Common Core Algebra I.Unit 3.Lesson 3.Graphs of Functions

Hello and welcome to another common core algebra one lesson. By E math instruction. My name is Kirk weiler, and today we're going to be doing unit three lesson three graphs of functions. Before we begin this topic, let me just remind you that you can find a worksheet and a homework assignment that go with this video by clicking on the video's description. As well, don't forget about those handy dandy QR codes that we have at the top of each one of our worksheets. Use your smartphone or your tablet to scan that code and come right to this video. All right, let's start to talk about graphs and functions. Remember, functions come in many different forms included in those are equations. Graphs, tables, and verbal descriptions. Today, we're going to get more practice with function notation, which we introduced in the last lesson. And we're going to look at how to use graphs and create graphs of functions. Now, it's very important. If you did not learn function notation, this and every other lesson that includes functions are going to be very, very confusing. So be sure that you've seen a lesson whether it's ours or somebody else's on this whole F parentheses ex parentheses thing. All right, let's jump right into it. Exercise number one says, given the function Y equals F of X defined by the graph below, answer the following questions. Find the value of each of the following. F of four equals. So remember, function notation, that four is the input. F of four over all is the output, the input is X and the output is Y so literally, when we see something like F of four equals what that is really saying is, hey, what is the Y coordinate when the X coordinate is four? So we go to this graph and we say, well, one, two, three, four, and we go right up here. And what's that Y coordinate? That's a Y coordinate of one. So F of four is equal to one. F of negative one, again, that's X equals negative one. Let's see. We're right here. And we go all the way up to here, and that is one, two, three, four, 5, 6. A Y coordinate of 6. All right? Inputs are X outputs or Y and let me again emphasize the fact that the output is considered the function itself. All right, so when we talk about the function, then we're really talking about its outputs. Now take a look at letter B, it says for what values of X does F of X equal negative two. In other words, in this case, we're going backwards. We know Y is equal to negative two. We want to know the X values. Pause the video right now and see if you can figure this out. All right, let's do it. Well, for a lot of students, what they would do is they'd come down here and they'd go all right. Here's negative two. In fact, I can illustrate that really nicely. I'm going to go red here. If I take a line and I draw it across all along the negative two line. Then it's very easy to see places. Where I hit that negative two. Now what I'm looking for are X values. So that's an X value of one. And that's an X value of three. Okay. Now letter C is a very, very important problem. Read over that text by yourself for a moment. All right, it's a state the minimum and maximum values of the function. Notice how it bolded them, then a bunch of these things. It's important, the values of the function are Y values. Not X values. Those are the inputs. What a function is really concerned about more than anything else. Are the outputs, the Y values. So when you see the minimum maximum values of the function, or anything referred to of the function, we're talking about the Y values. So let's go with the minimum. All right. The minimum value. That's got to be the lowest. Let me actually put equals lowest. And that's equal to, well, let's write down here. What is that lowest value? The lowest value is negative three. So we might even write it like this. Why with a little subscript min equals negative three. The maximum value, right? The highest value well, let's see, that comes right at the top here, and we already counted up that. That was 6. So we might say that Y sub max equals 6. There they are. I think the trickiest thing about this though is the idea that the values of the function are the outputs or the Y values. Okay? So I'm going to clear the text down, pause the video now if you need to. All right, here it goes. Let's keep moving. All right. Exercise two. Consider a function given by the rule G of X equals two X plus three. All right. Now, let's make sure before we even start with tables and graphs and things like that that we understand what this rule is doing. Based on order of operations, this rule is taking the input, multiplying it by two, and then adding three to get the output. All right? So watch as we translate this equation into a graph. All right? Let's do it. Here, we're going to take our input of negative three. We're going to multiply it by two, and we're going to add three. I'm going to do this without my calculator, the numbers are relatively small. You should do it as well. So two times negative