Hey guys, hope you're doing well, so this is a grade 11 mathematics paper one exam and this one is approximately here we go two hours and 100 marks so wishing you all the best with this one if you want to download the paper Just check in the description below and then you can See what questions are in the paper and then you can you can try it yourself first Or you can you can fast forward to the specific question. You need help with all the best Determine the values of x for which the following expressions are defined. Right, so the main thing you need to understand about these questions is, if you have a square root, then this number must always be 0, the number inside here, or anything positive. Anything positive. if it as soon as you have a negative then it becomes uh non-real okay and that also means like it's not defined okay so for so so so mathematically what we can say is that whatever's inside here let's call it x must always be bigger than zero but it can also be equal to zero okay so here you'll just say that whatever's inside here Okay, maybe I shouldn't say x, maybe I should say whatever's inside here, whatever's inside here must be bigger than or equal to zero. So here we can say that this must be bigger than or equal to zero, take the three to the other side. And so there's the answer. When you're dealing with a fraction, the denominator, whatever's over there, must not. be equal to zero. It can be negative, that's okay, but it mustn't be zero, because if it's zero, then we say undefined. But they said, where are the expressions? Defined. So what we'll say is that this part here can be anything, but it must not be equal to zero. So if you then solve that, you would see that x can be anything except three. Now, you're not allowed to just leave the answer like that. So what you need to say then is that, x can be any real number except for x must not be equal to 3. For which values of x will this be non-real? Okay, now what does non-real mean? Well, let's talk about that. So when you have a 0, let's say you have like a 3 over 0. So when you have a 0 in the denominator, that is actually called undefined. When you have a negative number inside a square root, this is called non-real. So we are going to say, so where this expression is non-real is when whatever's inside, okay? So if you haven't, yeah, so whatever's inside, so we're just going to say. um 3x minus 1 over x minus 1 and we're going to make that smaller than 0 because when you say something is smaller than 0 that means that it is negative and when that expression is negative then it's non-real so now we need to go and solve but this is the part that they don't talk about too much in normal pure maths i have taught some learners a bit of ap maths and in ap math they do talk about it Normally, if this was a normal mathematical question, right, you would take this and you would multiply it to the other side and so it would end up just becoming 3x minus 1 equals to 0. But when you have an inequality sign and you have an x over here, you are not allowed to multiply that over. Okay, there's a whole reasoning behind it and it goes into the AP math students go into the history of it. But I'm just going to tell you, you're probably hardly ever going to get a question like this in grade 11 and 12 normal pure maths to be honest, I'm quite surprised that they've got this one here. But what you do is you leave everything on one side, and you need to go and find the critical values the critical values. Now to do that, you're going to make the numerator equal to zero, and you're going to make the denominator equal to zero. And then so we're going to say 3x minus 1 is equal to zero, x minus 1. These are not the answers, these are just our critical values. You're then going to solve each one. So this one would become 3x equals to 1, and so therefore x is a third. This one you're going to make it equal to zero. I mean you're gonna take the one over okay now these become your critical values that you put onto a number line there and there so a third is always first and then one now what we need to go and do because maybe you've seen my number line questions before where I use a parabola but the problem is that this is not a parabola so we need to go and do a different technique I'll show you this now We now have three categories in our number line or three areas. which is up to here, B, which is between the two, and then C. We need to go and examine between in section A, section B, and section C. Okay, let's just say here 1 over 3. So you choose any number you want over here. So I'm going to choose the number 0, because 0 would be over here. I'm going to plug 0 into this equation. So 3 times 0 minus 1 over 0 minus 1. and what we should get, it doesn't really matter what the answer is, it matters if the answer is negative or positive. Now, if you had to go do this one, you should get a positive answer, and so we put a little plus there, but it doesn't matter what the number was, that, you know, it doesn't matter at all. Now, we need to go and choose a number anywhere between these two, okay? So, I'm going to choose two-thirds, so, because I'm choosing a number inside here, right? So, I'm going to say three times two-thirds, take away one over two-thirds. Takeaway 1. Now the answer is going to be a negative. Okay, so that'll be a negative. You can go check it out for yourself. And then we need to choose a number here. Now you can choose anything. So I'm going to choose 10. So we're going to say 3 times 10 take away 1 over 10 take away 1. And if you had to go work this out, you should get a positive answer. Okay, we're nearly done. Now we just need to see. They asked us where is this expression smaller than 0. So where is that expression negative? So it is negative over here. And so your answer, you could either use interval notation. We're not going to include because they didn't include. We're going to say from a third. up to 1 and then if you prefer interval I mean set builder you're gonna say X is bigger than 1 3rd but smaller than 1 all right so this question we need to solve for x now this one's a very easy question some learners are going to try multiply the brackets out but just pause for a second and think about that um i don't literally mean pause the video because then it's awkward you're not going to hear what i'm about to say um what i mean is so like for example if you have x minus let's say let's say let me take you back to the beginning so back in the day you know that was a joke it's just back to the beginning of this type of question now if you had this if you had something like that You would obviously try factorize it right and that would give you like X minus 4 X plus 1 What would you do at this step? Ah You would say X minus 4 is equal to 0 or X plus 1 is equal to 0 you see So when you have brackets, you can go straight to getting your answers If you multiply these two brackets together, then you're actually going back to the beginning instead of going down to the finish line. Okay, so take your brackets straight away and just say X plus 1 is 0 or x minus a half is zero. So brackets are good. Brackets are where you want to be. Now you can solve by taking the one over, and then you can take the half over, and there would be your two answers. With this question, whenever they say correct to two decimal places, that is a hint that this formula, this equation cannot factorize, because if it could factorize, you wouldn't be getting decimals, right? So what they're trying to tell you, hint, hint, nudge, nudge, wink, wink, is that you must use this quadratic formula over here, you know, the one we use many, many times. Well, I hope you have, otherwise this is very awkward, like, what are you doing here in this video, bro? Like, are you grade 9 maybe? So that was just a joke, like if you are grade 9 and you're just curious about learning new stuff, that's absolutely awesome. So remember what a, b and c are. So a is always the number in front of x squared, b is always the number in front of x, and then c is always the leftovers. So b. so a is 1 because there's nothing there, right? So if there's nothing there, it's a 1. b is also 1 because there's nothing there, and then c is minus 1. So I'm just going to go fill that in now. So minus 1 plus minus the square root of 1. a is 1, c is minus 1. At the bottom, you have a as 1. Okay, go ahead, choose a plus or a minus. You can't choose both on your calculator. And go get your first one. So I'm going to choose a plus on my calculator for the first one. and so if you're around to two decimal places as they suggested or not suggested they told us it's not a suggestion