 So we're going to look a little bit more at how to use a graphing calculator. Specifically we're going to look at how you can use a graphing calculator to solve an equation. Now, your graphing calculators do have a built-in solver. If you press the math button, all of you right now, the math button is on the left side just below the function key, I believe. And you go up arrow one. I think the bottom is solver, is it not? It's terrible. It's loud. It's almost a waste of software. It's why there is a green solve button right above the equal sign, by the way. If you look on your calculator, bottom right corner, there is in green a solve button. It's in green on the TI-83. It's probably a different color on the 84s. I don't know. It's really a useless piece of software. I'm going to show you how to solve using the graph because if I'm going to type in the equation into the solver, why don't I type it into the graph and that way I get a visual representation, which might help me spot typos. For example, if I know it's supposed to be a parabola and it's a straight line, I know I forgot to hit the squared. Things like that. Okay. Amanda, you with me? You're good? Sure. Yeah. Okay. So here is how you can solve an equation. We're going to solve this equation here. So write down in your notes using the intersection method and write down example one, please. And you can do this either at the very back of your book where I had that little page called how to solve graph and how to use graph and calculators, or you can do this on a set piece of paper either or as fun. And we're going to use something called the intersection method. By the way, let's see how clever you are. What kind of an equation is this and how do I know? How do I know it's a quadratic? Because when I foil it out, what will I have? A squared. So you could, all of you, solve this by hand. You could foil everything out. Gather like turns. It's a quadratic. Make it equal to zero. And then I'm pretty sure this one doesn't factor. If it did it would be a complete flute, you'd have to use a quadratic formula. And I'll be honest. If I didn't have a graph and calculator, it's probably what I would do. Look at the left side of the graph. The left side of that equation, sorry. That's a quadratic. See how, Steph, if I multiply it out, I'd get a 2x squared. Not only is that a quadratic, it's a parabola opening upwards. Look at the right side of the equation. See if I multiplied that out, I'd get a negative x squared somewhere along the way. See it? The right side is a parabola, but it's opening which way? Because of that, okay? What I'm going to do is I'm going to graph y1 as the left side. And I'm going to graph y2 as the right side. I'm going to treat the left side and the right side as two completely separate equations. Why not? So I get up my graph and calculator, press y equals, I'll clear any graphs that I have there. Left side, 2x plus 3 bracket, bracket x plus 4, close bracket. The right side, bracket 5 minus x plus bracket, bracket x plus 3 plus bracket. And because once you have that typed, don't hit graph yet. Just get that typed in, right? Y1 and Y2, we're okay Shannon? And then before I graph, because I don't know what windows we had last time, I'm going to go zoom and standard. That gives me the negative 10 positive 10. So zoom 6, zoom standard. Gives you a negative 10 by positive 10 by negative 10 by positive 10 window. It's a pretty good window, works for most graphs. Amanda, can you see the two parabolas? Yeah. Do you know where the left side equals the right side? Where these two graphs do what? Gives you a hint in the title. Cross, where they cross? These two graphs, where they cross? That's where this left side equals the right side. Now they definitely cross there. And Steph, I'm positive just by looking at this. They cross up here somewhere. But that's off my screen. So I'm going to have to do an adjustment. I'm going to press my window button. And instead of my Y maximum height being 10, I'm going to make my Y maximum height 20. See if that'll fit. 30, if not, I'll try 40. But let's see if 20 fits at all. Now I'm going to hit graph. Did 20 fit both intersections OK on my screen? Good. Let me pause and make sure everybody has this part. If you don't, you want to tell me. Very important that you not try and fake your way through this and not learn this. Anybody didn't get this screen right now? I'll come and show you how to get it. Good. OK. So the method we're using, it's called the intersection method. It says, look, if you got an equation with stuff on both sides, graph the left side as one equation, graph the right side as two equations, and find where they cross. Now how do I find where they cross? It's built into my calculator. I want to do a calculation. I want to do a calculation. What does it say in yellow letters above the trace button? Calc for calculation. Second function trace. That brings you your calculation menu. Which one do you think I want to use here? Read what's the second function calculate. Second function trace. Look at the menus. Which one do you think I want to use here? Take an educated guess. Number five, intersection. Yes, so press the five and then pause. So you all get