three is negative 6 plus three. And that gives me an output also of negative three. So that's a little bit of a confusing one to begin with. When the input was negative three, the output was negative three. On the other hand, when I put an input of negative two N, two times negative two is negative four plus three. Negative four plus three is negative one. So we get two negative one. All right. What I'd like you to do is pause the video right now and finish filling out that table. All right, let's go through it. I'm going to go a little bit faster now, since you had some time, and you're really just checking your answers. In each case, though, we're just fulfilling what the rule says, and then understanding that the input is the X coordinate and the output is the Y coordinate. All right. So a little bit of calculation time here. Sorry. But especially for me, it's more of an issue of making sure that I write well enough that you can actually see it on the screen. But I'm almost there. In some future lessons, we'll learn how to use our calculators and our tables on calculators to get these values a lot faster. But there they are. Input is X output is Y, and then we get a coordinate pair. Let's plot them. Negative three negative three is down here. Negative two negative one here, negative one, one, zero, three, one, 5, two, 7, and three, 9. And now I think I'll connect them with a nice straight line. Oh, it's red. And there's my graph. Maybe I even throw some arrows on the end of it. Right? Unless there's a reason not to, and sometimes there is a reason not to. We should throw arrows on the end of our graph. We'll make sure to discuss when you shouldn't do that. But here we want it. All right. So we took a rule that was given an equation form. We translated it into some table values and translated those then into its graph. All right. Let's keep going. I'm going to clear out the text. So pause the video now if you need to. All right, here we go. All cleared. Okay. Let's take a look at what's called a quadratic functions. Quadratic functions which we're going to study in depth are ones that involve X to the second as its highest power. There's all sorts of really more complicated expressions. But X times X or X squared is the simplest of all quadratic functions. Let's take a look at what this thing looks like, okay? Again, we're going to translate this function rule. Right here, that says take the input and multiply it by itself. And we're going to translate that into a graph. So let's take a look. Negative three squared. Which is negative three times negative three is positive 9. So we have negative three comma 9. Remember a negative times a negative is positive. All right? Try to avoid using your calculator on this. Negative two times negative two is a positive four. Right? If I go with my negative one. My hope is that you wouldn't have to write out the second step after a while, but it is important that you get it right. It's important that you do it without your calculator. All right, reason being is you'll get more and more practice with your arithmetic. You'll get stronger with basic ideas about multiplying signed at numbers. By hand is going to be sore after this lesson so much right in. That's okay. All right. So there we have a whole series of values that we can now plot input output pairs. Always always always. Input comma output. X comma Y all right, let's do it. Negative 312-345-6789. Right there. Negative two, one, two, three, four. Right there. Negative one, one. Zero, zero. One, one, two, four, three, 9. We'll eventually talk about all sorts of things associated with a graph like this, including symmetry. Here we go. I am remarkably bad at drawing these kind of graphs. I guess I'm trying to translate the screen onto a tablet. I suppose it could have been worse, not much worse, but trust me. It could be. So that's actually what's known as a parabola. We're going to get into that more later on in the course. But it's kind of a cool graph. Notice, by the way, that something that we talked about a couple of lessons ago. Every input, every X value gets only one Y value. But certainly, the Y values can get repeated. So for instance, the 9 and the 9 are the same. The four and the four are the same. I'm going to run out of colors soon. So I can only do so many of these, but one and one are the same. So it's no problem for inputs to get for outputs to get repeated. What we can't have our inputs repeated. No X value that's the same having two different Y values. All right, I'm going to clear this out. So pause the video now if you need to. Okay. Now the last thing that we're going to talk about is actually quite complicated. It's what's known as a piecewise defined function. Now remember, a function is simply a rule that allows you to convert an input into an output. Nobody says the rule has to be simple. It's just got to be a clearly defined rule. So piecewise defined functions are functions that are made up by combining two or more rules together based on the values