 review the entire course all in one fell swoop by the way remember to write your sequences test there added that little memo projectors frozen remember to write your sequences test so what are you gonna see on the written the first question is going to be a transformations in fact probably the first two questions will be transformations they are on the provincial almost every year what kind of transformations questions will they ask you well they'll give you a generic slide stretch flip and then as a part B they'll almost always either give you a reciprocal absolute value with the twist or inverse with a twist or reciprocal with a twist so let's jog our memory here what they'll probably do is give you the original graph as a dotted line not all solid remember we it's gonna be a very very thin dotted line on your graph paper I was in a rush and I didn't have chance to do the fancy graphics I just pasted one this line is solid it would normally be dotted first thing we would check is we would make sure this is correct format it is what does this mean two what three what two left three down in fact I said to you if you wanted to remember two words everything's backwards oh a muck as long as you get the why stuff next to the why so if it's not next to the why it's no longer backwards so to left one two three down one two three to left one two three down one two three connect them to left one two three down one two three connect them to left one two three down one two three connect them to left one two three down one two three connect them to left 1, 2, 3.1, 2, 3. Connect them. Now you'll notice I did not do this. Why do I not do that? My original graph is not infinite. It's finite. They put dots there, which means I better stop where they stop. Don't get arrow happy. I'll be honest, that's probably too easy. That was to jog your memory. You'll have something much more along the lines maybe of this. First thing I ask is vertical or horizontal? This three. How do I know it's vertical and not horizontal? Because it was horizontal. What letter would it be next to? The X. It's not inside the function. So we have vertical expansion or compression? Vertical expansion by three. What does this mean? Two, write. I would be posting this online as a PDF file for those who are writing like mad. I would say the graphing ones don't write out when we get to the log stuff. Those are easier to write with the graph paper. You don't have graph paper in front of you anyways. I would take this point now. This is from the 2002 exam. They used to put the graph and the points. Now what they do on your grid paper is it would just be very, very light gray. You could barely see it as a dotted line. That's what you'd be moving around. We'll do this one here. Vertical expansion by two. Instead of two down, it'll be six down. By three, sorry. Instead of two down, it'll be six down. Then two right. That point would end up there. This point instead of one down, it'll be three down and one two right. That point would end up right there. If I can use zero zero, if I expand zero zero, it's still zero high. Two right. That point would end up right there. One one would end up one comma three and then two right. Right there. This one is two high. It would end up six high and two right. That blue graph is the image of the original. Right? B. How do I find an inverse? Switch the X and Y around. Instead of negative four comma negative two, negative two comma negative one, two, three, four. Instead of negative one comma negative one, negative one comma negative one. Okay. What was the fancy word for a point that didn't move? Do you remember? Invariant. Very nice. Nice comeback. Zero zero will still be zero zero. One one will still be one one strangely enough, but four comma two will be two comma four. The graph, the inverse would look like that. We said there was a built in test. We also said it was a reflection about the line Y equals X. If you did the line Y equals X, which goes through one one two two three three four four five six six seven seven, et cetera. You should find the graphs were reflections about that line and they are reciprocal. So this was from the 2002 exam. It was worth five marks. Now they've started upping the ante. They'll often add reciprocal with a twist, but we'll just do reciprocal for now. How did we do reciprocal? We said the first thing you wanted to look for were invariant points. Which reciprocal heights did not change? Anything how high or how high? Okay. Anywhere one high or negative one high isn't going anywhere. I always put big dots there because those are going to be my references. Then we said we looked for asymptotes. What gave us asymptotes? Anywhere that the graph was how high? Zero because you can't take the reciprocal of zero. One over zero is undefined. In other words, there's going to be a vertical asymptote right here and a vertical asymptote right here. Then I gave you the analogy I said amulc pretend you're a little bug walking on the graph starting on an invariant point. I'll start right here. As I walk to my left, the original graph is getting bigger shooting off to infinity. What's the reciprocal of getting bigger shooting off to infinity? Arrow on the end because there's an arrow on the end. Go back to my invariant point, Whitney. As I walk to the right, my original is getting smaller, getting closer to