you don't like choose if you want to use the advice okay so 0.62 then if you go use the negative on the calculator so just backspace backspace backspace backspace until you can choose the change the plus to a negative and this one is negative 1.62 so those are your two answers you here we have an inequality so with inequalities i find a lot of learners make mistakes so what they do is they do this this is the wrong i always have to write wrong here because you know what happens i mean i think this maybe happens imagine a learner is like okay yeah i understand the question um and then they they they fast forward a little bit and then they see they don't see that this part's wrong and then they're like they see what i'm doing here This is just made up by the way. And then they get to this part. and then they're like, no, but that's, I don't understand how Kevin's doing that, but they, because they fast forward, they didn't hear the part where I said, this part is wrong, you understand what I mean? So yeah, this part's wrong, nb, nb, this part's wrong, so x squared take away 2x plus 1, smaller than or equal to 0, so what a lot of learners do is they go factorize, now there's nothing wrong with that part, that's absolutely fine, you should get x minus 1, x minus 1, that's perfect, okay? What they then do is they pretend that this is an equal sign and they just say that x minus 1 is 0 and then x minus 1 is 0 and then they just, but they keep it like this and then they just go solve both of those. That's the wrong way to do it. What you want to rather do is, so this is now the correct way, is you can factorize it like that but keep it like this. put a little like box for yourself here and go get what we call your critical values so what you do is you can um it's like a little side thing that you're doing so make your teacher aware that this is not the part of the your this is not like your answer this is just a side little calculation um you make it equal to zero you go get your answers like you normally would so x minus one is zero or x minus one zero so obviously it's just going to be x equals to one and that is your critical value but that is not the answer. That is just a number that you can now put on your number line. Okay, so it's only a one. Then I want you to now look at this and think about what kind of graph is that? Exponential, hyperbola, log graph, if you're grade 12, parabola, straight line. Well, it's a parabola because it's got this leading coefficient. I mean, so it's got this leading parameter of x squared. Now, a parabola is happy or sad. See the sad face? See the happy face? Now, I hope you're looking like this today, by the way. If you have an exam tomorrow, then maybe, I don't know, shouldn't be like that, eh? Should be like that. Okay, Kevin, get going, dude. So, if this number here is a positive, it's happy. If it's negative, it's sad. So that's a positive one. So you draw a parabola. Now, normally you would draw a parabola that goes through two points, you know, like your two x-intercepts. But this one only has one, so that means this graph is on the x-axis, like that. Kevin, where's your y-axis? You don't need a y-axis for this technique. Okay, so as I said, normally you would have two points and then you'd go through both of those, but with this one, it's only one. Okay, now what they said is where is this parabola smaller than zero? I know it also says equal to, but for now I just want to look at the smaller. Smaller than zero means... underneath the x-axis bigger than zero means above so they want to know where is it below well it's never below see here there's no um this is your x-axis okay it's never below but they also said where is it equal to zero now what that means is touching touching the x-axis okay touching the x-axis so let's make a little summary over there so when it's above you That is where your parabola is bigger than zero. When it's below, that is when your parabola is smaller than zero. when it is on the x-axis, then that is where we say that it is equal to zero. Okay, so they said here, where is this parabola smaller than zero? Okay, there's none of those, we can't see anything underneath, but then also equal to zero. Now that's where it's touching the x-axis, and so it's there. So we can say it is at the point where x is one. You're not going to say where x is smaller than one, x is bigger than one, it is only touching at the point there where x is one. This looks like an awesome question five marks So what we want to do whenever you have a square root like this with these square root equations step one Get the square root alone or get the square root term alone what that means is that this is one term so that three that three can be friends with the square roots and they can move together for now okay maybe later we'll tell the three okay dude you're being a bit clingy but for now they can stay together so step one is just take that X over this X of here okay so you could if you wanted to divide by three but I don't like to do it what I do is a square at this moment of here And I know in some of my previous videos I have divided, but you know, I think this actually works out better. So you just square this part and you square this part, so it becomes nine and then this two. evaporates, I like that, evaporates the square root. And so that becomes x minus 1. And then on this side, please don't just put the 2 in there and put the 2 in there. But Kevin, that's what you did exactly what you did over here. Yes, I know that. But that's because here we have one term. There's no plus or minus in between. As soon as there's a plus or minus in between the two terms, then you have to do a double bracket. Okay, some learners... they know how to do this part without using a double bracket, that is okay, if you know how to do that, but I know when I was in grade 11 and 12, I wasn't comfortable with that, so I just used to use double bracket, Kevin, you're such a rookie, bro, I've been doing the anti-double bracket method since I was like in grade 9, bro, okay, well, that's great, I'm happy, but let's just carry on now, so 9x minus 9, is equal to, okay, now I'm going to multiply this out, so it becomes 1, and then minus x, minus x, and then plus x squared, so that becomes minus 2x, eventually, you might want to just make sure that you agree with me on that. Okay, now here we're just going to solve an equation, so I'm going to take everything to the right. The reason is that I just like it when this is positive, but you could have taken everything to the left. It really doesn't matter. So we're going to end up with that. I'm going to bring that 9x over, where it will then become a negative, and then plus 1 plus 9. And so now we end up with x squared take away 11x plus 10, because that's 9 plus 1, and that's negative 2 minus 9. And now we just factorize, or if you wanted to, you could use the quadratic. formula but this one will factorize pretty easily as that so now we can just say that x take away 10 is zero you see how i put the zero on the right now whereas here it was on the left that really doesn't matter but if you wanted to keep it all technical you could also do that it doesn't matter it The way you do it in the test they don't care so for that part so X is 10 Or you could say X take away 1 is 0 and so X is 1 now up till this point You will get 4 out of 5 so Kevin. How do you get 5 out of 5? Well to get 5 out of 5 you need to remember Whenever we have a square root in these equations, you need to do a little check at the end Okay, so how does that work? Well, you take each number, so let's take the 10 and you go plug it into all the places where you see x. Okay, so that'll become 10 plus 3 square root of 10 take away 1. And then on the other side, we just have a 1. So that's the left hand side. So that's the and then on the right-hand side, you just have a 1. And we want to see if the two sides are equal. So if you had to go work out this side, you're going to end up with 10 plus... You're going to end up with 19 eventually. And then on the right-hand side, you have a 1. So is 19 equal to 1? Nope. So that's a no solution. So this one is not actually part of our answers. Now we go do exactly the same with our other answer which is x equals to 1. We go plug it into all the places where you see x. So let's do it up here. Okay, and then that's the LHS or the left hand side and the RHS, which is the right hand side. This will be pretty random. I just said LHS. I actually used to go to a school in Ladysmith called LHS. So if you go to, if you, I didn't go there my whole life, but I was there in grade eight, Ladysmith High School. It's an old