to this screen, and it should say first curve, question mark in the bottom of your screen. Does it OK, Katie? Good. The reason it's pausing is it's possible that you might have three or four or five different graphs on your screen. And so it's not sure which ones you want to find, which ones they cross. And so it will always ask you to identify. You there? OK. Second function trace brings you the calculate. And then we want the intersection, which is option number five. Yes? We're there now? That says first curve question mark. Good, good, good, good. If I had three or four graphs, Katie, I'm going to tell it which graphs to find where they cross because I can't find where three or four graphs cross. I'm going to do two at a time. And so I could scroll up or down between the equations. How many equations do we have this time? Exactly how many have we graphed? Two. So the shortcut is just hit enter for the first one. And then it says, is that the second one right there? Hit enter. Isabel, how many intersection points are there? Can you see? Itzel. I said Isabel because she sits there last class. Sorry. Itzel, how many intersection points are there? Can you see? Two. Sometimes you'll find on a multiple choice test, there's only one answer that has two intersection points quit. Don't even bother finding them. You know that's the right answer. Everyone else has three. Or if there's other ones that have two, but one of them, for example, has a zero as an answer, no, they're not going right through it. I'm sure they're not going through it zero. You can often find just one answer and quit because these are slow. They're not quick. But how many intersections are there, Itzel? I need to tell it which one I want to find. And that's why it's saying guess question mark. It wants me to give it a guess. And it's going to use that guess to find a better guess. And it's going to use the better guess to find a better guess. It's going to do it about a thousand times. It'll quit when it's accurate to about 99 decimal places. It actually approximates the roots. It never actually finds them, but it finds them to 99 decimal places, which Stephanie is accurate enough for our purposes. I want to find the left hand root, the left hand intersection first. And to do that, I'm just going to use my arrow key and I'm going to move the cursor to the left until I'm fairly close to where they cross. About there. That's not the answer, but that Ryan is a good guess. So hit enter. And you should see that your calculator starts to think for a moment and now it actually does find the intersection point. My intersection point is negative 3.3072776. And the height is negative 2.5 blah blah. Who can't get that? Now is the chance to ask. Okay. So you went second function calculate already and you chose intersection and you hit first curve enter and second curve enter. And it says guess on your screen in the bottom right corner, bottom left corner. So just go left arrow, left arrow, left arrow until your cursor is fairly close to here. Yeah, left arrow. And when you're really close, hit enter. We get this. Okay. Now, yep. Things just died. Come over and replace the batteries. Katie, this gives me both an X and a Y. I mean the Y is the height, which sometimes will be handy. But in my equation right now, are there any Y's in the equation anywhere in the original equation? No. Then all I care about are the X's. One of my X values, X1 is negative 3.303. Is that okay, Reggie? Pretty good. Let's find the second one. We have to do this whole procedure over again. Second function, calculate. Second function, trace. Intersection, first curve, enter. Second curve, enter. Guess, now I'll go right arrow until I'm close to where they cross over there. About there. As long as I'm fairly close, it'll find it. Enter. Positive 3.03. 3.027756. The second X value is positive 0.30. Now, we used to actually ask students to solve questions using their graphing calculator. But we used to then say, here's how you had to show the answer. I'm going to show you how we used to do it in front of across this in university. I will never ask you to solve something using a graphing calculator on a written section of a test. What I'm doing though is teaching you how to check your answer. Because this is not quite the finished product of showing work. You've shown the roots, but now what they would ask you to do is to sketch the graph roughly. Draw a rough rectangle. I'm cheating in using my graphing, my little draw software, about the same size or the same dimensions as your graphing calculator screen. I would sketch the Y axis. I would sketch the X axis and you'll notice I'm free handing. And then I'm going to roughly trace this. Let's see, there's a parabola right about here. A little too low off my screen. Parabola right about here. And another parabola about there. The call close enough. I realized that this intersection is really further right than this actual close enough. Okay? If we were marking these back say eight years ago when we had these on the provincial exams, I would look for this, I would compare it to oh wait a minute though, the only way I could compare it to my graphing calculator they would have to give me one more