zero. What's the reciprocal of getting smaller, getting closer to zero? I think you said getting further away from zero shooting off to infinity, but it's hard for me to tell. If not, nod your head and no one will know. Same idea. Stand here as I walk left getting closer to zero shoot off to infinity. A little twist here. How high does this graph get right there, Julie? What's the height? What's the reciprocal of two? Careful. Don't get reciprocal mixed up with negative. I'm glad you did that. I was kind of hoping somebody would. It's think fractions. What is it for you? One half. Don't do that because as soon as you cross the x-axis when you shouldn't, I got to really hammer you for marks. It's instead two is going to become one half high. Both there, ish, ish. Gets bigger, gets smaller, curves there. You know what does that? Going to curve closer and curve closer. Closer to zero shoots off to infinity. Closer to zero, but from the negative side shoots off to infinity, but from the negative side shoots off to negative infinity. Shooting off to negative infinity, yeah, it's closer to zero. That would give you full marks. How could you lose marks? If you did the asymptotes as solid lines, those of you that are writing the provincial, if you draw the asymptote as a solid line, they'll take marks off. If you touch the asymptote, they'll take marks off. Or if you're sloppy when you're drawing because you have no artistic skill like me and you do this and you kind of start to curve back a little bit like that, they'll take marks off. You got to be a little fussier on the actual provincial. There is a marking scheme and there's certain things that they look for and they will be fussy on arrows. If you did not put an arrow there, they would take marks off because this does shoot off to infinity. Logarithms. Now you can write this down if you want to. Somewhere you'll be asked to solve a logarithmic equation. Log base 2 of 2 minus 2x plus log base 2 of 1 minus x equals 5. This logarithmic equation, there's two types of log equations. This kind and then don't write this down. Something like this. Log base 5 of x plus log base 5 of x minus 3. Log base 5, Mr. Dewick, equals log base 5 of 7. Equations that have logs in everything or equations that have logs in some stuff. If you only got logs in some stuff, get the log stuff to one side, get the other stuff to the other side. What will we do? We will try and write this as one log. Are my bases the same? Check. What's adding two logs the same as? Multiplying. Oh, you mean here's another place where and means multiply? Sort of. Yeah, don't stretch that too far. In other words, a mult you could write this. 2 minus 2x, 1 minus x, equals 5. Now, if I have log 5, then the logs would cancel. Well, technically taking the anti-log of both. Okay, the logs would cancel. Do I have logs everywhere? Say no. Logs don't cancel. Instead, I used my log definition. This number to the power of this number equals what's inside the log. It's how we define the logarithm in the first place. The base to that number equals what's inside the log. Remember doing this? Coming back. I know this is October. 2 to the 5th is 32. Foil. 2 minus 2x minus 2x plus 2x squared. I actually foiled that one out completely not in my head because it was in a weird order. Anytime it's in a weird order on a test, I'll get paranoid and just do the extra stuff. If the x's have been in front, I wouldn't have done it in my head because I've done that since grade 9. Gather like terms. Looks like I have this. 2x squared minus 4x minus 30 equals 0. Do you guys see where the 30 came from? I moved the 32 over. What kind of an equation is this quadratic? How do I know? It's kind of squared. This is actually not too bad a quadratic. At first I thought I had something in front of the x squared and I was lots of work. Do you see there's a 2 here and a 2 here and a 2 here and a 0 here? I can actually go divide by 2, divide by 2, divide by 2, divide by 2. Have I divided everything by 2? I haven't broken any equation solving rules. This equation is actually x squared minus 2x minus 15 equals 0. I think that does factor. Numbers that multiply to negative 15 can add to negative 2. x minus 5, x plus 3 equals 0. So my roots are 5 and negative 3. Ah, but remember, oh, who would I made of fun? I probably would have made fun of Colin in this class. Colin's date. Reject, was it you? Oh, guarantee. Thank you. You had me, somebody. We have to sometimes reject solutions. You have to check for extraneous. 5 and negative 3. If I put a 5 right there, what's 2 times 5? What's 2 take away 10? Oh, can't take the log of a negative. That one's extraneous. A little twist, by the way. Often the negative root was the extraneous one, but this time the positive one is, let's check the negative 3. Here I'd have negative 2 times negative 3. This would work out to a positive number and here I'd get a minus minus, which is a plus. This one is okay. Here's another kind of question you're going to get. This one I didn't find when I was cutting and pasting, but if I'm doing it, it's probably because you'll see it somewhere. Solve algebraically using logarithms 5 bracket 3 to the x plus 1 equals 15 to the 2 minus x. We need to make sure this actually works. We called this an exponential equation. The reason it was called an exponential equation is because your x is