A LEVEL MATHS: Graph Transformations LESSON 6

hey guys just to let you know that you can catch these lessons live every Thursday from 6PM until 7pm uh UK or BST time and I'll just allow you to ask any questions that you want and get some answers to them without further Ado let's get on with the lesson okay welcome back to another one of my free a-level maths tuition sessions this is the sixth one instead we're going to be covering sketching and Transformations so just so you're aware Transformations does come back later on in the course so this is mainly a for as okay this is like the as Transformations and there are a few more that you need to know for A2 but this is going to be a bit of a recap as well for what you did in GCSE really so whenever you have Transformations what they'll do is they'll give you like an equation so it might look something like this right or they'll say apologies for the terrible drawing they'll give you that this is the graph of y equals f of x right and it'd be like whatever kind of shape and then they'll ask you to transform it so it'll either look like this or it'll look like this either one is fine and then what they'll do is they'll say F of and then some kind of transformation okay so what we need to remember is inside the bracket so for example this could be something like f of 2x or f of x plus three for example so if it's inside that X bracket then you do the opposite you do the opposite okay and you only touch or affect the x coordinates okay so F of 2x you do the opposite so that says 2 times x so we're going to do divide by 2 and we only affect the X values okay so this would be a compression in the x-axis and then with X plus 3 it's saying plus 3 so we do minus three okay now if it's outside the bracket you so for example this could be like 2 f of x or it could be f of x plus two something like that you do what it says and it affects the Y coordinates okay so just let me check on the stream quickly yeah and it affects the y-coordinates only okay so what this would do for two f of x it would times all the Y coordinates by two f of x plus two you'd add 2 to all the white coordinates that should be fairly simple so far okay and this is a bit of a recap of what you did in GCSE but again you have to be able to do it to a level kind of graphs that you know so sine and cos and tan uh you have cubic graphs Etc and then just random graphs that they can give you but there is also some more stuff that you do in A2 which will come across um in another lesson so if we start just doing some of these questions as you can see you get quite a few marks for these and this is kind of a typical exam question so we didn't go over what sketching is but that's because I wanted a good example and this would be a good one so they tell me that f of x equals X plus three X plus two x minus 1. okay and we need to actually sketch what that curve is showing the points of intersection with the coordinate axes and that's three marks so I'm going to show you what you get the marks for now so if a part A we're just sketching let's draw this is like this your marks come from the y-intercept and all of the x-intercepts okay so this is actually going to be four coordinates for us but you only get three marks and you drop a mark for every single one that you don't uh get so to work out our y-intercept that is when x equals zero so if we do that now so y int we're just going to sub in x equals zero so then F of 0 is equal to X plus 3 now becomes 0 plus 3 so it becomes three X plus 2 now just becomes two and minus one so that is just going to be minus six so our coordinate there is going to be x is 0 Y is minus six we can even put that on the graph right now okay and again in terms of like where these values are as long as you keep it roughly proportional it's fine you don't have to be super accurate with it Now with an x-intercept that's the opposite that's when y equals zero so what we're going to do here is we're going to say 0 equals X plus 3 X plus 2 and x minus 1. okay now we know how to solve this already it's the same as solving a quadratic we set each of these brackets equal to zero so either we have well in this case we have all of them X plus 3 equals zero so minus three from both sides we get x equals minus three we have X plus two equals zero so then x equals minus two and the lastly we have x minus one equals zero so x equals one so if we plot these we would have one let's say let's spread out a bit we have minus three minus two and one now in terms of which way this comes so we have to this is a cubic graph right so in terms of which way this uh kind of comes from there are two ways you can do this what we can do is sub in a value of x that's less than minus three okay now if it's negative it means the graph starts from here if it's positive it starts from up here okay but there is also kind of a rule that you can remember as well so let's Sub in F of minus 4 and see what we get so if we just sub this in on our calculator to make it a bit quicker for us we would have minus four plus three then we have minus four plus two then we have minus four plus sorry minus one which gives me minus tens it's negative okay which means I start from the bottom and go up so we're going to go up and what it does is it kind of threads the needle like this okay let's move the one