Common Core Algebra I.Unit 3.Lesson 5.Exploring Functions on the Graphing Calculator

Hello and welcome to another E math instruction common core algebra one lesson. My name is Kirk weiler. And today, we're going to be doing unit three lesson number 5 on exploring functions using the graphing calculator. This is going to be an exciting lesson because we're going to be reviewing techniques that you can use specifically with the Texas Instruments 84 or the TI 84 plus. In order to look at functions in order to solve equations using graphs of functions and in order to just basically produce tables and explore general function behavior. Let me just remind you that you can find the worksheet and a homework that goes with this lesson by clicking on the description of the lesson or by visiting our website at WWW dot E math instruction dot com. As well on the top of every worksheet, don't forget that we've got our QR codes. These QR codes will allow you to take a smartphone or a tablet, scan the code and be directed to this video. All right, let's get right into it. The graphing calculator is an amazingly powerful tool, because it allows us to do the arithmetic of a problem to apply a function's rule very, very quickly. So what we're going to do here is we're going to look and exercise one at F of X equals one half X plus two, all right? Here's our rule. And we should automatically be looking at these rules now and saying, all right, I'm going to take the input. I'm going to multiply it by one half, and I'm going to add two to that result. Now we could certainly do that by hand. But the graphing calculator can do it very, very quickly. So what I'm going to do now for the first time this lesson is I'm going to be opening up the Texas Instruments or the TI 84 plus. Let's bring that up on the screen now. All right, so let's talk about how we're going to use tables to figure out letter a, where we're trying to figure out F of negative 6, F of zero, and F of 8. The first thing that we have to do is enter our equation as if we were going to graph it. So what we're going to do is we're going to hit the Y equals button. If we have any equations in Y one or Y two or anything like that, we're going to clear them out by hitting the clear button. And then hitting enter. All right. So in Y one, what I'm going to do is I'm going to put in one half X plus two. All right? Be very careful. Okay? So once we have the equation in there, what we can do is we can set up a table. Now in order to set up the table, what you have to do is go to that top row of buttons, and you have to go into what's called table setup. All right? So what I'm going to be doing is hitting second table to go into table setup. And there's two really important things when you set up a table. There's the X value at which you start. That's kind of called table start. And then there's that little triangle with a table, right? You've seen that little triangle before. It means the change in, right? Like in the slope formula or the average rate of change formula. So the change in the table is how much the table goes by does it go by one's disco by twos does it go by point 5s, right? So what we can do in letter X in this particular problem is we can make our table start at negative 6. Let's do that. And let's make our table go by ones. Okay? So we've got all that set up. Now normally we would probably graph the function at this point, but what we want to do is we want to look at its table. So we're going to hit second graph because that's where we get the table. So let's hit second graph. And it pops us into this table. Now it's a vertical table and what we can see on the left are the X values starting at negative 6. Now, if we had to go to negative 7 or negative 8, we could easily do that by hitting the up arrow. So hitting the up and down arrows will scroll through the table. And what's beautiful now is that we literally see a table that shows us the inputs to the function, the X values. And the output of the function, the Y values. So figuring out F of negative 6, F of zero and F of 8 is simple as long as you know that these things is the negative 6, the zero and the 8 are X values and what we're looking for are the Y values. So we look at our table and we look at what we have when we have an X value of negative 6. And we find we have an output of negative one. We have to scroll down a little bit until we get to an X value of zero, but once we get to the next value of zero, we see that the output is two. Now we have to scroll quite a bit more. But once we scroll enough, what we see is for an X value of 8, we're going to get a Y value or an output of 6. See how that easy that is? You could conceivably do this even in a situation where you had a multiple choice question and you were just trying to figure out F of whatever or G of whatever. Now letter B's kind of needed says explore the table to determine the value of X for which F of X equals 11. As a very important, I'm going to actually so important I'm going to change the color on my pet. All right? This is no longer an input. This is now an output. All right? It's a Y value. So