sitting in the exponent, so I called it an exponential equation. The temptation for some kids was to forget grade 8, forget grade 7. The temptation for some of you was to throw out your bed mass, your order of operations. Many of you wanted to write that as a 15 because you wanted to go, that's a 15. That's a 15 because if they're both 15s, I could equate the exponents and it'd be fairly easy to do. Can't do that here because that would be doing multiplying before the exponents that violates bed mass. Instead, why was that so helpful? Well, what can we do with exponents once they're inside a logarithm? Now, common mistake number two, kids want to go, you move that to the front. But if you move that x plus 1 to the front, you're implying that it's on the 5 and it's not. What's between the 5 and the 3 mathematically? What mathematical operation is happening right here? What's multiplying inside a log the same as? Can't hear you, sorry? No, you have to have that memorized. None of your log rules are on the formula. You have to have them all memorized. Log 5 plus x plus 1, log 3 equals 2 minus 2x brackets, brackets, brackets, brackets, brackets, log 15. So far so good. I'm going to simplify this question a little bit. We can fix it easy. I started out with a nasty one to give you an idea of how far to go but I'm not going to give you one this tough. Here's what I would like you to do. With your pen or your pencil? Just go cross that out, cross that out, cross that out, cross that out. Let's pretend there was no exponent on the right-hand side for now. Because I don't think they'd give you one quite this tough. You okay there? You sure? You'd never make fun of you. Ever. Here we would have log 5 plus x, log 3 plus 1, log 3 equals log 15. Now I'm regretting crossing that out but I'm not going to go back and change it. Is that right so far? Get rid of brackets. Get all your x's to one side. If you had an x over here you would move it to this side, move all your other stuff to the other side. Anyways I think I'm going to go like this. x log 3 equals log 15 minus log 5 minus log 3. How would I get the x by itself? Okay. What if you had more than one x here? Do you remember the trick? You factored out an x. So if you had something like this x log 3 plus 2x log 7 you would say hey that's x log 3 plus 2 log 7 and you would divide by that whole mess. So don't forget that trick there. Oh just jog my memory by the way going back up here for a second. Sometimes as a written question they'll ask you to find an inverse equation. How do I find an inverse equation? Switch the x and y around and get the y by itself. Okay, fair enough. Let's keep going. Looks like x is going to be log 15 minus log 5 minus log 3 all over log 3 which is going to be I have no idea. Make sure you put the whole top in brackets. Come on calculator come on up. Bracket log 15 close bracket minus log 5 close bracket minus log 3 close bracket close bracket for the top divided by log 3. Zero? Yeah I just figured out y too. Does anybody see y? What's subtracting logs the same as? What's 15 divided by 5 times 3 in the bottom? What's 15 divided by 15 because that's both those would end up on the bottom subtracting logs they both end up on the bottom. What's the log base anything of 1? Zero? My bad fluke sorry. Apologize. Half-life. Yeah it's still good review of the procedure. So radioactive substance has to blah blah 250 grams decays to 150 grams in 30 years. What's the half-life solve using logarithms? Here's what we said for word problems for logarithms I gave you this final amount this is not on your formula sheet you had to have this memorized equals initial amount times the growth rate to the power of total time divided by the growth period. Final amount initial initial amount growth rate total time growth period or half-life and since question this question wants me to find the half-life it's actually asking me to solve for p which means everything else hopefully will appear let's see what's the final amount that we're dealing with in this question 150 what's the initial amount 250 what's the growth rate well it's a half-life question so point five otherwise they'll say doubles use a two triples use a three if it's a percent and it's increase one plus the percentage if it's a percent and it's decrease one minus the percentage total time growth period how do I solve this first thing I would do is divide by 250 on both sides I like to try and get my exponent by itself if I can so if I do that this is 50 point six all right so I'll get 0.6 equals instead of the half I'll write 0.5 because probably I'll end up doing my calculator anyways 30 over p now what take the log of both sides very good log 0.6 equals the log of 0.5 but you know this 30 over p because it's inside a logarithm can move to the front and I think I yelled at you guys I said write it as a complete fraction don't write it a little fraction because all you're doing is increasing confusion if you write it like that probably now all of you are recognizing how you're gonna get the letter p by itself how are you gonna get the letter p by itself cross multiply if you read as a tiny fraction kids don't see that you're gonna get p log 0.6 equals 30 log 0.5 fact I think you're gonna get p equals I left the 30 the zero off 30 there we go p equals 30 log 0.5 divided by log 0.6 30 log 0.5 divided by log 0.6 