 So we're live, says this, graphing exponential and log function. We're going to start out looking at a couple of exponential functions. It says, consider the following function, y equals 2 to the x plus 2 minus 3. I'm going to argue that this is an exponential function. Amanda, how do I know this is an exponential function? Where is the x sitting? What's your base here? I'm going to argue this graph is based on this. And to graph either a log function or an exponential function, I'm always going to start out by doing a table of values for the basic graph. Then I'm going to do the transformations, then I'm going to sketch it. In other words, I'm going to plug in a zero right there. When x is zero, what's y? What's 2 to the zero power? What's 2 to the zero power? One, I know that when x is zero, y is one, that's on the original. I'm not going to put this on my graph paper because it's been moved around. Put a 1 there. What's 2 to the 1? 2? Put a 2 there. What's 2 to the 2? Put a 3 there. What's 2 to the 3rd? 8? In your head, put a 4 there. What's 2 to the 4th? 16, which is probably off my graph, which is why I didn't bother actually trying it. I usually try 0, 1, 2, 3, and I try negative 1. What's 2 to the negative 1? Elevator. Okay. And then there's one more thing. Kirsten, what does every single exponential graph look like? What I want you to realize, Kirsten, is your lower hand is curving closer and closer and closer and closer and closer to zero, but Kirsten, it's never touching zero. What I want you to realize is there's an asymptote. And what's the asymptote for an exponential starting out? It's the y-axis. Sorry. I said it wrong. Fixed that on the internet, Mr. It's the x-axis. This is the asymptote, which as an equation is this. Every single exponential graph has that. And then if there was a 3-dare, these numbers would be different, but the shape would be similar. Okay. I'm going to add two more things just for practice. Underneath where it says asymptote, I'm going to write domain and I'm going to write range. If every single exponential graph looks like this, what's the domain worth memorizing? What was the range of every single exponential? They were everything above, but not touching zero. I would, now it took us about three minutes to do this in your head quietly. It takes about 30 seconds when you get the hang of this. You jot this off in the margin somewhere. Now I wrote here, do the transformations. How has this graph been moved? What does that plus two do to the graph? What else? I think three down. Yes, because the three is no longer on the y's, so it's not backwards. Good review of transformations. Yes, see it? So, you ready? When I say left, what I really think is x values, x values, x values, I would take out a pencil or I'll change colors and I would say, move that to left. Move that to, by the way, to left is the same as subtracting two on a graph. Yes? Move this to left, move this to left, move this to left. Is that okay so far, Katie? What letters right here? Because I asked you to, what letters right here? I'll never get tired of that. A y, so I'm not going to touch that because I'm doing left and horizontal. What letters right here? X, so the domain would move to left, sorry, to left for you guys is this way, does that really change the domain? If you go from negative infinity to positive infinity and just move it a bit, okay, so that's not going to change, but I would look at it just to remind myself. And Katie, what letters right here? Because I'm not going to change it then. Now, down, vertical, I'm going to change anything with a y. Three down means subtract three. This is going to become a negative two, a negative one, a one, a five, one and a half, take away three, Kirsten is negative 2.5, okay, I can do a decimal. Kirsten, what letter is right here? Because I asked you, I'm never going to get tired of that joke. Since it's a y and we're doing a vertical transformation, I'll also subtract three from there, what letter is right here? I'm not going to touch that because I'm doing vertical stuff. What letter is right here? So I'll subtract three from there too. In fact, I can find the new range fairly easily. The new range is going to be everything above negative three. So if it was multiple choice and I just asked you, what's the new range? Do you now have a way you could get that without graphing? Okay, now let's graph this. I always graph the asymptote first. The asymptote is not y equals zero. The asymptote is y equals negative three. That's a horizontal line, negative three, high, so far so good. And then it looks like this graph is going to go through negative two, negative two, negative two, negative two right there. Negative one, negative one, zero comma one, one comma one, two, three, four, five. And also negative three, negative two and a half. I'm going to argue those five points if we remember what Kirsten's arms were doing are enough to figure out the rest because what does every single exponential graph look like? Or I'm pretty sure this way, by the way, that's why I had you doing that. It's