 So, turning your workbooks to where there's a blank page at the very, very back. It's got a bunch of blank pages at the very, very back. Find a page that looks kind of nice and has a little heading here. You can write how to use GCs, how to use GCs, the graphing calculators, okay? The basics, that's the on button, okay? By the way, if you don't have your graphing calculator in front of you, you get it in front of you. Second function on is the off button. Yours will turn off after, I think, eight minutes of inactivity automatically to save the battery. By the way, if yours is saying low battery, ignore that for as long as you can. You can usually go for about two months with that low battery symbol before replacing them. If you need to replace them, it's four triple A's, get them from your drugstore, get them from shoppers drug, wherever, nearest whatever store. So first thing you want to do is clear your screen, clear, clear, clear. If you're stuck in a menu somewhere and you can't get out, clear, clear, clear, or second function quit, which is second function mode, that bails you out of almost everything. So before you freak out, do that, okay? Let's do a mathematical operation together. I want to type in eight to the fifth minus 100 times negative three. How do I type that in? Well, I type eight. Where's the exponent button here? Exponent button is right above the divided by the little hat symbol thingy there. Here's your, to the power of five minus 100 bracket. Now the most common mistake, and I want you all to look up because I said this in my last class and sure enough three kids did it even though I said it, there is a huge difference on these calculators between your negative symbol, which is right here, and the subtraction symbol, which is right here. Now I want this to be a negative three. I'm going to pretend to make an error. I'm going to put a subtraction three. And if I hit enter, my calculator gives me an error, a syntax error. If you ever get this message, first of all, don't freak out means you type something wonky, but don't hit clear. One of the things I like about these calculators is if you pick option two, if I hit the number two, go to the error. It puts the cursor right where the glitch occurred. My cursor is flashing on that, it's like, I don't know what you mean there. That should be a negative. Look up, that's a negative, that's a subtraction. You can kind of see the difference, so a negative looks much smaller. Anyways, when you do that, you should get 33,068, yes? Let's take the square root of that. Here's your x squared button. The square root button is second function x squared. You'll notice as soon as you type a square root, it does an open bracket automatically, Holly. Did you see where the square root, by the way, I need you kind of with me here, second function x squared, it's a follow along, right there. I don't want to retype 33,068. Where's the answer button on my graphing calculator? Trevor, the answer button is second function negative, right above here you can see ants in yellow, anything in yellow is hit the yellow button. Close bracket, square root of it is 181.8460888, blah, blah, blah, blah, that's where the square root is. Brackets are here, what other stuff might you want? How do I find the sixth root of 5,430? I type 6, the math button right here, if you hit the math button, this has all your obscure mathematical functions, so if you press the math button, I can see option 3 is to cube, although I probably just go to the power of 3. Option 4 is cube root, option 5 is nth root, your generic root button. If I press option 5, and now I type 5,430, that will take the sixth root of that, and with that little x up there it's saying, look this number is really supposed to be there, but the graphics can't show that. The sixth root of 5,430 is 4.19264, blah, blah, blah, in fact, if I go 4.192438 to the power of 6, that should be really, really close to 5,430. So you okay where that is? 3 over 5 plus 7 over 8, how do I do fractions here? I literally go, don't worry about a common denominator, don't waste your time, that's math 8, I kind of like to assume that you know how to do it, but I'm also recognizing that I'm pressed for time, 3 slash 5 divided by 5 plus 7 divided by 8, enter. I should do it, that's a decimal, I want a fraction answer, no problem. One of the ones you'll use the most, especially in the probability section where we want our answers as fractions, if you press the math button again, the very first option goes from decimal to fraction, and so the really quick way to do that is to go hit the math button and then just go enter, enter, look what you have on your screen now, there it is, yes, enter, enter, 59 over 40, if I want to go 2 thirds plus 7 over 12, math, enter, enter, 5 over 4, nice and quick, type with me, 8 times 9 to the power of 3 minus 15, enter, it does the bed mass properly for you, 8 times 9 to the power of 3 minus 15, enter, you got 5, 8, 1, 7, don't, I made a mistake, I didn't want to put 8 times, I wanted to put 7 times, to bring back any line previous, if you go second function, enter, sorry, second function, enter, that brings back your last line of typing, and now I can go back, back, back, back, I said I wanted to change the 8 to a 7, I can delete the 7 with the delete button