piece of information. They would have to tell me what view windows they had used. They would have to tell me that they had gone from negative 10 to 10 to negative 10 to 20 scale one. Here's how you show that. Square bracket. Negative 10, 10, 1. That told me your X values that you typed in. Square bracket. Negative 10, 20, 1. Close square bracket. That told me your Y values. And then Tyson, I could recreate your graph. You told me what you'd graphed as Y1, what you'd graphed as Y2, what you'd change your view windows to like and where the intersections go. You can't use this method anymore. If you use this to solve an equation I'm giving you a zero. However, Kirsten you can use this to check your answer. In fact, example two you're going to get a question very very similar to this on your test. What kind of an equation is this? This is the brand new one that we learned a couple of days ago. Think where is the X and that tells you the name of this equation. What type of an equation is this? Expedential equation. How do I know the X is an exponent? So now we've got the quadratic from last year. We're building to our repertoire. How do I solve these? You don't know how to solve them by hand yet. You need something called a logarithm. That's going to be the next few lessons. However, you can check this on your graphing calculator. Graph the left side as Y1. Right side as Y2. So let's go to our graphing calculator. Let's clear any windows here. Clear any graphs. I mean, sorry I said windows. Clear your graphs. And you know what? Before we go further let's go zoom standard so we're back to our negative 10, positive 10, negative back to this window. Now let's graph 5 to the X as Y1 15 as Y2. Now, was it Kirsten I picked on last time? Kirsten, what does every single exponential graph look like? It was you I picked on for this, wasn't it? No? Who did I pick on for the? It was you. Come on. Work with me here girl. What does every single exponential graph look like? Come on. Like this. Or like that. So that Y1 I know it's going to look like this. Look at Y2. What does Y1 Y equals 15 look like? I gave you a hint. It's a line. All lines are straight. Thank you Captain Obvious. I know what you mean but you're more specific than straight. That's a straight line. I don't think you mean that. Did you? What does Y equals 15 look like? Okay. Pardon me. It's not a vertical line. It's a horizontal line. It's a horizontal line. How high? The line Y equals 15 says put dots 15 high a whole bunch of them and connect them. So if I graph this right now there's Kirsten's exponential. See it Kirsten? But the line Y equals 15 is how high? And how high is my graph right now? 10. So you know what? I'm going to go back and I'm going to change my view window so that my Y max is oh heck 16. So it fits. Graph. There's Y equals 5 to the X. There's the line Y equals 15. Where is the solution here? What's the answer to this equation where this graph and this graph do what? Interest. How many solutions are there by the way? One. And you know what? I can even tell you it's just below 2. The X value is going to be like 1.8 or something like that if I'm reading the scale properly. But let's find it officially. Second function. Calculate. Intersection. First curve. Enter. Second curve. Enter. Guess. Since there's only one intersection I'm not even going to move my cursor. I'm just going to enter for my guess because that's faster. I don't need to tell it which one to go find. And I get an answer of X equals 1.68 2 6 0 1.68 3. Amanda This is an exponential equation but the ones we did last time we wrote as a common base. Can I write 15 as 5 to some power though? No. 25 would have been fine. That would have been 5 squared. X equal, I would have equated the I would have said do I have one base equals one base? Yes. Are my bases the same? Yes. I can equate the exponents. What we're asking now is, well what if you can't? First of all, the answers are going to be decimals. Not nice numbers like we were doing last time. And we are going to find a way to solve this by hand. I've just shown you though on the test how you can check your answer after you solved it by hand. Okay? And I would totally do that before I had to test it. There's a second method that you can use to solve equation and it's called the zero method but it is not my favorite. It's tedious. Back when I wrote the software for this is not not good software. So, okay. Suppose they gave me a question like this. First of all, this is a polynomial and it's factored. X bracket, X plus three in brackets and X minus two in brackets. Because it's factored I can actually tell you what are the roots. I can tell you them right now. Negative three and plus two is incorrect. Ah, zero. Because there's an X in front. Negative three and positive two. There are three roots. What's this equation equal to, Carson? That's why this is called the zero method. It's when you make your equation equal to zero. If it wasn't equal to zero, I wouldn't waste time though. I'd grab left side, grab right side and find where they cross. But this is already ego. Okay, fine. So, I'm going to go to my graphing calculator. I'm going to clear anything that's in my Y equals and I'm going to