half life how many decimal places did they say here two forty point seven one by the way those you're writing the provincial if you wrote forty point seven zero they'll take marks off if you don't know how to round off properly so remember your math eight number three another word problem trig word problems are fairly popular and the reason is it's a word problem and it's an exponential equation they'll show up about half the time so the odds are pretty good that's what you'll see on your mock then about 30% of the time it's a log equation and only about 20% of the time is it a straight numbers algebraically exponential equation but if you can't do that you can't really do a log half life equation so that's why I spent so much time on those strontium blah blah blah has a half life of 28 days how many days will it take okay a equals a zero t t mr. do it c to the t over p it's really not important what letters you use as long as you put the right numbers in there yeah what's the final amount what's the initial amount it's a half life questions so one half or point five did they give me the half life oh they gave me this number they want me to find that try this one on your own I'll freeze the screen and do it slowly up here almost worked out bang even not quite but very close and I knew to go to days because how many days they didn't tell me to round off to whatever okay I'm going to your day 130 I hope trig so on the written for logarithms half life or word problem most likely logarithmic equation next most likely exponential equation okay on the written for transformations reciprocal almost always somewhere absolute value of the twist maybe once in a blue moon they'll give you an inverse equation with fractions and you'll have to get it by the wide by itself trig oh they're going to give you a quadratic trig equation almost certainly so here is your quadratic trig equation says solve two cos squared plus cos x minus one equals zero algebraically and it says over the set of real numbers give the general solution in other words they don't want me to go between zero and two pi they want me to find the answers between zero and two pi and then add the period which I think in this case is two pi because it's close so I'll rewrite it because I feel better to cos squared x plus cos x minus one equals zero remember I told you algebraically this was the same as two a squared plus a minus one equals zero if I temporarily replace the coses with a's you might like that a little better because it looks a little nicer I don't care how you find the roots to this equation quadratic formula or factor or the pq from Deutsch I don't care which method you use but you have to be able to find the roots because I know these factor I cheat hey I got a two-way I'm pretty sure it was that and that and they want a minus one a plus one plus one minus one I think when you foil that out you'll get a positive one there I just kind of intelligently guess my way which is probably the best way to do it but so here's what I really have what are the roots of this particular factor one half negative one except there's no letter a what was sitting where the a was originally okay so I'm gonna have this cos x equals one half cos x equals negative one and I'm gonna move this guy over I'm gonna solve both of these independently it did say exact values that's also my trigger phrase special triangles in unit circle so let's see how we do here this first factor cos x equals one half one over two do I have a special triangle it has a one and a two in it okay we're gonna use our cast rule cosine is positive which means I'm here or here this is the one two root three triangle because it's got a one and a two in it which of those angles has a cosine of one half the bottom one or the top one the top one how big is that top angle in radians please which means that's my reference angle this angle here is pi by 3 and this angle here is pi by 3 right so I'm going to have theta 1 equals pi by 3 and theta 2 equals pi pi by 3 because 6 pi by what because 6 pi by 3 was all the way around I'll do the general solution in a second how about over here Haley what does cosine of x equal alarm bell you know why because this is not when I have ones and zeros and negative ones this is unit circle not special triangles how did the unit circle work the unit circle said on the unit circle R was one by the way sign in terms of a graph is what over what y over r cosine colon is what over what in other words when it says cosine of x equals negative one it's really saying on the unit circle what is your x-coordinate negative one if I visualize my unit circle here on positive one you know what my x-coordinate is negative one right there if each of these is one long as an angle Shannon there's another way you get that as well you could also say I remember the cosine graph looked like that where this was one and this was negative one hey hey negative one what angle is right there well if that's too high you can also get it from our little sketch of the graph okay I won't be asking you to sketch a graph on your final on your mock although you might on the written sorry on the multiple choice be asked to recognize a graph totally which is why we practice sketching them anyways I think those are my three roots now I have to give the general solution plus multiples of 2 pi where n is an integer