it would be very, very easy to make a mistake and go down to X equals 11 and figure out what the Y is, but we're looking for a Y value. So let's scroll through our table and see if we can find a Y value of 11. Well, by the way, we notice that the Y values are increasing here, so we just got to keep at it. And there it is. You see it there on the table. And all we have to do is now sort of read the table backwards, if you will, and we find an X value of 18 gives us a Y value of 11. What we've really done is we've really effectively solved the equation one half X plus two equals 11. We found the value of X that gives us that 18. All right. Perhaps the most powerful thing, though, that we can do with these tables, is to create a table that we can then translate into a graph of a function. So let's actually fill this table out. Now, take a look at letter C for a second. In letter C, the table actually goes by twos. So why don't we use the power of the table setup to make the table go by twos? All right? So I'm going to go back into my table setup. All right, I'm going to still make my table start at negative 6. But now that Delta table, that little triangle TBL, now I'm going to set that to be two, because I want my values to go by twos. Consecutive even integers. And now it's so easy to fill out this table. Let's start at negative 6. And what do we see? We see a Y value of negative one, then we go to negative four, and we see Y value of zero. Actually, these Y values turn out to be quite nice, right? And X value of negative two, we have a Y value of one, and in fact, it's pretty quick to see the pattern. They just are consecutive integers at this point. But the table allows us to see that pattern. Now, of course, we compare these together to get coordinate points. Let me write these out really quick. And then we'll graph it. At this point in time, if you can enter a formula into Y one in your calculator, there's really almost no excuse for getting its graph roll. Because we can now easily find points and plot them. Without having to go through perhaps all of the arithmetic where we might make errors. I'm going to use the prefab graph in utility on this to graph the line. All right? And then I'm going to go back. I'm going to actually throw some arrows on it. Extend it there by hand. There we go. Wasn't that easy? I love the ability to use tables on my calculator to quickly generate outputs for inputs and then be able to graph functions. I'm going to pause for a minute, let you write down anything you need to. All right, I'm going to scrub out the text. During this lesson, I'm going to have to move that graph and calculator all over the place and move my face all over the place. Hopefully they won't run into each other. So please excuse me if I cover up portions of the screen I'll keep trying to let you see what's important. Okay? Let's move on. All right, take a look at exercise number one and we're going to continue this one. We're still looking at the function F of X equals one half X plus two. And I've got it graphed there. It says do the following by using your graphing calculator's table function. They now give me a new function. G of X equals 5 minus X oh, I like the green. All right, and what we want to do is we want to graph that and on the same set of axes we want to find the point where they intersect. Okay? So what we're going to do is we're going to go back into that TI 84 plus. We're going to go into Y equals now I could clear that first equation, but I'm going to actually leave it there for a moment, okay? And what I'm going to do is I'm going to put in Y two, 5 minus X, so let's do that. All right. We have the formula in there. We want to always make sure it's right. Make sure that that's 5 minus X and not 5 with a little negative X attached to it. That's a completely different expression. And now what I'm going to do is I'm going to pop into my table again. Now, before I do that, maybe what I'll do is I'll go into my table setup, and I'll make it go by ones again. So I'm going to go into my table setup. I might as well continue to start it at negative 6 because that's where my graph paper starts. But I'll make my delta table equal to one again. All right? Let me now go into my table. Okay. Now, if you notice out there at negative 6, let's scroll, make sure we see the negative 6. Out there at negative 6, this function is actually quite large. It's 11. So it doesn't even fit on my graph paper. I can't. I can't plot 11 because it would be somewhere up here. All right? But what I can do is I can scroll through the table until I find the first point that really fits, and that's where I have X equals negative one and Y equals 6. So I'm just going to immediately plot that. I'm not going to create a table right now. I'm going to plot right from the table I see. And then I see when X is equal to zero, Y is equal to 5. So I'll plot that. And when X is equal to one, Y is equal to four. When X is equal to two, Y is equal to three. When X is equal to three, Y is equal to two, et cetera. I think I see the pattern. Right. I think I'll graph this again. Using my