over here now because I'm and that's what our graph will look like okay so again it needs to be clear where the coordinates are if you've written them down then you're actually kind of you're fine it doesn't need to be too clear okay but in general positive cubics are start from the negative side and negative cubics shockingly dark positive okay so what I mean by that is if we look if we expanded this whole bracket the X cubed is going to be positive because it's x times x times x if there was a minus in front of one of the X's then it would start from the top which makes sense because it'd flip the um sign and that's it for part A so that's three marks so again you lose a mark to every um coordinate you don't have on there so we have three x x intercepts and one y intercept now if we scroll down a little bit we have part B part I and what it says is showing the coordinates of the points of intersection with the coordinate axis so sometimes they don't tell you that but essentially when you do Transformations all of these points you need all of the X and Y intercepts again okay sketch on separate diagrams the curve so first one is y equals f of x minus three we could do part two next to it here is exactly what I'm going to do actually because it makes a bit easier for me and Y equals F of minus X so I like to do is ask myself what's actually happening so for this First Transformation remember it's inside the bracket which means we do the opposite and it affects X so we're doing um x coordinates plus three so all of these x coordinates are now plusing three so instead of minus three we have zero instead of minus two we have one instead of one we have four we literally just dragging the whole thing across okay and again for f of minus X so again if it's minus X it's basically minus 1 times x so the opposite of timesing is dividing so you're dividing by minus one which is the same as timesing by minus one so all you're going to do basically is flip the sign of the x coordinates and just so you're aware what this basically is is a reflection in the Y axis okay so not too shabby I think so let's get started so if we start down here so we have our graph and again I made the right hand side slightly longer because if you notice all of my coordinates are now going to be shifted over and okay so this coordinate one is now going to be four this coordinate minus two is now going to be one which I'm going to put over here I've made this quite long [Music] um and that one's going to be zero okay now what you can do is actually make put the minus six on there too as well I should say so instead of being 0 minus six it's actually going to be three minus six so somewhere over here you don't need to do this though okay and basically it's still a positive cubics it's going to come up from the bottom and hopefully that makes sense right when I add three I'm just moving it all across which means it should still start from the bottom have we labeled all of the coordinates well if you look the y-intercept is just 0 0 in this case so we do have the y-intercept so we have one two three x intercepts and we have the y-intercept you don't need this point but I like to put it in there anyway okay but the action marks you get for one two three and then technically four even though it's only a two mark question okay because you don't actually have to work out the coordinate axes okay so let's get started with the next question we have y equals F of minus X so we're just flipping these signs so this would be plus three plus two and that'd be one and this is actually going to be a bit more interesting oh God okay so if we plot these now we have one at minus one and the other ones we have all right two and three now our y-intercept is zero minus six so if I times zero by minus one I still get zero so this actually stays perfectly the same so all my points are there that's good okay in general with these kinds of questions when it's a two Mark transformation you get one mark for getting the coordinates correct and you get one mark for the shape being correct okay now in this question however if we reflect this in the y-axis can you see it's going to start from the top so we actually need to start from here and if you look it's not perfect but this graph and this graph are just reflections of each other so where this one ends with an upwards flick we've started with an upwards flick and then other than that it looks the same okay so that's where the shape mark would come into play if you went from the to be fair if you have a look if I started from the bottom right so I started doing this how can I now go through minor six without crossing the x-axis it's impossible so I know it goes from the top anyway okay hopefully that will make sense and let's go on to the next one so again something else that they can ask you to do when it comes to sketching is actually solving certain equations as well so first off we need to sketch on the same diagram x squared plus 5x and Y equals minus 1 over X so I would actually say sketching the minus one over X is easier first for those of you that may not know the Y equals one over X graph looks like this okay so it goes like that where these the X and Y axis are actually asymptotes which means the graph will never cross them so make sure you don't make it touch um and so if it's a