school, man. My parents went there, my grandparents went there. Well, yeah, some of them. So yeah, I think that's yeah, in Ladysmith, just outside, like, I think, like, two hours outside of Durban, yep, I was there in grade eight, and then we moved to the Western Cape, okay, so, if you work out the left hand side, then you end up with one, if you had to go work that out, and then on the right hand side, you also end up with one, and so, these two sides are equal, and so, x equals to one is a valid answer. okay so with this question a lot of learners when they look at this they freak out what some learners would even do and i'd probably do this when i was in grade like earlier grade um i'll just cancel the threes but that is wrong okay so what think about this carefully if i said what is x plus x plus x if i just said that to you you'd say 3x wouldn't you um if you said x3 you just got to think about that carefully we're not multiplying so what is this plus this plus this well how many thing how many of those are there well there's three of those weird things. Does that make sense? There's three of these weird units over here. Okay. And then you make that equal to three to the four. Okay. Now, I like that there's threes everywhere, but what are we going to have to do? What would you do here? X3 multiplied by X4. What would that become? Well, X7, right? You add. So you don't, and notice, did I change these x's to a y? No, they stay x. So what could I do here? Here these are the same. So what do I do with these exponents? Well, just like you did over here, you add them. But then do I change this to a 9? No, you keep it a 3, just like you kept it an x over here. But Kevin, 3 times 3 is 9. I hear you, but we're not multiplying 3 times 3. We're multiplying 3 to the power of 1 and 3 to the power of x. it's very different. Okay, so we end up with 1 plus x. So we just added the exponents, just like we did over here. And then we have 3 to the 4. Now we are in a good position, because on the left and the right, we have the same base. So we can cancel the base. Boom. And then you can just take x to the, I mean, the 1 to the other side, and so x is 3. Here we need to solve simultaneously. The simultaneous ones, you should smile when you see them because they're six marks and they, once you've practiced a few of them and you get the habits or the techniques, it's pretty straightforward. So what I like to do is I always choose the easiest looking equation. So which is the easiest looking? Well first of all, let's call this equation one, equation two. Now which one is the easiest one, the most simple? Well, that's definitely this one. Now what you do is you see, can I get X or Y? alone. So I'm definitely not going to go for this 2y, because it's got a 2 in the front, and I don't like to work with fractions. So I'm definitely going to look at this x, and I'm going to try get it alone. I'm actually going to take it to the right hand side. Oops, that's minus 6 equals to x. You see, so I took that over there, and I took that over there. And so I'll call this equation number 3. So what this is now telling us is that we have x. x is equal to 2y minus 6, okay? So what I'm now going to do is I'm going to take that 2y minus 6, and I'm going to go substitute it into the other equation in all of the x places. There, oh no, there's no other one, okay? So I'm going to tell my teacher that I'm going to sub number 3 into number 1. and so that's going to give 2y minus 6 to the power of 2, plus 2, and then x is 2y minus 6, and then there's a y, that y over there, and then on the other side we have 3y squared. So what we've done now is we only have one variable, and that is y. Okay, and so now we need to now be careful, whenever there's a 2 here, to be safe, put two brackets. There we go. Now some learners get really awkward with this, this and this. What do you do? Well, remember, you can just put this y in the front there because when you multiply, order doesn't matter. So just put it in the front, then it looks normal, like something you've worked with before. So now you're just going to take this 2y and you're just going to multiply it to each part. And so that becomes 4y squared, take away 12y equals to 3y squared. Then what you're going to do is you're going to multiply these two brackets together. So that'll become 4y to the power of 2, take away 12y, take away 12y, add 36. Then we have 4y squared, take away 12y equals to 3y squared. And now it's just a... matter of putting everything together. So I'm going to put all the y squares together. So that's eight. And then I'm going to bring this three to the left. So that eventually becomes five. So I'm going to say five y squared. Then I'm going to do negative 12 y negative 12 y negative 12 y again. So that's negative 36 y, not 36 y to the power of three, you're not timesing, we're just plussing and minusing these numbers here. And then plus 36. Okay, now I would definitely use the quadratic formula here, but if you want to try factorize you can. So the quadratic formula is the negative b plus minus b squared take away 4ac over 2a. And so remember that a is whatever's in front of the square term, b is this one, and then c. So that would end up being negative, and then there's a negative 36 for b. negative 36, I'm going to run out of space aren't I? Now a is, what did we say, 5, c is 36, and then all over 2 times 5. Okay, go ahead, choose either a plus or minus for your calculator. I just realized I should change this to a y over here. Okay, so we actually end up with nice answers, and this was a y, it doesn't really matter. So y is 6, or Y is 6 over 5. Okay, now we're not done. So those are the Y answers. So what you do now is you come back to this beautiful equation number three where you have x by itself because now you can just say x is going to be equal to two and then y is six. take away 6, and so x is 6 as well, so 2 times 6 is 12, yeah, and then you do the same with this one, so we say x equals 2, 2 multiplied by 6 over 5, take away 6, and so x is negative 18 over 5. Simplify without using a calculator. Now guys, we are going to use a calculator, but we're going to do it in a way that doesn't look suspicious. So what you do, okay, is let me try to explain carefully how this all works. So you see this number here, okay? Well, let me first talk about this. What is the square root of 5? Quickly, tell me, what is the square root of 5? Exactly, you don't know because we don't know the square root of 5. What's the square root of 7? Come on, you can do better than that. Okay, I'm joking. Obviously we don't know what the square root of 7 is. But what is the square root of 4? Now most of us know that that's 2. What's the square root of 9? 3. What's the square root of 16? 4. What's the square root of 81? 9. You see, these are the nice numbers. We like to use these numbers, okay? So the goal is, and of course we can go a bit more, typically we need to know all the way up to... 11, have we done everything? Oh, the square root of 1 is 1, but we won't really use that one. 2, 3, yes, 4, 5, 25, Kevin, what about 36, brah? Come on, what about 7? Square root of 49. Okay, so those are the nice numbers, these ones here. 25s, the 36s. Okay, the goal is... Then, and I'll show you how we'll use the calculator shortly, but you want to take this number, right? This number over here. And the goal is to be able to rewrite that number using one of these, just one of those. So what do I mean? Four, you can say four times three. See? So you don't have to, I know this one is not a square root number, I know that, but this one is. And so that is all that you do. And then in the next step, what is the square root of 4? Well, it's 2. What is the square root of 3? We don't know, so we just leave it on the inside. That is what we are going to try to do. So for example, 75. we could change that to this number, 25 times three. So you see how we're doing that? So that is what the, oh, and then in the next step, what is the square root of 25? It's five. And then on the inside, you're just left with three. So you can do it by yourself like that if you want to, but the calculator does it as well, because the calculator, so, because some learners, they struggle to find the right number. So I'm gonna show you a cool technique that your calculator can do it for you. So. leave this line open for now. Okay, go to the very end and put the 3. Then type square root 12 on your calculator. So square root 12 on your calculator would be 2 square root 3. We already knew that, but... So we're just going to say times 2 square root 3. Then minus. Type this on your calculator. So that's going to be 5 