what if it was sine also still plus multiples of 2 pi hey what if it was tangent what was the period of tangent pi and then don't forget your reciprocals I'll go with their originals okay coming back to you remembering some of this brushing up the cobwebs what else might they ask on the written always a trig identity I'm telling you right now is going to be a trig identity on your mock you'll see one Thursday almost always so trig identity always hundred percent but they always ask two trig questions about seventy percent of the time it's a quadratic trig of some type about 30 percent of the time it's a trig word problem ferrous wheel tides I don't have an example of one of those look at your reviews ah but I do have a couple of trig identities for us to work through you know you probably want your formula sheets in front of you because I don't have all my trig stuff memorized so here's one suddenly worried that this might be the one you see on Thursday I hope not I picked these from old provincials but of course I picked the ones that you guys are doing often from old provincials well yeah whatever so many trig identities out there that you can make up before you start to get repetitive so I have sign of 2x all over 1 plus coast 2x equals secant squared x minus 1 all over tan x I love these remember our approach start with the uglier side you know what I'm look both these sides are ugly I'm pretty sure I'm gonna be doing work on both of these okay then we said look for the Pythagorean identities first I look for squares and right now I'm kind of twigging on secant squared minus 1 is that something from the top line there I think we have a secant squared in one sorry oh are you so this is tan squared x all over tan x oh good gosh that's gonna fall apart what's tan squared divided by tan tan x and what is tan x sign over coast I'll remember that I'll probably turn this into sign over coast and then I'll let it equal tangent okay I'm pretty much done on the right hand side left side now this is another one that we always did a substitution sign of 2x there's only one option is double angle what is the sign of 2x all over 1 plus now coast sign of 2x I got three options but I'm gonna give you a hint I would love to get rid of this positive one so pick the one that's got a negative one in it just two coast squared minus one now if you picked one of the other ones wouldn't be wrong you just be adding an extra line of work because the nice thing here then is I have one plus negative one yay in fact what I have is this 2 sin x cos x all over to coast square oh is there two on the top is there two on the bottom can I cancel is it factored it is oh not only that how many coasts on top how many coasts on the bottom you got your nerdy adrenaline rush yet I think we're almost there because I think I get this sine x over coast x now you can't stop here even though you know that's tangent you would have to for the marker say hey that's tangent question easily done or QED or full it's just nerdy so I do it you know how many marks that was in 2002 the trig identity is always the last question and I'm gonna say it Shanna if you nail that last question that is a great feeling especially when the provincials were mandatory I loved it when my kids could come out of the provincial hey mr. do it I know I got some wrong but I spiked that last question down that's a great feeling often the math exam was their last ever exam to just a nice way to end your school career your high school career ooh another one do you want to try this one on your own I'll make a little larger so that certain people can see it that big enough ooh I think you're pulling out the conjugate here you know I can tell just by glancing at it I noticed a binomial denominator with no squares and a one I suspect you're gonna be pulling out the conjugate along the way here can I shrink it back down you guys got it written down I'll try this here you go I'll go see if we end up in the same spot that's one way of doing it probably others conjugate is when you multiply instead of one minus one plus cost over one plus cost that we called that the conjugate that was our last resort but I'd gotten as far as I could with this one there was no more squared nothing was going to cancel better go over here and try the conjugate that okay now the only thing I didn't put in here is complex fraction when you had to multiply top and bottom by a common denominator to get rid of the complex fraction make sure you can handle those okay so that's straight there may be some combinatorics on the written section what kind of questions will they ask for combinatorics two main types the first type is solving some kind of factorial equation either they'll give it to you and choose form like I did on your test or they'll give it to you in actual factorial form solve algebraically this one here how do we do this well first of all we need to get rid of the factorials you can't cancel exclamation marks what you can do is expand one of them until the other term appears and then you can cancel and we did that by asking ourselves hey which one's bigger okay by the way little hint little hint little hint little hint little hint what if I did this it's the same question right I don't know if I put them over right away but eventually somewhere along the line I'd be cross multiplying I have that on one