so that, in fact, I would even argue this. Kirsten, if all I could find were those three, I think if I know it's an exponential and I know it's supposed to look like this, I think I could say, oh, it's going to go like that. Five is nicer to have, no denying. And let's bring that point back, there we go. So it's going to look like this, that's how I would graph this exponential graph. What if there had been a negative in front that would also give me a vertical reflection that would flip that, change the sign, change the sign, change the sign, change the sign, change the sign, ignore that and instead of greater than, you know what it would become? Below, less than, that'd be a vertical reflection. Example two we're going to come back to, turn the page, example three. I'm going to argue that this is an exponential graph. You know how I can tell just by glancing at it? Yeah, I got it. What's this graph based on? Well, I gave you a hint, Holly, pause. What don't you like about this one? What's bugging you about this one? Why is this one looking tougher than the first one? Fraction base, it's not a fraction base, yes it is. Steph, I'm telling you it's not a fraction base. Mr. Dewick, I'm telling you I see a one-third there. Steph, I'm telling you I actually see this, what did I do? Elevator and now it's no longer fractional. I said, look, if there's a fraction, that's just like having a negative in front of your external. This graph is based on y equals three to the x. I did tell you to memorize certain exponents, this is why. Because if I put a zero there, what's three to the zero? One, what's three to the one? Three, what's three squared? Nine, what's three to the third? I doubt I'm fitting that on my graph. So I'll write it down anticipating I won't use that. Oh, what's three to the negative ones, Steph? Three to the negative one, I agree. Oh, what does every single exponential graph look like? It has as an asymptote, that there. And then just for practice, we're going to add the domain and we're going to add the range. So the domain for every exponential we said was all reals. The range for every, we said was everything above zero. Now let's list our transformations. What's going on here? Oh, correct order, please. Expansion compressions, do you see any? Where? Ah, way up here. There is a coefficient in front of the x. Horizontal or vertical and how do I know? Because it's an x and timesing by two, is that a horizontal expansion by two or a compression by a half? So over here I'm going to write horizontal comp. A half. Why over here? Because there's room, you don't have room to do it over here. Like I did in the previous one. Are there any other expansion compressions? No, now if there was a coefficient in front here, I can handle it. Be a vertical stretch. Big whoop. What did we do after expansion compressions? Remember reflections? Are there any? Ah, yes. Now there wasn't in the original, but I introduced one by rewriting it a nicer way. Horizontal, how do I know it's a horizontal reflection? Because where is the negative next to the x? Any other reflections? No, slides. To what? Now remember Cassandra, anything with horizontal only affects stuff with an x in it. Anything with vertical only affects stuff with a y in it. So now I'm going to fix my points. Horizontal compression by a half, that's the same as timesing by a half, or holly dividing by two, what's zero divided by two stays the same. That's going to become a one half. Yeah, decimal, I'll deal with it though. That's going to become a one. That's going to become a 1.5. That's going to become a negative one half. Raggedy, what letter is right here? Oh, not going to touch that. What letter is right here? Now if your domain is all reals from negative infinity to positive infinity, and you do this, what's your domain still? I think it's still all reals, even if you've compressed, so that's not going to change. Oh, and what letter is here, Raggedy? It's not going to touch it. Check. Horizontal reflection, anything with an x is going to become negative. What's negative zero? Oh, zero. Negative, negative, negative. Oh, this one's going to become positive. And the range isn't going to change because they take your range and you flip it. I'm pretty sure that still ends up with negative infinity, positive infinity, yes. Verticals, two up, three, five, eleven, maybe off my graph, but I can eyeball it. Yeah, you know what? This 27 is not going to fit on my graph because 29, I'm not going to bother. Ooh, two up, this is going to be two and a third, which I can eyeball. Roughly, what letter right here, what letter is that? That's going to get moved two up, what letter is that? That's going to get moved two up, is that okay? Will I always graph first? Yes, until, oh, two up, Mr. Dewick. And then this graph goes through zero comma three. Negative a half comma one, two, three, four, five. Negative one comma eleven, which is one off my graph, but I'm going to argue those three points. Christian, what does every single exponential graph look like? Or, and I think it's that second one, starting