square root 3. Then come back to this middle step, and now you can look at the number that the calculator gave on the outside, and you can think about what was that number when it was inside the square root. Okay? So the way that you do that is, okay, this just stays the same, 3. So this number here is a 2, so on the inside it must have been a 4. And then this number is a 5, so on the inside it must have been 25. Because the square root of 25 is 5. Okay, I hope that that makes sense. So you could then show your teacher that step. So everything looks good, you're a good student, you didn't use your calculator. Now moving into the next step, 3 times 2 is 6. Now, 6 minus 5 is 1, so we have, well, 6 square root 3s minus 5 square root 3s means we have 1 square root 3s left over, and that is the answer. Without using a calculator, show the following. Okay, now I've shown many questions like this before. Whenever you get a question where it's like 2 to the power of, I don't know, like 1,007 plus 2 to the 1,012, just take out a square root, take out a common factor, and always choose the lowest number. Okay, that shouldn't be a surprise to you. If I asked you to factorize x squared plus x7, what would you take out? x squared you always take the lowest number okay so take that out and then you'd be left with one for this part, and then plus 2 to the power of 5, because it was 1012, and then, it's like this, guys, it's like this, this shouldn't surprise you, if you have x squared plus x7, and you take out x squared, then you're left with 1 plus x5, see how we just minused, or these two added together is the original, so these two added together is the original, okay. And so this is the technique I want you to be comfortable with. So I don't care about this stuff. When I see that, I think straight away, ah, factorize. Thank you very much. Factorize. Okay, so let's just quickly write that down. So 2 to the 2, what was it, 1, 2010, plus 2 to the 2012, equals to 5 times by 2 to the 2010. Okay, so take out that common factor of 2 to the 2010, and then you'd be left with 1 plus 2 to the 2, um oh and we're trying to show oh we're trying to show that this is equal to this okay well that's fine we can keep writing this down as well it doesn't matter um and then look at this guys what is that going to equal well one plus four is five and so this can just be five and then two to the 2010 which is that part and then we can say that and look at it left hand side is the same as the right-hand side. Okay, so we did that. That was a nice easy one. Ah, hence, they love to add in that, eh? Hence, calculate the value of this. Well, what we do now is we see straight away that that is over here. And so what we will do is we will just replace it with that. There was a specific reason they wanted us to know that this part can be replaced with that part. And so we now end up with so I'm gonna replace I'm gonna replace it so we end up with five times two to the 2010 Plus ten and then at the bottom we just have this Now there's nothing you can do at the bottom, but at the top you could factorize a five out. So You'd be left with that Oh, but no, I see what they're doing. I like it. Okay, but let's just do it like this for now so we're still not getting to where we need to be. So what I would say is the following. Change this to 2 to the 1, because it is 2 to the 1, and then take out a common factor from those two, just like I told you to do here. So now you're just going to take out the smallest one, which is this one. So now we end up with 5. and then over here we have 2 to the 2010 plus, oh no, we need to take out a common factor. So we take out a common factor of 2 to the 1, and then you're left with 2 to the 2009 plus 1. and then at the bottom you're still left with 2 to the 2009 plus 1, then this 2 can just come out and multiply, so we end up with 10, and then we've got 2 to the 2009, whoopsie, plus 1, at the bottom, 2 to the 2009, plus 1, and then those can cancel, so we're just left with 10. Okay, so for 3 marks, it says we must simplify. So all you're going to do is you're just going to say, we know that that is the same as 1 over x, and the other one is the same as 1 over y. Here you are adding two fractions, so you need to get a common denominator. So that would just be xy. So that means that this one is going to be multiplied by y, but what you do to the bottom, you do to the top. This one's going to be multiplied with x. What you do to the bottom, you do to the top. and so we end up with y over xy plus x over xy, and so that just ends up becoming y plus x over xy, and that is the answer. x plus y or y plus x doesn't really matter. y plus x over xy. Okay, so it says, hence determine the sum of the reciprocals. Now, these are called reciprocals. A reciprocal, so like if you have the number x, right, then what is the reciprocal of that? 1 over x, and that is what this is. So this is a reciprocal, and this is a reciprocal, and it says, hence determine the sum of the reciprocals. That means plus. So this is the sum of reciprocals, and in the previous question, we just worked out that that is equal to this. Okay, now it says of two variables, let's just make the two variables x and y. If their sum, if their sum, okay, so that's when you're plussing things together, and their product where you're multiplying is that. So check this out. Here's the sum. and here's product because you're multiplying so we can literally just do this we can literally say that this is going to be equal to y plus x over x y and so that's going to be their sum which is 10 and their product which is 20 and if you solve it you get a half Now some of you are like, yeah, but sir, how's that worth four marks? So the way, I don't know why I always talk like that, but the way that this works is that you got one mark for that, you got a second mark for that, you got a third mark for that one there, and then you got a fourth mark for the answer. With this question, here's all the questions that we are going to be doing, but I will split them up over different pages just so we have a bit more space to work with. So, here we go. So we are given the following number pattern, okay? The first question for two marks says write down the next two terms. So we need to just try to check, is this... linear or is it quadratic so if you have one uh let's write a little bit smaller over here um well no we're probably gonna have to oh in the next question we're gonna have to work out the formula okay so 1 minus 3 minus 9 minus 17 so we need to see what the differences will be to work out the difference you always take the number on the right minus the one on the left not the left minus the right it's the right minus the left So that's going to be negative 4. Here you're going to say minus 9. Be careful, minus 9 minus minus 3. So that becomes minus 6. And then this one will be minus 17 minus minus 9. And that becomes minus 8. To go down to the second difference, you take this one minus this one. So that'll be minus 6 minus minus 4. And so that's going to be... Um... negative two and then this one would also end up getting negative two so because the second row is remaining constant this is a quadratic number pattern so the first question says write down the next two terms so what you do is you just carry on so you see how it goes minus four minus six minus eight well the next one would then be um minus ten and then minus 12. So that'll help you to work out this value and this value. So minus 17 minus 10 is going to give us minus 27 and then minus 27 minus 12 is going to be negative 39. So the next two terms would be negative 27 and negative 39. This question says determine the general formula or general term. Now you need to be getting full marks for this question because this is a very easy question, very basic, where we've done many examples of this. OK, of course, you could make a little mistake in the test where you make like a silly error or whatever. But in general, like something like this, you need to be smashing. OK, so we know that there's these formulas we use. So 2A equals 3A plus B equals. and then a plus b plus c equals so let's go write those all down so um well let's start you always start at the bottom here so um whenever i say that i always want to be like side of the bottom now yeah but i do that all the time so i think it's getting a bit old now but i can't help it my mind just does it okay um and then for some reason whenever i say that my mind's like another one by um what's that guy DJ Khaled. I don't know. My mind just, it's been looping on that for like years. Whenever I do these maths questions, certain parts just trigger like a