of the inputs. Okay? And we'll see exactly what that means in the next exercise. But it's really kind of cool, right? It just means that we're going to combine two or more rules into a more complicated rule. That's all right. You know, we do that all the time in the real world. For instance, let's say I created a function that simply said the time that I wake up in the morning based on the day of the week. I might say something like this. Well, if the day of the week is Monday through Wednesday, then I get up at, let's say, 5 a.m.. On the other hand, if the day of the week is either Saturday or Sunday, then maybe I wake up at 7 30 a.m. I'll stretch it that far off my kids will let me sleep that late. All right, but it's a little bit more of a complicated rule because the rule that you use depends on your input. All right? So let's take a look at how that works. Holy cow. Look at this. That's the way piecewise functions are oftentimes written. So let's try to understand what that notation means. Nothing really here. Nothing to worry about here. Nothing to worry about. What this really says is that whenever X is less than zero, we're going to use this rule. Two times X plus 6. Whenever X is greater than or equal to zero, we're going to use this rule. 6 minus X that's it. That's how you interpret it. And I got lots of stuff written here, so I'm going to kind of get rid of it. But every time we apply this rule, we have to figure out which of the two smaller rules sub rules, whatever, we're going to use. So F of four, all right, here's what we do. We look at that input four, and then we look at which one of these it falls into. Now, since four is greater than or equal to zero, we use this rule, right? And we just do 6 minus four, and we get two. So F of four is two. All right. Here, though, the negative three, well, that falls into this rule, right? Because negative three is less than zero. So we'll get two times negative three. Plus 6, which gives me negative 6 plus 6, which gives me zero. All right. Now, that's just how we evaluate it based on values of X and based on the formula we're given. Now let's create a table. All right? So we always want to keep in mind which thing we're using. Now for every X value for every X value less than zero, we have to use two X plus 6. So for every X value less than zero, and that's going to be all of these, we have to use the rule two X plus 6. And we already kind of did it for negative three, but let me just do it again. And we're going to get negative three comma zero. Likewise, two times negative two. Plus 6. Gives me negative four, plus 6. Gives me two. So negative two comma two. Let's see two times negative one. Plus 6. Negative two plus 6, which is four. So I get negative one comma four. But now let me go with a different color. Maybe that'll help us out. Any time X is greater than or equal to zero, we use this formula. That's going to be for all of these. So what does that formula say? It just says do 6 minus the input. That's going to be easy. So we get zero 6 here, we'll have 6 minus one, so that'll give me 5. That'll be one comma 5. Then I'll do 6 minus two. Then I'll give me four. So two comma four. And finally, 6 minus three, which gives me three. So three comma three. I'm going to go back to blue. And let's start plotting. Negative three comma zero would be right there. Okay. Negative two, comma two, right there. Negative one, comma four would be right there. Now I'm going to go back to red. Zero comma 6 would be right here. One comma 5 right there. Two comma four would be right there. And three comma three would be right there. So I'm going to do something really quick. Let's go with blue here. Blue is going to be everywhere here. And let me throw a little arrow on there. All right. And red. Here. There's going to be right here. And again. All right, look at that. Now I will say for the record. In case you're wondering. They don't always match up. Okay. So in other words, the cutoff point. Which is the zero. The cutoff point. Oh, I got a fly in the room. How'd that happen? At the cutoff point, the two formulas won't always meet up. But they do in this case, and I'm going to keep them meeting up quite a bit at the beginning when we look at piecewise functions. Because they can be pretty complicated if they, if they sort of don't meet up the two lines or the two pieces. All right, but for now, they do meet up. All right? I'm going to clear out the text. Our little friend the fly went away. So that's good. Here we go. All right. Let's finish up. So today, we looked at how to translate a formula for a function into the graph of a function. Possibly the most important thing about functions and graphs is that the X coordinates are the inputs and the Y coordinates of the outputs. If you know that, then it's not too tough to do that translation either way. Thank you for joining me for another common core algebra one lesson by email instruction. My name is Kirk weiler, and until next time, keep thinking, and keep solving problems.