bigger, getting smaller. Can you see it just from those three points? I'm positive it's going to go like this and it's going to get closer and closer to that asymptote without touching. Okay, oh, how could we check our answer? Get your graphing calculators out. Clear any graphs you have in there? Clear, clear, clear. I'm going to type in this equation, the original. Now I've got to type very carefully. First of all, I'm going to have to put the one third in brackets to the power of, and since there's more than one thing in the exponent, I'm going to have to put the whole exponent in brackets, bracket two x closed bracket, plus two, enter. My graph paper I think goes from 10 to 10 and 10 to negative 10, yes, which is zoom standard, which is why I try and use that when I'm typing, so I'm just going to make sure, I'm going to go zoom standard. Now, does that sort of look like our answer? We can be even more specific. So once you get that graph to appear, press the trace button. This is what allowed us to move around on the graph, but it also allowed us to enter, oh, first of all, mine started right at zero three, which I had on my graph. What's another point I had on my graph? What went with the five? What went with the five? Oh, you know what? I'm going to type in negative one half, and I'm going to see if that goes with the five. Does it? I have to be right. I'll check one more just because I'll check the negative one, and it goes with an 11. So you can use the trace button to type in values to see if you're correct, except on the test you won't be able to, but in your homework, because I don't have an answer key made up for this one. This is kind of how you'll check your answers. Is that all right? Yeah. Did you press your trace button? Press graph. Give me one second. I'll come over there and pause. Let's go back to example two, log graphs. Example two, Katie, what's the base of this log? Two. What that means is this is the inverse of this. Yes? Okay. I know that this graph goes through zero, one, one, two, two, four, three, eight, four, 60, I'll do my powers of two. Oh, and I'll even add one more, one, sorry, not one half, Mr. Deweyck, negative one comma one half. I deliberately didn't put those in here. I jotted these off in the margin. I know my exponents. How do I find an inverse? I'm going to argue the key points for a log base two are one half negative one, one, zero, two, one, four, two, eight, three. How did I get those Trevor? I literally switched the X's and Y's around. Oh, did I do a big rant on inverses last unit? Oh, what would the asymptote of this guy be? What was the asymptote of every single exponential graph? What are we right at the bottom of the previous question there? Here, yes? No, your answer was correct. I mean, if the asymptote for the exponential is Y equals, the asymptote for the log should be X equals, going to switch the X and Y around. Oh, I'm even going to go further. Won't it be all reels for Y and X greater than zero for domain? Switching the X's and Y's are. Kirsten, are you seeing why I kept having you do that stupid dance? I figure if you know this, you can derive the log one in 30 seconds of thinking. You can, by the way, feel free to memorize it, Jessica. If you want to memorize the logs inside, no, go ahead. Every year I have some kids that do that. I usually just, it's an inverse. And, oh, did I have you write out all your exponents on a page and we know our exponents? Now, let's do the transformations. What's going on here? Expansions, compressions, reflections, slides? What's that mean? Four what? What's this mean? One what? Now, again, when I think left, I think X's, X's, X's, Oop, X there, Oop, X there. When I think down, I think Y's, Y's, Y's, Y's. Four left means take this number and minus four from it. That'll be negative 3.5. Yeah, it's a decimal. Shut up and deal with it. Negative three, negative two, zero, and four. Oh, look, look, look, look, look. Asymptote, not X equals zero, X equals negative four, four left, yes? Range, oh, look, look, look, look, look. Not X bigger than zero, X bigger than negative four, yes? Oh, oh, that's a Y. One down, negative two, negative one, zero, one, two. Oh, and the range is all reels vertically. If I move all reels one down, I think I'm still all reels. What will this look like? What did I always grab first as my guide railing? And that's now X equals negative four. That's a vertical line way over here at negative four. And then it's going to go through negative 3.5, negative two, negative three, negative one. Well, that scared me. Negative two, zero, zero, one, four, two. Can you figure out what this graph must look like? I think it does. Now I'm going to need to fix my asymptote just a little bit there. I think it does this. I think it shoots nearly vertically down forever and ever and ever and ever. And it shoots to the right. And if you think along and hard, Kirsten, if you were to reflect yourself about the line Y equals X, that is what you would look like when you did that. Ooh, should we check? Let's see if we're right. Mr. Dick, how the heck do