little brain synapse in my head. Sorry. So, so Kevin, you're so weird. So 2A is equal to negative 2. Then A would be negative 1. Okay. Now you come up to this next. part of a year. And so that's going to be three a plus b equals to negative four. And so it's a stupid song in my head right now. Mark gonna sing it. Don't worry. Three times a it's that okay, it's that song that is so embarrassing. How does it go? It's like came up to a dinner. I did it. I did it. I did it Did it did it did it I don't know the exact words, but I did it. I did it I don't like the song at all, but it's just in my head right now. I Think it's quite yeah, I don't like this. I'm stupid I mean it for me. This is boring doesn't have any like doesn't hit me So, you know I sound like I'm probably like Like that song is pretty old, but like, I don't know, it's still in my head. Okay, so 3 times a is negative 1. and then plus b equals to negative 4, and so now we just solve, so we end up with negative 4, plus 3, because this becomes negative 3, you take it over to the other side, and so b would end up being negative 1, okay, negative 1. Now you come up to the last step, so you go a plus b plus c equals to 1, and so a is negative 1, b is negative 1, equals to 1. Okay, now if you had to go take c over to the other, I mean, if you had to go take the minus 1s over, you should eventually end up with c as 3. Now, we know that a quadratic formula is a n squared plus b n plus c, and so let's just write that a little bit better. Okay, so A is going to be negative one, B is negative one, and C is three. Now always go and check yourself. So the way that you check yourself is you choose one of these. Now don't choose the one in the beginning. There's just a higher chance that you might get the same answer, but it's actually going to be wrong. Rather choose something more complex. So let's choose this one. Now that is at position one, two, three, and four. So what you do is you go plug in position 4 into this formula, so that means all the ends become a 4. Remember that minus 1 is not part of the end, so it stays on the outside like that, okay? And then 4 and then plus 3. And then if you have to work it out, we end up with negative 17. So negative 17. and so because we're getting the same, we're good. Okay, so let's move on. Okay, so let's just remember what our formula was. It was negative n squared, take away n, add 3. So it says determine the value of the 30th term, okay? So now some learners don't know, do we plug 30 here, or do we plug 30 in the place of n? Well, let me quickly show you a sequence. So let me show you this. These numbers. These are TNs, these are TN numbers. The position, number 1, number 2, number 3, number 4, so let's put that at the top, that's 1, 2, 3, 4. Those are N values. So where would 30 go? Well, they're saying it's the 30th term, so that's an N value. They're not saying this number's a 30. and so so you're gonna go plug in 30 into the place of n okay and so that's gonna give us t30 so you're gonna put a negative then you're gonna put 30 don't put the negative in this in the square it's not part of it and then negative 30 and then plus 3 and that should give you negative 927 this one says which term is equal to that so now you can use your formula again but now all of a sudden, they're saying it has a value of negative 7479, so that's the value, so you put that here, negative 7479, and then the n, we don't know, we don't know what position it is, so now we just have some type of quadratic formula, so I'm going to take everything to the right, so you're going to end up with that, and then I'm just going to simplify, So we're just going to plus these two together, so that would be 7482. I'm then going to divide everything by negative. I just really don't like working with a negative over there, so that ends up becoming n squared plus n minus 7482. Anyone feel like factorizing? Nope, I don't. I want to use the quadratic formula, which is then going to be the minus b plus minus square root of b squared minus 4ac over 2a. Okay, now A would be 1 because A is always the number in the front over there. B is going to be 1 because that's the number in the front over there. And then C is going to be negative 7482. So if you had to go and work that all out and plug that all into the formula and do all of that stuff. Ah, it's nice when you end up with a whole number, then you know you've probably done it right. So you end up with n equals to 86 or n equals to negative 87. Now remember, n is the position, like position 1, position 2, position 3, position 4. So you cannot have a negative position. And so for this one, we'll just put a little cross, cross, and so n is 86. consider the following pattern below that emerges when odd numbers are added together so like for example one um and then one plus three because those are odd numbers one plus three plus five uh see they're adding all the odd numbers together and that seems to be giving us a certain uh pattern over there can you see what the pattern is that's quite interesting it's one four 9, 16. Why is this an interesting pattern? Because this is position one, this is position two, this is position three, and this is position four. If you square that number, you get this one. If you square this number, you get that one. If you square that number, you get that one. And if you square that number, you get that one. Now remember that these position numbers over here, those are called n. these values, those are normally called Tn. So what we can say immediately from these is we could make a formula. We could say that to work out the value of Tn, you just take whatever n is, and what do you do with it? You square it. So, and if you really wanted to, if you didn't see that, you could also have done the following. You could have said 1, 4, 9, 16. you could have said that that's 3, that's 5, that's 7, that's 2, that's 2. You would have realized that this is quadratic, and you could have done the normal technique that we normally do, which is 2a equals to that, 3a plus b is equal to that, and then a plus b plus c is equal to that. And you would have gone and solved your a, b, and c, and you would have found out that this is your only answer. So a would have been 1, b would have been... 0 and C would have been 0. Okay, so that can also work. Now, I haven't even read the question, but I got excited. Hence, calculate the value of this pattern. So we know that these numbers have the formula, so these numbers have the formula tn is equal to n squared. But now they want to know what is 1 plus 3 plus 5 plus 7 plus dot dot dot 1001, and we need to work out the answer. So let me just quickly write that so it's in a better order. Just hold on. So let's write it there. No, 1001 is here. Okay, so you get the idea but we don't know what This is so we don't know how many rows down we are This is row 1 row 2 row 3 row 4, but we don't know where we're gonna be over here. What row? Are we gonna be when you reach a thousand and one? However, if you look at these numbers They are going up in twos. So they are going 1 3 5 7 So we need to just find out after how many terms, this is term 1, term 2, term 3, term 4, after how many terms would we be at 1001? Because then we know what row we are in, and then we know what number we can plug in over there. So luckily this is arithmetic, well linear, sorry. We typically call this linear in grade 11. So we know that now there's different ways of finding a linear formula in grade 12 you'll learn a certain way but Typically, I think in grade 11 a lot of schools use this method. Maybe your teacher uses a different letter That doesn't matter. We need to go find a and b. So to go find a and b What we can do is the following. So many, okay, so A is term zero. So it could be whatever this number is. But some learners don't like to do it that way. So that would be negative one, by the way. But if you don't like to do it that way, then what you do is the following. B is always this difference, okay? And then N. Then you can just choose any one of these. I'm going to choose the seven. So you're going to put the seven over there. the 7 is at position number 4, so you're going to put a 4 over there, and then if you go solve for a, you would eventually get minus 1. So this linear formula is going to be tn equals to minus 1 plus 2n. So now we can say, okay, what position is 1001? then if you had to go work it out, you'd say 2n equals to 1002, because I took that minus 1 over. Then if you divide everything by 2, you end up with n equals to 501. So n is 501. So that means you would have to go to row 1, row 2, row 3, row 4, skip a whole bunch of rows, and