right here, don't, second function insert, which is shift delete, put a 7 there, as a matter of fact, if you go second function, enter, second function, keep hitting second function, enter, it remembers the last 20 lines I believe that you've typed, so if you just brought a calculator from me, it's like you're getting secret messages from whoever used this last year, you can see what their last equations were, some of those might be from the final exam, or maybe not, I wonder, oh, I don't know, maybe, oh, I don't know, okay, it remembers your last 20 operations, that can be really handy for something like the quadratic formula, suppose you're doing the quadratic equation and you go bracket, negative b, 8 plus the square root of 8 squared minus 4 times a times c, I'm just making up numbers, close bracket, close bracket, all over 2a, what about 4a, if you did your quadratic equation with the plus sign, then all you do is second function, enter, backspace, backspace, backspace, change the plus to a minus, and there's your plus or minus b squared minus 4acl over 2a, very nice, or if you're in chemistry and you're doing the same mathematical operation over and over to fill in a chart, but you just need to change one number each time, second function, enter, change the number, how to graph, press y equals, right here, you might have some graphs on your screen already, if you have a y1 or a y2 or a y3, just press clear, down arrow, clear, down arrow, clear, until everything is cleared, so clear any graphs that you have there already, the other thing you want to make sure is you want to make sure that plot 1, plot 2, and plot 3 are not highlighted, if they are, just go up arrow and hit enter, that toggles highlighting on and off, you want to make sure none of them are highlighted, those are stats graphs, if they're highlighted it'll try and graph those as well, you don't want some strange graph coming out of nowhere showing up on your screen, let's do our old standard, the parabola, so y1 equals, where is the x button, you guys feel okay Holly, good, yes, or question, by the way if you fall behind or you can't do something, ask, we're good, Carly, Hailey, I said Holly twice, Hailey, sorry, Hailey, we're good, twice, yes, happy, okay, y equals, where's the x button, look up, right next to the alpha, right there, x squared, boom, right here, x squared, to graph it, hit graph, now I'm willing to bet your graph might not look like mine, it might, the reason is we have to tell the graph in calculator which area of the graph to focus on, we could have all different windows, so press the window button right here, this tells the graph in calculator how much of the screen or what dimensions of the screen to display to you, x min stands for the minimum x value, it's how far left it's going to graph, x max stands for how far right the graph is going to go, so I have my graph set up to go from negative 10 to positive 10, x school stands for my x scale, one on this graph, each hash mark is going to go up by ones, y min, this graph is going to go to negative 10, y max, this graph is going to go to positive 10, y scale, one, so if your numbers don't match mine, just type them in and hit enter as you go along to get to the next line, the only one you don't want to change is this one here, x res, don't change that, okay, if you're getting an error, if you have a question, now's the chance to ask, okay, we were good? No, yes, no, yes for sure, because otherwise you can't do the next step, you can't do that, something wonky, yes, that's what I was trying to get at, then, so now hit graph and all of our graphs look identical, yes, if they don't, now's the chance to raise your hand, good, yeah, the last important one that we're going to talk about for now is the mode button, which is right here, give or press your mode button, for math 12, you want everything in the first column to be highlighted, except right now you want to be in degrees, if you borrow a calculator, you're probably in radians, just use your cursor key to move around until degrees is highlighted and press enter to toggle it on and off, one more thing, you can make your screen lighter or darker, I don't know how your calculator is set, if you go second function down arrow, keep going second function down arrow, second function down arrow, second function down arrow, that makes your screen either lighter or darker, I can't remember what, which one does it do, lighter, go second function up arrow, until you're at a darkness that you like, all depends on how far you hold the calculator away from you and what angle, if you ever have a calculator and it's not going on, it might not be the battery, it might be so light you can't see the screen, so before you do anything else try going second function up arrow, up arrow, up arrow, bunch of thumbs, try making it darker, now keep your graphing calculators out, we're going to be using them later on, but first the homework from last day, lesson two, page 98, which ones would you like me to go over, number seven, yep any before that, this one here, someone already got the candy by the way, now I did say for 4f, this is not going to be on your test, however this trick that I'm going to show you is going to come in handy after Christmas, so let's do 4f, can you read me the hint