prefab graphing program that didn't work all that well. Uh oh. And I sure didn't want to flip the page there. Here we go. So there's my function G of X all right. So I've got a nice graph of it, maybe throw a few arrows on the end. I probably should have an arrow on the end here too. And then it also asks me to find the intersection point. Well, that's sitting. Right here. So they intersect at two comma three. That's what's called the solution to that system. A system is just two equations. With two variables, then the kind of system that we've got here is two equations with two variables. And solving it just means to find where they intersect. You might remember that from previous courses. Anyway, letter E says show that the point that we found in D, our friend right here, is the solution to both equations. Now remember what a solution is is something that makes the equations check. So let's check this one. Let's put three in for Y let's put two in for X and see if this is actually true. Well, as we should know, one half times two is one. And yes. So it's a solution. It tells the truth. Let's do this one. This is going to be even easier. Let's put three in for Y and two infra X and again, three is equal to three. All right. So we have it. Wow. It's amazing how much easier tables make graphing. So always consider that whenever you have to produce a graph of anything consider using tables on your calculator, there's nothing wrong with having the calculator. Do the rote arithmetic for us. Okay? So we're going to clear out the text. And then actually go on hopefully to exercise two. All right, let's do it. Exercise two. Oh, more complicated function. Consider the function Y equals. X minus one, quantity squared, minus four, over the interval negative one to four. Now this is important whenever a problem tells you that you're between two X values, that's it. Just assume that there's nothing sort of like out here in terms of other X values or out here. All right, letter a says, create a table of values for this function over the specified interval. So here we go again with the TI 84 plus. Let's hit Y equals. Okay. Let's clear out the equations that we had from the previous exercise. Now, Y one, we need to be very, very careful. And we're going to put in X minus one in parentheses. And we're going to hit the squaring button. Get back out of that. And then we're going to do minus four. Always look your equation over carefully. There's nothing that can be done if you put the equation in it correctly. All right? X minus one squared minus four. Now we want to create a table of values. So we're going to go into our table setup. And we're going to make our table start at negative one. Makes sense. All right, now I'm just going to draw because of the space I was given. I'm going to draw my table out like this. I know in the calculator, it's vertical. But we can handle this, right? What do we see? At negative one, we have a value of zero. At zero, we have a value of negative three. At one, we have a value of negative four. At two, we have a value of negative three. At three, we have a value of zero. And then four, we have a value of 5. All right? There you go. That quick. That used to take a half a period a long time ago. I mean, I've been teaching since like the 1800s. So I remember even like slide rules and abaca or abacus, or I don't know. However, that's produced. Anyway, let's plot this thing. We've got negative one zero. All right. We've got zero negative three. We've got one negative four. Two negative three. Three zero. And four, 5. And now I'm going to attempt to draw a nice curve through here. Okay, that was that was horrible. But there is no parabola program on here. Okay, so we've got our curve. Not the greatest thing. We're not going to put errors. The reason we're not going to put arrows is because we've been told, we're only thinking about this function from negative one to four, so it starts when X is negative one, it ends when X is four. Letter C, what are the functions minimum, and maximum values. Now remember, the functions, right? Those are the Y values, not the X values. Why would they ask us the largest and smallest X values? They're right there. So let's go with the minimum Y value. I'll do a little subscript here. Why minimum? Let me change the color. Here's the lowest point on the graph, I can come right over here and I can see that the minimum value is negative four. All right? Let's do the maximum value. All right, the highest Y value. Again, let me go with red. Here it is. Go over one, two, three, four, 5. The maximum value is 5. All right. Over what interval is the function negative, remember. Negative is going to mean below the X axis. That makes sense below the X axis, right? That's everywhere down here. I'm going to, I'm going to be marking this graph up like mad. Okay? But the interval is all about the domain, the X values, all right? So as X goes from negative one, all the way to three, this function is negative. Now you might wonder why I didn't put the equals there, right? The reason I didn't put the equals there is at X equals negative one, the