then this would be row 501. So, if you remember what we said earlier, to find this value, to find this value, you just say 1 squared. To find this value, you just say 2 squared, because it's row 2. This is row 3, this is row 4. So to find this value, you're just going to use row 501. So you're going to say 501 squared, and if you work that out, you end up with 251,001 as your final answer. All right, so with this question, I am sorry, but I'm actually going to leave this question out. It doesn't make any sense. The person who set up this question, yeah, I don't think they... I thought this one perfectly through because it doesn't make any sense. I tried to think of a way I could explain it, but it just does not make sense at the end of the day. So yeah, you can try showing it to your teacher. Maybe I'm missing something. yeah, I don't know. This question just makes no mathematical sense. It cannot really be solved. I'm going to leave it out, okay? You can leave it out too. Trust me, it's not a good question. It doesn't make any sense at all. So here are all the questions that we will do in this question, but I will split them up into individual questions. So here's the first one. It says that the graph of f, which is 3x, which is a 3 to the power of x, sorry, which is an exponential graph. is drawn below, x is an element of r, that just means that x can be anything, but we can see that, okay, x can be any number, point a is the y-intercept, the first question, write down the coordinates of a, or to find a y-intercept, you make x equal to zero, so we're going to take this equation, make x equal to zero, and that's going to give you a one, so when you write out the coordinates, you would say zero and one. This question says that a new graph, which is called G, is formed when this graph that we can see over here is reflected in the y-axis. Write down the new equation. Okay, so here's the y-axis. If you take anything in life, I mean any point, you know, like, Kevin, this is maths, bro. We're not talking about life here. Okay, so if you take this point, for example, let's say the coordinates of this point are 2 and 3. If you reflected across the y-axis, what would this coordinate be? Want to try to figure that out? Well, it would be 3 as the y-value because that doesn't change. You see how we still have the same height, okay? We're only moving horizontally. We're not moving vertically. So the y, I mean the x-value will just become negative 2. You see? Here it was negative 2. Here it was positive 2. Okay, so what changes? Well, it is the x value that becomes the opposite of what it was. So if it was negative, it becomes positive. If it was negative, it was positive, it becomes negative, it was negative, it becomes positive. So what we then do, we know that the equation of f originally was this. Okay, now, G is gonna become the following. What you're gonna do is you are gonna say all of the Xs on this graph need to become the negative of what they were. So you just go like this and you put that as a negative of what it was. Now, you are allowed to leave the answer like that or if you wanted to, if you think about, for example, X to the negative two, do you know what that becomes? Well, if you wanted to write it with positive exponents, then you would write it as. 1 over x to the power of 2. So if we then look at this one, it's going to be 1 over 3 to the power of x. Okay, so you could write the answer like that or like that. This question says, sketch the graph of G in your answer book and clearly indicate all intercepts with the axes. Or there aren't going to be any intercepts with the x-axis. Because all that we're going to do is we're going to take this graph and we're literally just going to flip it over the y-axis. So this point is going to go here. This point is going to go there. this point won't go anywhere because it already is on the y-axis so you can't reflect over then this point here would go there and i mean you can just however many points you want maybe this one over here would just go there and so now you can just put a line going through all of that and that is what the equation of g would look like. So you could just say that that is g of x. This, on your answer book, this would still be 0 and 1. You can't, it says sketch the graph of g, clearly indicate all intercepts. That is the only intercept that you would have. Now, some of you might be wondering, do you need to show any coordinates? And the answer is no. I went and looked at the memo. They literally just wanted you to label the y-intercept. They wanted you to let it come down like that over there. It mustn't cross this line. And then they wanted it to have the right shape. And that's it. And so here we are at this question. It says that, okay, so let's just see what we have so far. So we already knew that this was 0 and 1. I don't think we're going to need graph G. I think that was just a once-off. So it says that the graph of K, geez, all these different graphs, F, G, and now K, is formed as a result of the transformation of F. Describe the transformation to get to that. Okay, so this is quite a cool one. So we know that F, let's go right out F so long, F of X was equal to 3 to the power of X. Now we have k of x, which is equal to this. Right. So each part, okay, so we're going to go look at each part. So what I want us to do here first is get this to a part that the x is not negative. So the way that you do that is you do the following. You take out negative as a common factor, and then you'd be left with x take away 1, and then you have plus 2. Now we need to go and examine what is this due to a graph, what is this part due to a graph and what is this due to a graph. Okay, so let us begin here, putting that little negative in front of this X part over here. I explained this to you in the previous question. When you reflect something in the y-axis, what happens to the x values? Will they become the negative of what they are? So if you have a graph that is 3x and then it becomes 3 to the negative x, then you see how the x values have become the negative of what they were. That is a reflection in the y-axis. So because there is this negative over here, we can say that there was a reflection in the y-axis. Okay, now mathematically, if you have something like, let's say it's a hyperbola, for example, 3 over x minus 4, what does that do to a graph? Does it move the graph left and right, or is it up and down? Well, because it's with the x's, it's the horizontal part, it's left and right. But now you've got to remember that when we are talking about horizontal, minus means right and plus means left. So if we look over here, then that is going to be one place to the right. So we can say that it is translated one unit right. And then... we need to just analyze this part. Now that part's easy. That's just plussing 2 at the end, so that just moves the graph 2 units up. When we move up and down, it's easy. Plus means up, minus means down. So 2 units up. Here are all the questions that we're going to do in this question, but I'm going to split it up so we have more space. So here's the first question. It says that the graph of G and the graph of F. Okay, so G is a parabola, but they've also labeled it for us over there. And then F is a hyperbola, and there they've labeled it. It's got this part and it's got this part. Now remember that hyperbolas have asymptotes, and so these dotted lines are the asymptotes. for the hyperbola. Point Q is the turning point. Point D are the X intercepts of G. The horizontal and vertical asymptotes intersect the graph of G at point Q and at point D. So it's pretty straightforward what they're showing us there. Now, the first question for 4Mark says, show that G can be represented like this. Okay, so guys, we have G of X currently like this. How do you transform a normal trinomial equation to something that is rather in this type of format over here? The technique is called completing the square. That is how you go from a trinomial like this to a turn, we call this the turning point form of a parabola. Okay, and completing the square can be quite challenging, it's a bit weird. But step one is you need to get this x squared as with a one in front of it. Okay, so we're not going to divide everything by minus two because this is not equal to zero on the other side. you have a g of x on the other side. So if you divided everything by minus 2, you would literally have to divide this by minus 2, and things are going to get weird. So instead, we take out a common factor, but we don't get rid of it. You see how it's still chilling there later on anyways? Okay, so then you're going to be left