Brett, sorry as what, not 6x, that's completely different, I got to be fussy here because I got to beat a bad habit out of you, okay, 6 to the power of x, okay, since we're doing exponents all this unit you better be fussy on the difference between the 6x and the power of x, here's what I'm going to do, temporarily I'm going to let that be a variable and replace it in here, now watch what happens before you write down, just watch what happens, look up, watch what happens, have patience, I'll pause in a second, watch what happens when I do that, now the zero stays the same, the 72 stays the same, the plus stays the same, but I'm going to write this as 74a, because 6 to the x I'm replacing with an a, and I'm going to write this as 2a squared, why a squared, well this is 6 to the 2x which is really 6 to the x all squared, I'm replacing every 6 to the x with an a, write that down, then we'll talk about why, Joran did I give you your candy for getting this one, I did, Joran was the one who already got this one so pull your socks up everybody else, you okay with that so far, what kind of an equation is this, quadratic, how do I know, it's got a squared, have I written it as a quadratic equation, replacing the 6 to the x with a variable, yes, I think the first thing that I would do is I would go divide by 2, divide by 2, divide by 2, divide by 2, and it gives me a squared minus 37a plus 36 equals 0, I'm much nicer quadratic because now Dominique there's no coefficient in the front, this factors, what are two numbers that multiplied a positive 36 and add to negative 37, Justin, I think you said, did you, negative 36 and negative 1, this factors into a minus 36, a minus 1 equals 0, what are the roots, what are the roots, now hold on, don't write this, just watch for a second, I was gonna go a equals and a equals but what's a actually, what was the substitution that I did at the very beginning, you know what I'm gonna write, here's the equations that I get out of here, 6 to the x equals 36 and 6 to the x equals 1 and I can solve these in my head, 6 to what equals 36, what does x have to be for this to be true, 6 to what power equals 36, 6 to the third is 36, 6 times 6 times 6 is 36, really, you're tired right now, everyone else, huh, yes, and a little bit of a trick question, 6 to what power equals 1, ah, there's your roots, there's your real solutions from this equation, it's a quadratic, I got two answers in disguise because it wasn't really obvious that there was a squared in there but there is a squared in there, this trick of taking something ugly and substituting it with something that gives you a prettier equation, that's the trick I want you to remember, after Christmas we'll be solving quadratic trig equations so that we'll have 2 sine squared plus 2 sine minus 5, I'm gonna replace the sines with a's and we'll solve the quadratic 2a squared plus 2a plus 5 and then bring the sine back later, easier, any before number seven, yes, this one here, yep, first of all, what base am I gonna write everything as, pretty sure, and I'm gonna do the right hand side first because I know I'm gonna get it right and it's no longer blank and I'll feel better, I know this is 3 squared, I've done half the equation, yay, and I know that this is 3 to the x plus 1, yay, and I know that this is 3 cubed to the 2x minus 1, yay, what's this thing as an exponent, cool, I now have a power to a power on the bottom and I have a power to a power to a power on the top and multiply everything, in the interest of saving space I'm gonna go three times a third first because you know what three times a third is, these guys would end up just canceling, on the top I would get three to the 2x minus 1 all over, on the bottom I would get three x times, oh fractions is the easiest it's top times top, bottom times bottom, it's x times 1 which is x 1 times it's x over 3 plus 1 over 3 equals 3 squared, now what, I'm not gonna panic I'm gonna be stubborn and clever, by the way I would consider this probably a little harder than I feel comfy on the test, now what, how would I get rid of the fractions using a math 8 trick, no, no, no, even easier than that, maybe we're not talking about the same fractions, I'm talking about the great big fraction on the left, yeah you do, what can I do here, absolutely I can do it here, it looks ugly I can still do it here, I heard it finally, what, no, I can't, no, cross multiply is this not, one fraction equals, it's one ugly fraction, yes, but it's one fraction equals one fraction, the great 8 trick still has to work, what's the bottom of this fraction over here by the way, gotta still work, cross multiply and I'll get 3 to the 2x minus 1 equals 3 squared times 3 to the x over 3 plus a third, still ugly but that definitely actually looks better, do I have one base equals one base, say no, I don't have one base, I have one base equals two bases, so you ready, do I have one base equals one base, say no, well then I better write that right-hand side as one base, what's, oh, is that a 3, is that a 3, my bases here are the same and I'm multiplying, what do I do with the exponents, add them, ah, look up, adding fractions is a bit more work, that's not a 2, it's a 2, okay, I'll drop the left side down, is there an x in here, so in terms of like terms there's gonna be an x over 3 plus 