a table of values. Let's hit Y equals. All right, if there's any equations in Y one, Y two, et cetera, clear them out. So I'm going to get rid of anything I've got in there now. All right. And then Y one, I'm going to put that equation. So I'm going to type in zero point one zero X don't want to have ten X plus 5.0 zero. All right. Now, sometimes we'll use the graphing calculator to actually graph. To produce a graph on the screen. But here we're going to use it to create a table. And I think I noticed something. I think there's really nice numbers. Every ten songs that we download because songs cost ten cents apiece. So let's go into our table setup. Remember, we do that by hitting second window. Let's start our table at zero. And let's go make it go by tens. Now tens, that's kind of cool. Because that means every ten downloads, I'm going to see something else. All right, let's go into my table. And look at that. Now you don't have to have your table set up exactly like me or like mine. But it is kind of nice, right? When there were zero downloads, we saw that RY output was 5. Ten downloads or output was 6. 20 downloads or output was 7. 30 downloads or output was 8. And 40 downloads are outputs 9. So we can now graph this, you can probably graph it a little bit better than me, but I'll have zero 5. Ten. Oh, what are these going by? 50 6. They're going by 50 cents each. So at ten, we're at 6, which is right there. At 20, we're at 7, which is right there. 30. We're going to be at $8. A little bit tricky to see. We're right there. And 40 will be at $9. All right? I think I'm going to actually try to use my line command. And there it is. And there's my functions graph. All right. Now, technically, this is actually, if you remember something from algebra one, this is the case of a discrete variable. So really, I shouldn't be connecting these with a nice solid line because obviously the number of downloads would have to be an integer. It would have to be an integer. But sometimes we'll draw a graph is continuous, even when it's not, because it's too difficult to show just the dots instead of connected with a nice solid curve. Okay, I'm going to be clearing this out in a moment, getting rid of the TI calculator. So write down what you need to. All right, I'm going to clear it out. Let's get rid of our calculator. It'll be back later. One more time. All right, let's take a look at exercise two. That took us a while to get through exercise one. Exercise two involves something very important in terms of the graphs of relationships. Not functions, relationships. One of the following graphs shows a relationship where Y is a function of X and one does not. Let array says draw the vertical line whose equation is X equals three on both graphs. Again, I think that we're going to use our nice line utility on that. Here's X equals three on that graph. And here's X equals three on that graph. So X equals three. X equals three. Letter B says give all output values for each graph at an input of three. Well, a, we have an output there, right? So that's at the .3 comma four. And we also have an output here, and that's at three negative four. So the outputs are negative four. And four. Relationship B well, there, we had three common negative two. So only negative two. The letter C asks us to explain which of these relationships is a function. And why? Well, only B and the reason is because it has only one output. For a given input. This leads to what's called the vertical line test, the vertical line test. All students who study functions have heard of this guy, the vertical line test. And the vertical line test is relatively easy. It just says, look. If I draw a vertical line like I did in relationship a and it hits more than once, then it's not a function. But if it hits only once, then it is a function. All right, the vertical line test. It's helpful. It's helpful to quickly tell whether a graph is the graph of a function or not. All right, pause the video now, write down whatever you need to. Okay, I'm going to clear this out, and then we'll move on. Exercise three. The graph of the function Y equals X squared minus four X plus one is shown below. State this function's Y intercept. I bet you can figure this one out. What's Y intercept? Y intercept is the Y coordinate where we cross the Y axis. And that's at Y equals one. Between what two consecutive integers, consecutive means right in a row integers whole numbers. Does the larger X intercept lie? Well, here's an X intercept and here's an X intercept. This is the larger one, so that must lie between three and four. Letter C, draw the horizontal line Y equals negative two on this graph. Let me do that in a different color. There it is. That's Y equals negative two. Great. Using these two graphs find all values of X that solve the equation below. Remember, when I'm graph solving an equation graphically, and I have this equation graphed and I have this equation graphed, then all I have to do is come up with the X values in this case X equals one and X equals three, one, and three, where they intersect. Letter E says verify that these values of X's XR solutions by using the store on your graphing calculator. Well, we could use store. We could use a table. We could use a variety of things, but let me use store since that's what the direction said. Let me open up the TI 84 plus again. Great. Now let me use store. Now how am I going to do this? Well, what I'm going to do is I'm going to store X equals I'm going to store one into X so let's do that one. Store X and then what I'm going to do is I'm going to type out the left hand side of that expression. So watch X squared minus four X plus one. Now I'm going to hit enter. And it tells me negative two. Which tells me it's a solution. I could do the same thing for three, but I'm not for X equals three, but I don't think I'm going to. I just wanted to show you how easy it is to use stored check to see if something is the solution to an equation very, very simple. All right? Anyhow, I'm going to clear out the text and get rid of the calculator, so pause the video if you need to. All right, let's go on. Let's get rid of the calculator. It's gone as well. And let's move on. So in today's lesson, we laid the groundwork for maybe the most important idea in all of mathematics. The idea of a function. Its definition is simple. If I put an input in, there can only be one output out. This has a lot of different ramifications, and one of the things it gives rise to is that vertical line test, right? So use that on the homework to test whether or not graphs are, in fact, graphs of functions. We're going to see functions a lot this year. So take the homework seriously, really try to understand them upfront. All right, because you'll need them a lot later. For now, let me thank you for joining me for another common core algebra two lesson by E math instruction. My name is Kirk weiler, and until next time, keep thinking, and keep solving problems.