of my mocks I don't know if it's the one you're getting tomorrow but the number of kids that panic no now Colin you said n minus one was larger I agree so I'm going to expand that it's n minus one and then one less and then one less and I'll stop writing because the denominator just appeared can I cancel this is factored and I get n minus one n minus two equals 30 a lovely quadratic foil n squared minus three n plus two equals 30 n squared minus three n minus 28 equals zero numbers have multiplied a negative 28 and add to negative 3 7 and 4 you say oh I agree with that how about n minus 7 and plus 4 equals 0 my roots are 7 and negative 4 extraneous why because negative 4 take away 1 is the factor of a negative or you can just think of it as you can't really have negative 4 objects if you're talking about counting that's the first type of wow big young that's the first type of combinatorics written question the second is the committee at most at least kinds of questions so a class has 30 students how many ways can a committee of three people be selected from the class have they mentioned boys and girls or anything they mentioned order then it's real simple from 30 choose 3 how do I know it's not a permutation how do I know it's not letter P they didn't mention order and I glanced at part B where they did mention order and okay what is 30 choose 3 I've no idea 4060 how many ways can an executive committee consisting of three people be selected from the class now there's two ways to do this I fall back on fundamental counting principle when I'm in doubt I could go hey one two three how many choices do I have for the president 30 29 28 now that happens to be 30 permutate three because that's how we derive the permutation equation either of those you get full marks what do you get see if there are 10 boys and 20 girls in the class how many ways can a committee of three people be selected if it must contain one boy and two girls this sounds like a bucket boys girls 10 20 1 2 right what's my equation gonna be 10 choose 1 and 20 choose to 1900 what if they said at least two girls or three so it'd be this or 10 choose 0 20 choose 3 and evaluate probability what kind of probability shows up about 80% of the time it's some kind of conditional with marbles or factories defective or given that it's a black marble what's probably came from bag a or something like that about 30% of the time but it's showing up more often now is binomial with conditional so here we go probability of winning a game is point seven you play three games how many games I probably could tree this but it's right on the edge and I think this seems to be suggesting that the probability is winning is always point seven never changing I think this is a binomial probability distribution function so I could tree or if I win all three games several ways to do this I could go by an own PDF three games point seven three wins or can you visualize the tree ready what's the odds of winning the first game and what's the odds of winning the second game yet see this branch would be the far left hand branch where it was success success success so I could almost visualize it just that way as a tree as well anyways it's gonna be that which is what sorry point two four three okay that's what I couldn't hear Shannon can you read for me the first word of part C given yeah what the heck happened to part B I don't remember if given given given if you win at least twice what's the probability that you won three times given see I put that together okay which is going to be both over the given one now let's see if we can do the both if you win at least twice that means you've won two or three times if you win three times that means you've won three times what's the overlap between two or three and three three yet fact that statement all three and at least twice just simplifies to all three which I think we did in part A what was the probability of winning all three at least twice that means that means what I want the cases that means what Savannah two or three two would be by no PDF of three comma point seven comma two or sorry three is right on my limit but yeah see the three the three three branch tree would have eight branches at the bottom four on the two on the first four on the second eight on the on the next one and you have to go at least twice you'd be walking down a lot of branches but yeah I'm what I'm saying Kayla is I've also seen this as with seven games let's say where they forced you to use a binomial conditional okay so that's why I'm doing this one that's two or what did you say the other case was Savannah oh I'll just put the three four three there I'll leave the top for now I would work out the denominator on my calculators so I would go by no PDF of point not a point three of three comma point seven comma there's two wins or point three four three there's three wins the odds of winning two or three is point seven eight four the odds if you know that you've won at least two what are the odds that you've won all three point three four three divided by point seven eight four point four three seven five conditional okay and then the last one we had one on your test as well some kind of a jar marbles question there's three minutes left I'm not gonna try and work this one and I'm just gonna say make sure you know how to do this one that's pretty much your written