6 thirds plus 1 third, by the way have I done anything new so far, nope, stubborn clever, do I have one base equals one base, are my bases the same, then I could equate the exponents, the equation I'm gonna spend my rest of my time solving now is this bad boy, 2x minus 1 equals x over 3 plus 7 over 3, now what don't I like about this equation, fractions, grade 9 you learned a wonderful trick that can make the fractions in an equation vanish, multiply everything by the common denominator which in this case is 3, I'm gonna put little 3 right there, little 3 right there, little 3 right there, little 3 right there and the nice thing is you know what happens to these 3's Ellen and to these 3 they cancel, in fact I get this, so much fun, 6x minus 3 equals x plus 7, now what minus x from both sides all the 5x plus 3 to both sides all the 10, what's x, oh that was fun, so much fun, any others before number 7, I like 4c, I like 4c, 4c and 4d is also very very similar, okay, is this exponent on the 49, then let's move the 49 over, yeah I'm gonna do this first thing, I'm gonna write this as 7 over 12 to the 2x equals 144 over 49, what do I want to write this as, I'm pretty sure sandally since I can't break a 7 down or a 12 down it's not, they're not some number to my photo, I'm pretty sure I want to write this as right hand side as 7 over 12 to some power, problem, what's that in disguise, a 12 on top, where's the 12 here, what's this in disguise on the bottom, where would I prefer it to be, it'll be wonderful if there was some kind of mathematical operation that acted, oh I don't know, like an elevator as it were and would cause things to change levels, okay and this is 12 to the what, and this is 49 to the what, I think the whole thing is gonna look like this then, if I want to get the 7 on top and the 12 on the bottom, it can't be a 2, it's got to be a negative 2, do I have one base equals one base or my base is the same, then I can equate the exponents, 2x equals negative 2, I think x is going to get a 1 in the 7 though, little question Number is 7, 7, Dominic what am I gonna write everything as, a power of what, 5? No, 8? No, 2, yes so what's 8 written as a 2, 2 to the what, what's 1 over 8 written as a 2, okay this first one is 2 to the negative 3 to the x minus 3 equals, I'm happy with that 2, what's 16 as a 2, 2 to the 4th to the 2x plus 1, power to power, power to power, I'm gonna get 2 to the junk junk negative 3x plus 9 equals, I'm happy with that 2 there, 2 to the junk junk 8x plus 4, do I have one base equals one base, not yet, oh on the right hand side I got 2 bases, what's the base here, what's the base here, the bases are the same and I'm multiplying, what do I do with exponents when my bases are the same and I'm multiplying, what is the exponent on the 2 here it's invisible, so you're saying it's that, in fact call me silly but if I add the exponents don't I get, yes, are my bases the same, do I have one base equals one base, then I can equate the exponents, the actual equation I'm gonna solve is negative 3x plus 9 equals 8x plus 5, I think you'll get 4 equals 11x when you minus 5 from both sides plus 3x to both sides, I think you'll get x equals 4 over 11 and they want to the nearest hundred, I used to know it 4 over, is it 0.36363, okay, doesn't seem right to me, gosh I forgot I mean mixed numerals, oh it is 0.36, I'm right, wow, is that okay, once again, did I really do anything new there, nope, stubborn clever, careful, turn to lesson 3, lesson 3, the exponential function, the exponential function, now in my last class I didn't get through the whole lesson, I got through part of it, so if that happens here, we'll do two lessons on Monday which works okay anyways because the second lesson fits into this one, if you have not finished the homework from last day, you really really really really really really want to finish it because I like almost everything that I gave you last day, woman number one, Mallory who's a medical research scientist discovered a new bacteria culture which can help strengthen a person's immune system, to find the growth rate she isolated five cells of the culture and here's what she noticed, after one hour there were 10 cells, after two hours there were 20 cells, after three hours there were 40 cells, how many cells would there be after four hours, it's about the pattern, 80, okay, what's happening every hour, A says let T represent the time in hours, okay, and N of T represent the number of cells after T hours, the formula can be written like this, where T is an exponent, find the values of A and B, so as an equation what they're saying is there is an equation for which when I put in a zero I get a five back, when I put in a one I get a ten back, when I put in a two I get a twenty back, when I put in a three I get a forty back, when I put in a four I get an eighty back, can anybody spot the equation that looks like this, I always, I rarely, every once in a while I have a kid who can see it, otherwise I'll tell it to you but I always pause, anybody come up with the equation, sorry, is it a parabola, no, even weirder, so here's the equation and here's how I get it, Brett how many bacteria did I start out with, start out with that, start out with what you start out with, Brett, Brett, Brett, there's a Freudian slip, say one