I graph log base two? Base change log. Log of X plus four, closed bracket, divided by log two. Isn't that how we did base two on our calculator? Minus one. That's how I would type that in. Put what's inside the log and brackets and divide by the base. Or you could have, if you really wanted to be scary, K, could have used LN and LN. That would work too, but I usually use log base two. Pick graph, what do you get? And you know what? You don't quite get our graph. Now, who has a TI-84? You do? Your graph might look a bit different because the software was improved. Does yours start in midair over here too? Oh, yours does? You see, we're there? Yes. The graphing calculators have a heck of a time with that nearly vertical line. It does not know what to do there. I guarantee this is wrong. The graph does not start out of nowhere. It has a range of all reels. It continues going on down there, but your graphing calculator can't handle it. Let's check a couple of our points. Shannon will press the trace button very firmly. Well, how about negative three? Negative three goes with negative one? Check. How about negative two? Negative two goes with zero? Check. I'm pretty sure I'm right. Let's do another one. Example four. Yep. Yep. This is about as tricky as I'll throw at you probably. Graph that. Well, this graph is the inverse of that. Steph, what do you like about that one there? Okay, you know what? It's really the inverse of that, which is this with a horizontal reflection. Is that okay? So in my mind, over here in the margin, I'm going to do this. Let's see. If I put a zero in, I'll get a one. If I put a one in, I'll get a four. If I put a two in, I'll get a 16. You know what? I'm not going past that. In fact, I doubt that 16 is even going to fit my navy. Oh, and I always try negative one. Negative one, what's the four to the negative one is going to be Katie? That's that. However, that's not what we started with. Steph, we started with one quarter to the X or a negative, oh, you know what? Positive, negative, negative. See how I got that? What are my key points? Now switch your X's and Y's around. One quarter goes with one. One goes with zero. Four goes with negative one. Sixteen goes with negative two. Oh, asymptote, X equals zero. Yes? Oh, domain, range, domain X greater than zero, range Y all reels. Oh, then it says do the transformations. Now let's go back to our original here. Expansions, compressions, are there any? Yep, vertical or horizontal? If it was horizontal, where would it be? Isn't that next to the X? Vertical and it's not backwards. This is a vert expansion by two. Any other expansions? No, nothing in front of the X here, check. Reflections, is there a negative there? Is there a negative there? We had one in our invert, we did that. We're coming back to this guy now. Slides, one, one what? So vertical means Y's expand by two. Two, what's zero times two, still zero. Negative two, negative four. Shut the door. Oh, and this would expand by two because that's vertical because it's got a Y in front of it, Holly. But when you expand this, it's not all reels. Slides, one left. One quarter, take away one, negative three quarters. That'll be a bit tricky, but maybe. Oh, but this one I can do zero. This one I can do three. This one, 15. I don't think that's going to fit on my graph. Oh, but this is an X here as well. This is an X here as well. This is an X here as well. This is going to move left as well. Negative one. Oh, and the domain is going to become everything greater than negative one. Graph this, what did I graph first? And then it's going to go through. You know, I'll save this fraction one for last if I need it. It definitely goes through zero, zero and three negative two. Okay, and negative three quarters, positive two. Oh, you know, the only Kirsten shape that fits here, I think, is this. It's the only way I can get it to work, and we're right. Do you want to check? Well, really quickly I would go, okay, clear whatever I have here. Two, log of X plus one, close bracket, divided by log of a quarter. There's how I would type that in. It looks kind of like what I got. Trace negative three quarters. Yay, it goes through two, zero. Yay, it goes through zero, three. Yay, it goes through negative two. I am right. Take a look at the second sheet that I gave you, this one here. I'm going to give you about four of each, four logs and four exponentials to try. This part here is actually a lesson. So if you want to turn to where it says examples, does number two fit nicely? Let me see, three times two to the power of two bracket X minus one, close bracket, close bracket minus five. Yeah, okay, try number two, then skip, skip, skip, skip, skip. Oh, heck, you know what, nuke number two, go where it says practice questions. Find this page right here. I think you can try number one. That's tougher than I'll probably ask question two. So one, two, three and four and then practice questions one, two, three and four. I'm looking at this now and I'm not liking the way some of those graphs will look. They won't fit very well, but that's okay.