graph X bracket X plus three close bracket, bracket X minus two close bracket. And again, because I think we changed our window, I'm going to go zoom standard. So, go zoom six and then hit graph. Oh, it graphs automatically. Our windows should all look the same and I'll pause and make sure you all get there. And if you can't get there now is the chance to tell me. Everybody there? Alright. And again for what it's worth here, Matthias, I can actually read the roots. I'm pretty sure looking at the hash marks that that's negative three. That's definitely zero on that, too. But I want to show you, because I know the answers ahead of time, I can show you how we can find them mathematically on a graphing calculator. Okay. Are we all there on the screen? I've got a graph looking like this because I'm seeing a bunch of heads down. Okay. Once again, we're going to do a calculation. So once again, it's second function calculate second function trace. This is the zero method. Which option do you think I want to choose? Option two. How many roots? How many x-intercepts? How many zeros are there? Count. How many are there? Three. I need to tell my calculator which one to look for. And that's what this left bound question mark is asking. What I'm going to do is I'm going to move my cursor using my left arrow until I'm just to the left of the first root about there just past it to the left and I'm going to hit enter and when I do that a little arrow appears above the cursor pointing to the right and it's saying okay I'm going to look for an x-intercept to the right of this arrow. You've given me my left hand boundary. What does it say right here, Sabrina? So now I'm going to move just to the right of the root. How about right about there? I am just to the right of it, yes? And hit enter and now I've given it the right hand boundary. See how the two arrows appear and it's saying I'm going to look for a root below in this area in this little column. I'm going to look for an x-intercept. There better be one. Is that okay so far? Then it wants me to make a guess. You know what? Just hit enter because it puts a guess in there anyways and it's a good guess. So if you hit enter it'll find this first root. There it is, negative three. How do I find the middle root? I start this whole thing over. Second function calculate, zero. Now it wants the left bound. I want the middle root so I'm going to move the cursor until I'm just to the left of the middle root. How about right about there, Ryan? Is that okay? I'm going to hit enter. I'm going to move the cursor until I'm just to the right of this middle root. How about about there? Enter. It's saying I'm going to look for a root in this area. And there's how I can get the zero as a root. Yeah? Find the third one on your own. See if you can get the two. There's my two. I'm not a fan of this. You know why? That's a lot of pecking away on the keyboard. I'll be honest if they gave me this and I really wanted to use my, first of all if they gave me this I wouldn't need to use my graphic calculator because it's factored. If they gave it to me in unfactored form I would use the intersection method. I would graph that as y1, the left-hand side. You know what I would graph as y2? What's the right-hand side here of this equation? I would graph that. Graph. Now I can't see that it's graphed y equals zero because that's technically the x-axis, but now I would go second function, calculate, intersection. First curve, second curve, guess. There's the first one. Second function, calculate, intersection. First curve, second curve, guess. There's that one. Second function, calculate, intersection. First curve, second curve, guess. I think that's faster. So I rarely, if ever, use this. However, I did want to teach you that left-bound, right-bound thing because it's handy for something else. Now, this is really for those of you that are in Calculus, which is none of you because Calculus class is this block here. That's okay. You can use this to find maximums and minimums. Maximums and minimums. So, go back to y equals clear whatever you got there and type in this equation, 0.1x cubed minus x plus 2 and hit graph. And you get a little squiggly kind of this thing, yes? And you don't get that? We call this area right here, you see how it's the top of a hill? We call it a local maximum. You know what we call this section right here? Local minimum. Why do we say local? Because this is not the highest point on the graph. The highest point on the graph is infinity. It's not the lowest point on the graph. The lowest point of the graph is negative infinity. This graph has a domain of all reels and a range of all reels. But locally in that area, that's the lowest point. Locally in that area, that's the highest point. Show you how you can find that. Sometimes, remember last year when you did the parabola, when you were doing those optimization word problems, or the max... The vertex of a parabola is a maximum or a minimum. You could actually find it using this software instead if you graph it. Second function, calculate. Andrew, what do you want to find first? The maximum or the minimum? Okay, let's pick option 4. Now the maximum is right there. So it says again left bound