thing and do the other, Brett what's happening here, the growth rate is two and the T is an exponent, in fact it's not a parabola, Ellen, it's an exponential equation for the first time, T the variable is sitting up here, we looked at it a little bit last day, I'll prove to you that this works by the way, Ellen put a zero in, what's anything to the zero power, times five is, you mean zero gives you a five, yes it does, let's try putting a two in, what's two squared times five, does two give you a 20, yes it does, let's try putting a three in, what's two cubed times five, does three give you a 40, it works, that's an exponential equation, call that because the variable isn't exploding, B says use this formula to determine how many cells there were after eight hours, what they're really saying is find N when T is eight, N of eight and it's going to be straight calculator five times two to the eighth, on your calculator, on your graphing calculator, get used to it, go five times two to the eighth, can't remember where the exponent button is, find it again, we have a little hat thing, what do you get, hang on, hang on, I can do this in my head, two, four, eight, sixteen, thirty, two, sixty, four hundred, twenty-six, twenty-six times five is going to be a thousand, plus two hundred, plus two hundred and fifty, plus eight, one thousand, two hundred and eighty, after twelve hours there's twenty thousand, four hundred and eighty cells, when was there, how many hours ago, how many hours did it take to get half that many, after twelve hours there's twenty thousand, four hundred and eighty, how many hours did it take to get half that amount, you guys are smart, every year I have a bunch of kids that say six, eleven because if it doubles every hour it had half as many one hour ago, eleven, I have a little talk, I don't think I'll be able to show it in class I want to time, but the guy who's giving the talk says he argues one of the biggest issues in society, one of the biggest reasons that we are so terrible at planning the future is we don't have a good understanding of how exponential growth works, put your pencils down and look up, let's suppose this is the earth, there's the earth and let's suppose right now, let's suppose that the population is that big, we have a huge amount of room, you want to burn some oil go ahead, you want to make some landfills, pile up some garbage, go ahead we got lots of room, you want to fish the oceans, go ahead and the population doubles, still lots of room, still lots of room, we would have most people because they'd understand exponential growth works, they'd be saying oh for a pizza, it's global warming, are you stupid, look at all the room, please, we double one more time, we got a problem, we've run out of everything, most of the experts that I've read think we're here right now and sometime in the next 20 years, you're starting to see bits of it already, have gas prices gone way up in the past 10 years, yeah more people and more cars, electricity a couple of years ago California was actually having rolling blackouts, so your town wouldn't have power for two hours at certain times of the day because they were running out, but 10 years ago California was get more air conditioners, lots of room, build more houses, lots of room, we don't understand how exponential growth works, far too many citizens, far too many politicians for part C would answer oh six hours ago, lots of time, lots of time, D says use a graphing calculator to graph this, okay get your graphing calculators out, press y equals and clear whatever equation you got there, now I want to graph this, my calculator does not have an n, instead I've got to put a y there, my calculator does not have a t, what letter am I going to put there instead, x, I'll adjust, so I'm going to go five times two to the power of and I'm going to press my x button, now if you hit graph you're going to get some nonsense, nothing terribly entertaining, we want to adjust this window, so press window, now remember x is actually time, look at my chart, what's the smallest amount of time that appears in my chart, so I'm going to let x min be zero, what's the maximum amount of time that this question mentions anywhere, not three, what's the maximum amount of time this question mentions anywhere in the question, when I get the whole thing on there, 12, I think going up by ones there would work okay because 12 hash marks in an area of screen that long should be okay, so scale one, now y is the number of bacteria, what's the fewest number of bacteria that this question mentions, five, I could go five, I like to see zeros, I'm actually going to start out at zero high, I always like to see the x axis, what's the largest number of bacteria that this question mentions, 20,480, so I'm going to go to a nice roundish number, I'm going to go to 21,000, if I'm going from zero to 21,000 to the space this big, would a scale of one make, going up by ones with hash marks make much sense, Kara's shaking her head, like you know what Kara, I'm going to let my scale be about 5,000, now hit graph, all of our graphs should now look the same, and we should be able to see way more stuff going on, is there anybody who didn't get that now is your chance to ask, usually it means a little typo, I can fix it right away, as the