with x squared minus 2x minus 8. Okay? That's the first step, get this x squared to be a positive 1. Now we can do the completing of the square technique where you're going to do the following. You're going to write out the x squared minus 2x. Okay, the minus eight will still be at the end. Now, you're gonna say plus every single time. You're then gonna open up a bracket, you're gonna square that bracket. The number that you're gonna put inside here is always gonna be this number over two. So that number divided by two, which is negative one. Now because mathematically we've just added this, you're not allowed to do that. So to reverse our operation, we are gonna now subtract it again. And that's the most difficult part. now we're going to go to the next part which is minus 2 and then these three parts over here they are going to end up becoming one bracket so you're going to take everything that is to the power of 2 so you're going to take the X and you're going to take the minus 1 and you're going to put them to the power of 2 okay then this part here you're just going to put all together on your calculator include this negative it's also part of it and so that becomes minus 9 Now what we do, last step, is take this minus 2 and multiply it into this part and this part. So what I mean by that is just multiply it into the front over there. So we're going to end up with g of x is equal to negative 2 and then bracket x minus 1 squared. And then you're going to jump it over to here where it becomes positive 18. And there we go. We have generated exactly that. So here's the next question. Hence, okay, so I've got the previous answer that we had, where we had to change this into this. It says, hence, determine the coordinates of point Q. Now here's the interesting part. It says, hence, or otherwise, okay. Oh no, that wasn't the interesting part. I wanna quickly talk about how to find the turning point of a parabola. There are two ways, okay. The first way is when your equation is written normally. So minus two X squared plus four X plus 16. The second way is when it's written in this completed square form. Alright, when you have it in this form over here, to find the turning point, you are going to use minus b over 2a. That finds the x value, and then once you have that x value, you plug it back in to get the y value. Okay? That's that way. If it's written in turning point form, then the turning point, this is literally called turning point form. Do you know why? Because by looking at this number and this number, you instantly get your turning point. I'll explain. So if you have x minus 1, it means that the graph has moved one place to the right. Okay, so it's all graphs start at the origin. Okay, now this graph moved one place to the right, so one place to the right, and then 18 units up. and that is how it ended up over there, where we see it in position Q. Kevin, why is it going down and not up? Why isn't it smiling? Well, that is because this number in the front is a negative, so it's a sad parabola. Okay, so this is called turning point form because it instantly gives you the turning point without having to go use negative B over 2A. So in the question, they are saying, hence, what that means is use this formula, but if you don't want to, that's the... Or otherwise, then you could also use this. You would get the exact same answer. So, do whatever method you feel comfortable with, but... Look, the way I would do it is if I have access, I just wanna see something quickly. If I have access to this equation, I use that because, I mean, it literally gives me the answer immediately. But if I have that, then I don't mind having to use x equals to negative b over two a. Okay, so, Be careful though, what does x minus 1 mean? It means that the graph moved one unit to the right. So the x value is going to be 1, and then the y value, so don't put negative 1 over here, and then the y value would be 18. So this coordinate of q is 1 and 18. Now keep in mind, that is also a point on this dotted line, so that's going to help us a little bit later, because now we know the coordinates of this dotted line. This question says, calculate the coordinates of D and E. Okay, so D and E are the X-intercepts of this equation. Some of you might be like, yeah, it's also the dotted line of the hyperbola. I agree, it is correct, but we can't use that because we only have P and Q over there. Okay, so we can use this equation. If you wanna use this one, you can also use that one. It doesn't matter. To find X-intercepts, we make Y equal to zero. I'm then going to go straight to the quadratic formula, x equals to negative b plus minus b squared minus 4ac over 2. And I'm not going to show you how I sub it in, but I'll tell you that a is negative 2 because it's the number in the front. b. is positive 4, and then C is 16. So if you had to go plug all of that in, you end up with x equals to 4, and then x equals to negative 2. So what that means then is that E is 4, and D is negative 2. So when you give the coordinates, you would say that point D is... Negative 2 for x and 0 for y, you must give the y as well. And then e is 4 and 0. This question says, determine the equations of the asymptotes of F and then state the values of P and Q. Okay, so you see here we know the X value on this dotted line, or at least over here we know the X value, right? It's minus 2. But the X value at any point on that line is negative 2. It is critical that you understand that. Because think about it, this point over here is not further left or further right than... this point down here. So their x values must be the same. So the x value on this dotted line is minus 2. So that line there is called x equals 2, negative 2. So that'll be our first answer. Now, if we look at this horizontal line, we know that... the y values stay the same. Think about it. If this y value is 5, for example, then this y value would also be 5 because it has not gone further up or further down. Okay, so all the y values stay constant on that dotted line. So what is the y value over here? It is 18. So that means the y value here is 18, and here, and here, and here. So we will say y is... 18. Now these numbers help us to work out p and q. So if we write out f of x, equals to 12 over x plus p plus q. We need to understand that this part of a graph moves it up and down. So a normal hyperbola, the most basic hyperbola you could ever get, would have asymptotes as the y-axis, and it would have asymptotes as the x-axis, and the graph would look literally like that. But, as soon as you start to move the graph up or down, the asymptote also goes up in that scenario, or it could go down in that scenario. So if we look at... this asymptote over here, we can see that it has moved, because this is the original asymptote on the x-axis, but now it's over here. So it's moved up a distance of 18, so that means this value is 18. So we could say q is 18. So remember, I told you that the original asymptotes are the y-axis and the x-axis. So if you want to look how far a graph has moved right or left, what you need to do is you need to look at this asymptote over here. So this asymptote was originally on the y-axis, but now it has gone two places to the left, and that is why you can see it over here. So this graph has moved two places left. Be careful though, does that mean that... At the bottom here, we're going to say x minus 2? No. Because remember, minus means right, plus means left when you are talking about horizontal movement. When you are talking about vertical movement, then up means plus, down means minus. And so then, if we think about what their original equation was, they said x plus p. Okay, so they said that this was a plus already. Sometimes they'll put this as a minus. You've got to be careful. So if you just compare them, we can easily say then that P is 2, okay? P is going to be 2. So we can say Q is 18, P is 2. This question says, determine the values of X for which the graph of G will decrease. So the graph of G is the parabola. So that is this one over here. So, the way to do this is you need to go from left to right. So if you're moving from left to right, where is the graph going down? So if you look here, this area, this graph is going uphill. You can think about someone going for a walk. Here they're busy walking uphill. Okay, but then as soon as they go past this point, suddenly they're going to start going downhill. Okay, and so where is the graph going downhill or decreasing? It is from this point onwards. So when you give your answer, use the x values. So you could say that it is when x becomes bigger than one, because this x value is one. And as soon as the x becomes larger than one, then the graph starts to go. Down. You could also use interval notation and you could say from 1 up to infinity. This question says write down the range of G. Now remember G is the parabola. and range is y values. So you're always going to look at the lowest y value, and then the highest y value. So if you look down here, what is the lowest y value? Well, it actually just keeps going on. So the lowest y value would be negative infinity, so you could say y is an element. I'll also show set builder notation shortly. Now we need to look at the highest value that that yellow graph can get to, and that highest value is over here. So what is the y value there? 