minus one over X graph it means that it's going to actually be flipped okay so it's actually going to be over here and here okay so let's start with part a I'm going to make this quite big just so I can uh I'm sure so again this was one over X so minus one of X means I'm flipping the x coordinate I'm reflecting it in the y-axis so we're going to have that top right is now in the top left and that bottom left is now in the bottom right now you might be wondering well I'm not putting the asymptotes on here it's because we have x squared plus 5x as well which is not going to look sorry it's going to cross the x-axis so we can't have asymptotes so now with y equals x squared plus 5x if you remember rightly we need to have x-intercept and y-intercepts okay so if I want to work out the y-intercept that'd be the easiest one to do I sub in x equals zero so I have y equals 0 squared 0 plus 5 times 0 is 0. so in other words it crosses at the origin x-intercept I have y equals zero and to solve that we need to factorize so the biggest thing I have to take out is X so either we have x equals 0 which we do because that's this point over here or we have X plus 5 equals zero so x equals minus five so we can make minus five let's say over here uh actually let's move it over a little bit because what we're going to have is a big U shape and it's a u shape because it's a positive quadratic so yeah that's not bad not bad and that'll be four marks okay so you get a mark for the shape of both of these right so a mark for here a mark for here a mark for having the x-intercept and then what Mark for having the other x-intercept which is also the y-intercept for marks too bad now here's where things get a bit interesting so this was in your GCSE spec but I it has never really come up in the exams so you may not remember it so I have an equation like this it says State giving a reason the number of real solutions to the equation now This Is A Part B and it's also only two marks and it also says giving a reason which means we should be able to get to the answer pretty quickly now if I have a look x squared plus 5X was Y in this equation now I can get one over X I have one over X here I can get minus one of x if I move this to the other side so if I have x squared I can actually get both of my equations by doing this moving the one over x to the other side now if you look the left hand side is that quadratic and the right hand side is the other bit this isn't a coincidence this is why they wanted it on the same diagram because you can see you can actually read off the answers to this equation by just reading off the coordinates where they cross okay and every time they cross there is a solution to this equation how many times do they cross it crosses once so there's only one solution so one real solution as the curves cross once or you can say intersect which is sometimes like a better word to use so one solution is one Mark a good reason is the second mark okay not bad okay let's keep pushing forwards okay the sketch shows the graph of y equals f of x the curve has a minimum at minus three minus two and a maximum of four seven showing the coordinates of the points of the intersection with the coordinate axes sketch on separate diagrams the curves f of x plus 2 and minus f of x okie doke okay so we are going to keep referring back I don't really need to put the number three so f of x plus two let's think about what that does what it does is it adds to to the Y coordinates in other words it's going to shift all of this stuff upwards that's it drag and drop so let's do this again your axes don't have to be huge but I prefer to make them decently large just to make sure that I get the um points and the first thing I do is plot the point so instead of minus three minus 2 add 2 to the y coordinate makes it minus three zero so it's actually that is the um x-intercept and lastly we have this so instead of four seven it's going to be four nine okay and it just looks like a big like snakey boy they're like that yeah that's not gonna work okay it would be a lot easier if I um could look at the question at the same time right that is absolutely perfect okay and those are the two marks okay so if you notice in the actual question just in case you're wondering why I'm not putting the y-intercept we don't have it um and we can't really work it out you can work it out but it says the intersection with the coordinate axis right Axis is one axes would be both so it's only the ones that actually uh cross okay and for part two uh we can probably just leave it like this we have y equals F whoops minus f of x so what it does is it's timesing the Y values by minus one so it's just flipping the sign of the Y values so flip sign of Y coordinates okay and that's all you need to do so instead of minus three minus two we have minus three plus two so again let's draw our little axes out okay minus three minus two now becomes minus three plus two and one thing to note is that when you have a minus f of x it's the same as a reflection in the x-axis so that may sound a bit fancy but all you're doing is you're taking it and flipping it over okay and then you also have four minus seven so four minus seven so again now instead of it going down and then up it's going to go up and then down because again we're flipping this rotating at 180 