graph, it looks like this, the further right you go, the steeper it gets, the further left you go, the closer it gets to zero, but never quite touches, in fact the x axis is a horizontal asymptote, looks like this, if we get our disco now, because it's getting bigger as we move to the right, we call it exponential growth, you know what an exponential decay graph looks like, gets smaller as we move to the right, Chernobyl is a city in the former Soviet Union, what happened at Chernobyl, why is Chernobyl famous, Eric, more specific there are plenty of places that have nuclear power plants that aren't famous, what happened at Chernobyl, sorry didn't blow up, didn't actually leak, a leak is like, ah but it had a complete core meltdown, it had a nuclear meltdown, the worst nuclear power plant disaster that we've had, it happened in the height of the cold war, and so the Russians at that time denied it, all the world nations were detecting in 1986 a huge radioactive cloud circled the planet three or four times before it finally faded into nothingness, and all of our instruments were detecting it, and when we traced our way backwards to the cloud's origin it kept starting somewhere in central Russia, so we, the US, Canada, all the countries phoned Russia and said you've got a problem, Russia said yes, you don't, finally after a couple of weeks they fessed up and they admitted yeah we've had a very, very serious nuclear reactor meltdown, if you want to read just a tragic story google chernobyl and just find how horrible it was because the way they fixed it was they sent, they ordered soldiers to their death, for example they couldn't get the fire to stop, so they flew helicopters and dropped liquid concrete from the helicopters into the reactor, the pilots all died horrible deaths a couple of months later of radiation, even to this day people that live in the area have a high incidence of leukemia, kids in that area are born without, with birth defects, they moved the town of Chernobyl, but there are still some people that live there, horrible the way it was handled, so it was partly the fact that it was a nuclear meltdown and probably the fact that they botched it so badly, the one that happened that almost happened in Japan with the earthquake, nothing compared to this, I don't think that Japan will need much radiation at all, so radiation, radioactive, radiation decays, radioactivity decays, now again just like in the previous question Carly, we've cooked the numbers to make the numbers nice mathematically because I'm going to tell you that cells don't divide, don't multiply by two on every single hour, they would multiply by 1.67 every three hours, 43 minutes and 19 seconds, like you never get nice numbers like that, they've done the same thing here, so it says initially they measured 32,768 units of radioactive iodine, after one week there was that many, two weeks there was that many, two weeks there was that many, what's the pattern here, someone said something I think, what's the pattern, what's going on here, Justin, did you say it, I like the way you said it, almost every year people say dividing by two and I say well I don't want to divide because here I multiplied instead of dividing by two, what number would I multiply by, but you said it right away, yeah as it turns out the growth rate here is a half, says use the formula to determine how many units of radioactive iodine were left after 14 weeks, they want me to find n of 14, I'm going to go, 30, oh turn my calculator around, clear clear clear to get out of the graph menu, 32,768 times bracket, one half closed bracket to the power of, it says two, this is how I know they cook these answers, these are fake answers, but okay, paradigm, use a graphing calculator to graph this, okay y equals, clear the equation you have there and the equation is this bad boy, so 32,768 times bracket, one half closed bracket to the power of, oh I can't do t, what am I going to do instead, x, where's the x, oh yeah, right there Henry, oh except they also gave me the view window that they want me to use so that all of our graphs look the same, so press window and carefully type in this view window, I guess on the y axis we're going to go up by 10,000 which makes sense, once you do that hit graph and you should get a decay graph, a graph that starts up steep and gets shallower and shallower and closer and closer and closer, like that, like this, some handy features, how much iodine is left after five weeks, don't type it in, instead you have the graph in front of you, press the trace button right here and once you press the trace button a little cursor should appear and what that's saying is you can use your left and right arrows to move around and it gives you y values that go with, so after five weeks I can get 4.936 weeks and I can get 5.03131949 weeks, there's a better way to trace, so you're all in the trace menu, instead type 5, if you type the number 5 when you're in the trace menu at the bottom x equals appears hit enter, oh after five weeks how much iodine was left, 1024 after seven weeks how much iodine was left, 256 after 10 weeks how much iodine was left, you get an error, why, what's the biggest x value that I was graphing, eight I'd have to change my window if I wanted to get the 10 maybe easy enough to