function is zero, which isn't negative. That's just a neutral number. At X equals three, it's also zero. You could also express this in interval notation. Now, if your teachers didn't teach you interval notation, this might look kind of confusing. So if you see this and you have no idea if you're thinking to yourself, why you put a coordinate point in town, especially one that's not on here. Don't worry about it. This is what's known as interval notation if you don't use it in your class, that's okay. Common core standards are unclear about what kind of notation student should learn. And when they should learn it. Anyhow, letter E says, for your graph, state the interval over which the function is increasing. Again, let's keep going with more and more colors. Maybe I'll use black. The function is increasing all in here. All right? It starts to increase when X is one. And I guess it just plainly flat out stops when X is four. And that time I am going to include the equal signs. And if you use interval notation, you would use what's called square brackets. All right. Finally, letter F says, how can this graph help us to solve this equation? X squared minus X minus one squared minus four equals negative three. Well, remember, forget about the negative three just for a second. This is what we just graphed. All right? So ultimately, what we're doing when we solve this equation is we're finding the X values that will give us a Y value of negative three. Well, here's a Y value of negative three. Here's a Y value of negative three. So that looks like it happens at an X value of zero. And an X value of two. You can also see that in the table, I'm going to change to red real quick. We're looking for Y values of negative three. Well, here's one, and here's one. And the X values that give us those things which are what we're solving for are zero and two. Wow. That's a great problem. I like that problem a lot. Anyway, I'm going to pause the video right now, or I'm going to let you pause the video right now before I scrub the text. All right, let's get rid of the text. I bet my head was really small in this picture because there's just nowhere for me to put me. Here it goes. All right, gone. Okay. So one last problem, pretty simple. Nice multiple choice. Which of the following is a point where Y equals three halves X plus 7 and Y equals negative 5 X -6 intersect. All right, this is really cool. Normally this would take some kind of algebraic approach or you'd have to graph both of these lines. But we can do this completely with tables. So let's open up that TI 84 plus one more time. Let's hit Y equals on it. Let's clear out any equations that might be in there, probably that quadratic from the last problem. And now in Y one, let's put in three halves X plus 7. So that's going to be going in Y one. Three halves X plus 7. And then in Y two, let's put in negative 5 X -6. Again, watch yourself. That's a negative sign. And that's a subtraction. If you get any errors, that could be Y all right, let's make sure those are correct. Look at both equations carefully. Okay. Now let's go into the table, set up, all right, where should we start our table? It's questionable. We could look at the multiple choice answers and we can say, well, the smallest X value there's negative two. I'm just going to start it at negative 5 and make it go by one. Okay? So let's pop into the table. All right. And what we're really doing now is we're looking for a point that's the same on both functions. Now what that really means is I'm looking for a place where the Y coordinate is the same for both. All right, now if I scroll down, I pretty much see the correct choice first. What I see in my table is something that looks like this. X, Y one, Y two, and we see a Y coordinate of four from X coordinate of negative two on both lines. It's right there. Do you see it? Now, just for a moment, we know that three is the correct choice, but let's just take a look at choice one and see why it doesn't work. If we go down, pan down that table until we get to X equals zero. What we see is that Y one, the output is 7, but Y two, the output is negative 6. So that means that they don't intersect at X equals zero. So tables are very powerful way of figuring out where to curves. They don't have to both be lines like in this case. But where two equations share common points. All right. I'm going to clear out that text. And it's gone. All right, let's finish up. So I hope that today's lesson was helpful in terms of reviewing or teaching you simply how to use specifically a TI 84 plus, but this would work with any of the TI models. How to use those to set up a table, how to graph a function using that table, and perhaps how to even solve some equations and solve some systems by using tables. All right. We'll be using the graph and calculator a lot in this course to do graphing. So it's an important thing for you to get up to speed on. For now though, let me just remind you that this has been another E map instruction common core output for one lesson. My name is Kurt weiler and until next time. Keep thinking. I keep solving problems.