18. So we can go up to 18, And is the graph touching 18? Yes it is, so we will include that with square brackets. If you prefer set, I mean, yeah, set builder, you could say that, you could simply say that y is any number that is smaller than 18, but also equal to 18. On the memo, quite interestingly, they said that this part is a round bracket. But guys, just trust me. Memos do make mistakes. That is definitely a mistake. I'm guaranteed. Look, if I'm not 100% sure on something that I'm saying is wrong compared to the memo, I wouldn't put the video up for you to see. Because that is not right. Because you guys are taking what I'm saying as fact. So I will only put facts available to you guys. So if I'm not 100% sure, I won't put it up, okay? So this I am 100% sure about. That number 18 is included because the graph is literally touching that position. The memo is incorrect there, but that does happen. Usually they release a little document. It's called an errata document after the paper where they list all of the memo mistakes, but I couldn't find one for this paper, but I can guarantee you it is a mistake. This question says, write down the domain of f. So f is the hyperbola this time. Let me explain any hyperbola. So I'm going to draw a random hyperbola now. So let's say it had a dotted line over here, which is x equals to 3. And let's say it had a dotted line here, which was y equals to negative 2. And the graph, let's say that it went like this. and like this. So domain is X. So I want you to start on the left here, okay? And let's start on the graph itself, okay? Now, I want you to try get from this side to this side without lifting up your pen. So I'll use my highlighter. See, now, but you've got to follow the graph. Now, look what happens here. What is going on? If you know what these graphs do, it's just going to keep going up and I'm never going to be able to get to this position over here. So unfortunately I'm going to have to jump over this ugly dotted line because it's getting in the way. And then where does the graph carry on? On the other side. Or here it is here. And then it just carries on again. And then everything seems to be okay. So it looks like x could be, and this graph can go all the way to the left, so it looks like x could be absolutely anything. But the only problem was that we weren't able to jump, we weren't able to, this line got in the way. So x was not allowed to be this line because it had to jump over. So we can say that x can be any number, but it simply must not be equal to 3. Okay, so that's what that answer would be for that one. Now let's go to our graph. With our graph, we can see that the graph could go all the way here, here, here. Let me get a highlighter. So it could go here, here, here, here, here, here, here, here, here. Oh no, it gets to the dotted line, has to jump over the dotted line and then carry on over here. So x could be anything, but it can't be negative 2. Like that. The sketch below represents the graph of f of x, okay, where d is the turning point. Okay, the graph intersects the x-axis at p and q. First question, for five marks, it's a lot of marks for such a basic amount of information. It says determine the values of b and c. Okay, so one of the ways we could handle this is we know that to find the x value of a turning point when it's written like this, you use the formula x equals to minus b over 2a. Okay, so we can use, but we know, so we can say we know what the x value of the turning point is at 1. So we could say 1 is equal to minus b, which we don't know, you see that, over 2a. Now a is negative 2. Okay, and so that means 1 is equal to negative b over negative 4. I'm going to cancel out those negatives until we end up with that. To get b alone, you are going to multiply this 4 over to the other side, and so you end up with 4 is equal to b. And so we have the value of b now, plus 4 times x plus c. What we can now go do is let's go plug this point into the equation. So the 8 is a y value and the 1 is an x value. And then we can solve for c. So 8 is equal to negative 2 plus 4 plus c. And if you solve for c, you should eventually get 6. And so there we have it. So b is equal to 4 and then c is 6. The other way you could do this question is to remember that there is a turning point formula that we have which goes like this. Okay, now remember that the p and the q are the turning points. Now we have those. Okay, so we could go full in p. We know that the graph has moved one unit right. So you say x minus one, not x plus one. And then this has moved eight units up. So we say plus eight. A we already have because we know that a comes from ax squared plus bx plus c. So a is the number in front of x squared. But look, we have it. So we can fill in a as negative two. And so there we have an equation. We can then go multiply this equation out. So remember the best way to do that, don't put this minus 2 inside. You cannot because this first needs to be double bracketed. Aha, there we go. Now multiply these two together so long. So that would become x squared, take away x, take away x, which is going to be take away 2x, and then plus 1. Now multiply the minus 2 in, so negative 2x squared plus 4x, take away 2, plus 8. And so then if you go solve, y is equal to negative 2x plus 4x plus 6. So now we can look at this, and we can look at this, and we can say that b is 4. And c is 6. this last part says that the graph of G represents F. Okay, wait. So, okay, okay. So we have F at the moment, okay? So now they're talking about some random graph G, which is a new one. So it says that the graph of G represents F. Okay, so the graph G is F when F is translated two units left and three units up. Determine the equation of G. Okay, so I've showed you two ways of writing the equation of F. Now, there's this way. But then there was also the turning point method, sorry this is graph F, which was negative 2 and then we had x minus 1 and then we said plus 8. This is a better option to use now because what we're going to do, they said that the graph of G is the same as F when F is translated two units to the left. Okay, so how do you move something two units to the left? Well, you're going to change this over here. Now when you go left, that means plus. So you are going to do this. So you're going to say that g of x is going to be negative 2, x minus 1. Now when you go left, you go plus, okay? And then it says 3 units up, so we're going to say plus 8 and then plus 3. And so you're going to end up with the following. Now this part here becomes x plus 1 plus 11. And so that would be the equation of g. Now, for those of you that didn't want to use the turning points method, then what you would do is you would use this equation and it also works out perfectly. You could just do the following. Whenever you see x. open up a bracket, okay, so wherever you see x, open up a bracket, and then change it from x, and rather change it to x plus 2, because we're going to go two units left, and then do the same over here, x plus 2, then to change the graph three units up, you're going to say plus 6, and then you're going to just add 3 over there, then you're just going to go simplify now, okay, and so x plus 2, I want you to double bracket that, remember to double bracket that, and then this, I'm just going to leave it like this for now. and then we might as well just put these two together as plus 9. And so this becomes g of x is equal to negative 2. Now we're going to multiply these two brackets together, so it would eventually become x squared plus 4x plus 4, and then plus 4x plus 8 plus 9. Now we're just going to neaten and clean this all up, so it's going to become negative 2x squared. I'm multiplying the negative 2 into the bracket, minus 8x minus 8, plus 4x plus 17. Okay, and now we're just going to neaten up a little bit more, so it's going to be minus 2x squared, minus 8 plus 4 is minus 4. And then minus 8 plus 17 is plus 9. And that is also a valid answer.