do, sometimes if I'm doing the same equation over and over and over and over and over but just with different numbers like maybe in chemistry I type the equation, oh go to, I type the equation in here pick a big window and just use my trace feature it's nice we call this exponential decay turn the page exponential functions an exponential function is an equation that looks like that where the x is an exponent where b is the base, b is the base, b is the base, I wonder how I can remember that b is the base and a is the coefficient in fact Trevor a is a vertical stretch if I go back to last unit and it could even be a vertical reflection if I go back to last unit we have to add a couple of restrictions though a can't be zero why can't a be zero if a was zero I'd have zero times b to the x what's zero times anything that's not going to give me this shape guarantee it oh and we add one more thing b has to be positive you can't have a negative base put pencils down here's why you couldn't have this don't write this down when this was even your answer would be positive when this was odd your answer would be you'd have a graph that was positive the negative positive negative positive negative positive I guarantee that's not giving you this shape that's jumping all over the place what would that graph look like try it on your calculator if you board sometimes oh let's try that right now just because y equals clear clear clear negative three to the power to the power of x I'd like look up here's your next shortcut which is why I'm doing this I want to graph from negative 10 to positive 10 from negative 10 to positive 10 it's such a handy window if you press zoom these are built in windows and zoom standard option six gives you negative 10 to positive 10 negative 10 to positive 10 what appeared your calculator doesn't even know how to graph it maybe that's too big let's try a smaller number your calculator doesn't even know how to graph it so the base has to be positive let's look at two graphs and we're going to do these in our head we're going to compare two to the x and one half to the x we're going to do these pencil and paper not with technology first thing it says is state the values of a and b for both of these graphs what's a here it's not two it's the number in front what number is in front here it's invisible and also a equals one here in other words we're saying look ignore vertical stretches while we start you out what's be the base here to what's be the base here b says sketch the graph of two to the x so I'm going to graph y equals two to the x and the way I'm going to do that is I'm going to plug in numbers and I always start with zero what's two to the zero zero goes with one on the graph zero goes with one zero comma one is a point what's two to the one when x is one y is two what's two to the two when x is two y is is what four two over four up what's two to the third when x is three y is a four five six seven eight what's two to the fourth two times two times sixteen off my graph forget it let's go in the other direction what's two to the negative one elevator one over two what's two to the negative two elevator one over four what's two to the negative three elevator one over eight now I can't graph negative three comma one eighth very easily I can't graph negative two comma one quarter very easily I can and graph negative one comma a half and this gives us the shape. It looks like that. Eric, you know what it looks like? Put your pencils down, Eric. Looks like this. Looks like this. Can you do that with your hands, Eric? One hand here and one hand here. Disco man. Oh, you might hit Brianna in the head accidentally. That's just a risk she's gonna take. Okay? So you got that? I'll be picking on you all day, all year for this one. I don't have a basketball player in the back row. Could go with Trevor. Doesn't got long arms. Okay, hands back down. Let's graph this next one. I'm gonna do this in red. Y equals one half to the X. And like always, I start out with zero. What's any of them to the zero power? One, right? What's one half to the one? A half? What's one half squared? Okay, we're getting smaller. Let's go in the other direction. What's one half to the negative one? What's the elevator when the fraction's already on the bottom? It's gonna make it... What's one half to the negative one? Two. And you know what one half to the negative two is? Four. Keep the pattern going. Eight. This is gonna go through negative three, eight high. Negative two, four high. Negative one, two high. Zero, one high. In fact, I can't look like that. Eric, my friends, are you penciled down? Every single exponential graph looks like this or like this. Do that for me, please. Lean forwards. Like this. Lean forwards. Or like this. Slide forwards. The chair forwards. Every exponential graph looks like this or like this. Like this or like this. You're supposed to hit them in the heads, Eric. No, as a graph, okay? Now it works. It's your one chance to smack somebody up the head. Now, since they, oh, oh, oh, by the way, put your pencils down. Don't write this down, but watch, watch, watch, watch, watch, watch, watch. Look, look, look, look, look. Is this true? Is one half to the X the same as that? Why? You're right. It is the same. Why? Where the negative come from? Oh, elevator. Look