which means I'm going to go left arrow, left arrow until I'm to the left of the highest point. How about there? Enter. I'm going to go right arrow, right arrow, right arrow, right arrow, right arrow, right arrow, until I'm to the right of the highest point. How about there? Enter. I'm going to look for the highest point in this area. When x equals negative 1.825743, the maximum height is 3.2171612. Get that okay, Nicole? Let's try the minimum instead. That way you can try it with me. Ready, Nicole? You get it now. I really need you to stop hitting keys and look up so I have your attention for this one. That's what I'm getting at. Let's find the bottom trough, the minimum right there. Nicole, got the graph on your screen? Second function, calculate. Minimum. I'm going to go just to the left of this valley with my cursor key. How about right about there? I'm going to hit enter. It says right bound question mark. I'm going to go to the right of this valley over here somewhere and hit enter. Then it wants me to make a guess. Just hit enter again. The lowest point is that. When is that handy? Well, to be honest, I'd use it for something like this. Here's a parabola. What's the vertex, the lowest point? Second function, calculate. Minimum. Left bound, right there. Right bound. Over there. The vertex of this parabola that I just made up is 0.6666666585 and 0.666666 By the way, I'm willing to bet the vertex is 2 thirds comma 2 thirds and it just can't quite get that. Mr. Dirk. Yes, Jessica. Why don't they just give us these in grade eight and not teach us math at all? Good point. Because these are stupid. There's basic questions that they can't solve. Let me give you an example. Don't write this down. What if I said solve x minus 3 all squared equals 0? What are the roots? All of you. 3 is correct. Is 0 a root? If 0 was a root, what would I have to have in front of everything? Next. There's only one root. We called it a double root last year. Right? It's also a parabola with 3, right? If I try going, second function, first section, first curve, second curve, yes. It won't find it. Your graphing calculators cannot find where graphs touch. They can only find where graphs cross. And a lot of the time you'll have graphs that kiss each other but never cross. They can't find any of those. The software does not. That's a very basic limitation. There's plenty of others. It turns to the homework from last day. We left off on page 105. Page 105. Kirsten, what did every single exponential graph look like? That's your cue. Or? You gotta be hitting them in the head if you want to. I gave you permission to hit them in the head whenever you do that. We said this, Kirsten. Every exponential graph has a domain of what? What about the range? It was everything above 0 but not touching. Okay? We said that every single exponential graph had no x-intercept. The y-intercept was 0, 1. Actually, it was 0, a because it's 0, 1 if there's an invisible 1 in front of everything but whatever a is, that's your vertical stretch. If a is positive when b is bigger than 1, for example, we call this a growth function. And when 0 less than b less than 1 when b is between 0 and 1, fancy word for that is when b is a fraction, it's a decay function. Growth function looks like this. Decay function looks like this. Growth and decay. Don't worry about this, let's jump straight to example 1. Example 1 says, describe how the graph of the second function compares to the graph of the first function. So here's my first function. Financial graph looks something like this. How has this graph been moved? Hmm. I see a 2 in front of everything. Vertical or horizontal and how do I know without hesitating at all? Why do I know it's vertical? If it was horizontal, where would it be? There is an exponent. Vertical what? What's that 2 do? Steph, you're right. Bye. Is there a horizontal expansion? No. Reflections, when I say reflections, what are we really looking for? Negatives, are there? Slides, oh! To what? To write. So, imagine Lea, this graph expanded by 4 and then moved to right. How has this graph been moved? Expansions, compressions, are there any? No. Horizontal or vertical? Is there a horizontal, or is there a vertical expansion? Nope. Otherwise, there'd be something in front of the 2 in brackets. Is there a horizontal expansion compression? Yup. Oh, is it next to the x where it belongs? Why, that would mean everything's backwards. Sorry. Anyone? What's that over 5 do? Ah, nothing. So, maybe you needed that. Okay, we got a horizontal expansion by 5. Any other expansion compressions? Nope. Reflections, then what? Example 2 says explain using transformations by the graph of y equals 1 third to the x is a reflection in the y axis. It's a horizontal reflection of this graph. Why is this a reflection of this? Well, what's another way to write 1 third as 3 to some power? I hear you saying this, Tyson, which is correct. What can I do with those 2 exponents though when I have a power to a power? How does that compare to that? You know what? The graph of y equals 1 third to the x is a horizontal reflection of y equals 3 to the x. It is. What's your homework? 1 is good, 2 is good. 4 all skip D 5 is good. Skip 6 7 is good, 8 is good.