at this, look at this, look at this. Didn't I replace X with negative X? Isn't that a horizontal reflection? Oh, there's another way to think about it. Instead of fractions, you can think about it as, it's actually a horizontal reflection, which it is. It's a nice tie-in with last unit, whatever. Here's what I want you to notice. Here's what they have in common. Eric, what does every single exponential graph look like? Come on, come on. This is your cue every time, buddy. Don't let me down here, here or? What's the domain right now? By the way, not straight up, curving up, right? Getting steeper, yes. Not this though, this levels. Okay, what's your domain? Point, sorry? What's the domain of Eric? What's the domain of this? We've never said infinity all year, what we called that. So in this chart, put your hands down now, Eric. Domain, all reels, and it's the same for both of them. So I'm just gonna write it big and put it in both. What's the range? How high do these graphs go? Forever, how low do these graphs go? Zero, but not touching, closer and closer to zero. Everything above, but not touching zero. How will I write that? Everything above zero, but not touching, so no or equal to. What's the x-intercept of an exponential graph? It's a trick question. An x-intercept of an exponential graph. It's a trick question, isn't it one? It gets closer and closer and closer and closer and closer and closer and closer and closer and closer and closer and closer to the x-axis, never touches. What's the y-intercept of an exponential graph? Zero comma one. And what I said to you earlier was, both of these graphs have as an asymptote, the x-axis, what's the equation of the x-axis align horizontally zero high? Nice, y equals zero. That is true of every single exponential graph that doesn't have a coefficient for anything else done to it. Put your pencils down and look up and turn your brains on. On Monday, I'm going to introduce you to the log graph, the logarithmic graph. You know what, all I'm gonna tell you is it's the inverse of the exponential. It's the inverse of the exponential. So just by telling you that and looking at this chart, you ready? If logs are inverses of exponential, what's the domain of every log graph? What's about it? X is greater than zero, because the range is gonna become a domain, it's what's x and y. You know what the range of every log graph is? Oh, what's the x-intercept of a log graph? One comma zero, it's the y-intercept. Does it have any y-intercepts? Nope, does it have an asymptote? Yes, vertical asymptote, no. Horizontal asymptote? Sorry, horizontal asymptote, no. Has a vertical asymptote, y equals zero. So even though you've never met the graph, you've seen parts of it before. You have to go earlier or something? Okay. Oh. Growth decay. Dear Mr. Dewitt, you are my favorite math teacher. I've attached $5 here, excellent. Turn the page. What do you think happens as my base gets bigger? Let's find out, get your calculators out. We're gonna do three equations. So clear, clear everything, and go zoom standard right away so that our window is the same. Zoom button, option six. Now here are the three we're going to graph. Two to the power of x, enter. Three to the power of x, enter. And then five to the power of x, enter. But before we hit graph, I'm gonna show you a few more things you can do. You see, if we graph these, they're all gonna look the same. If you go up arrow and to the left of y2 with your cursor, so that you're right here, and you hit enter once, that makes this graph a thick line, so you can tell it apart. If you go down arrow, so that you're to the left of y3, enter once would make it a thick line, we already got that. Enter again, colors everything above the graph, I don't wanna do that. Enter again, colors everything below the graph, I don't wanna do that. What I do want is this one, which is a follow the bouncing ball. Now hit graph. There's two to the x, there's three to the x, there's four to the x. As this number gets bigger, what happens to the steepness? Get steeped, which makes sense. Here's what you need to memorize. For any exponential function without a vertical stretch, with no a, x intercept, none, y intercept, zero comma one, domain, all reels, range, everything above, not touching zero, equation of the asymptote, y equals zero. So the page, warmup number five, wants us to explore what happens when you put an a in front. I already know that's a vertical stretch. So if normally my y intercept is zero comma one, what would the y intercept of this one be? What's my vertical stretch right here? It's a vertical stretch by a factor of two, it'll be zero comma two. In fact, turn the page, page 105. This is the page we want to be on. The y intercept is normally zero comma a. Oh, and if a is one, like the first bunch that we looked at is zero comma one. There is blank x intercept. There is blank x intercept. What word goes there? No x intercept. The x-axis is a horizontal asymptote. The domain is all reels, and the range is everything above zero. Reminder for Halloween for 100 straight drives may end in three. Cringes plus non-perishable to run, you can do your block A cost. If you have a straighter, bring them to Mr. and Classroom. The winning home. Cringes plus non-perishable to run, you can do your block A cost.