degrees right so it's going to go up uh like that yeah oh and that would be it so those are the new coordinates that we have and in terms of the Y intercept if you notice it's actually stayed the same um because all we're doing is reflecting it so uh it might actually cost a bit lower yeah okay eight more questions so pretty beefy so broke yourselves and this is four we have equals x squared okay and it says sketch it in the form this now this form always means completing the square okay now there's a trick with completing the square you probably already know and that it gives you the coordinates of the minimum point of any graph okay so that's exactly what they're asking us to do here now I'll explain how the trick works right but you can just memorize it but I'll explain why it works as well now with the completing the square if you remember you have x squared plus BX plus C that now becomes X plus b over 2 squared minus B over 2 squared plus C on the N so let's now sub our thing in we have X 4 over 2 is going to be 2 so X plus 2 squared minus 2 squared to five so we have X plus 2 squared now 2 squared is 4 minus 4 and then plus 5 so we have plus one so basically the trick on the minimum point is so if we write min is all you do is you flip the sign of the inside of this and that gives you the x coordinate and then you keep the y coordinate the same that's it now the reason why this works is if we look at this as a function right what is the smallest thing I can make this overall expression well that might be a bit hard to work out what's the smallest thing I can make this bracket since it's squared if it's negative inside it will then become positive once it's squared right which means the smallest thing I can make that bracket is zero now how can I make it equal zero well if the inside equals zero and I'm plusing two x has to equal minus two so that's why that works now if I do make that bracket equals zero then I have a plus one so the smallest value f of x can give me is plus one which is where the one comes from so that's where that comes from okay I hope that makes sense okay perfect so you may know that this is not going to have an x-intercept now because the lowest value can be on the y-axis is one so now if we uh for Part B it says we need to sketch it yeah this is three marks by the way so all it is is completing the square and that's it but this is our answer and that's another way we managed to get to our answer we need to actually sketch the graph now X y okay so we have the minimum point there are no x-intercepts right and if you try to work it out with this one you'd get a math error because you have to put the one to the other side and square root it you can't square root the negative number but the y-intercept though all we need to do is something x equals zero and if we look at that expression I'm hoping you can see we have x squared is 0 squared to zero four times zero is zero plus five is just five right other way to do it is 0 plus 2 is 2 squared is four plus one is five either way so it crosses at five let's make it relatively high and then we've got minus two one as the minimum point so it looks something along the lines of that okay now of course we I didn't really need the x axis to be that long I could have kept it more in the negative so you don't have to make it even if that makes sense you can like skew it to one side so I could have cut off maybe this much of the x-axis but it's completely up to you um but that's three marks so you get a mark for the shape the coordinate and the y-intercept because there is no x-intercept so really um not too bad really then all it says is find the minimum points of these curves now this is interesting because if we just need the minimum point I don't need to sketch it okay and for the first one it says y equals two f of x so what does this do it times is so times y chord by two so if I'm just finding the minimum point I know that the x value is still minus two because we haven't touched the x value and what we're doing to the Y value we're doubling it so minus 2 2. cool that that's literally it not too bad not too bad at all and then for part two it says do the same thing but for y equals F of 2x now remember if it's inside we do the opposite we fact X so we divide X chord by two so now the we have the second value it's going to be one the first value is going to be half of -2 which is minus one that's it not bad for four marks I would say for part C so hopefully this is making some sense at the moment okay finally My Stream okay and tips okay so 5A minus 5x okay and we need to sketch this curve showing the points of intersection with the coordinate axis so I can already tell you we need to factorize this but remember what we need we need the X and Y intercepts the y-intercepts Gen generally are far easier to work out you just need something x equals zero so what we have is f of zero and if you notice there's X in every single one of these terms so F of zero zero so our y-intercept is zero zero done now for the x-intercept where we need f of x to equals zero or y equals zero we have this so if I do it over here we have X cubed plus 4X minus five x equals zero so for x squared sorry so we need to do some factorization so pull out One X and we get x squared plus 4X minus 5 equals zero so that X is what gave us the x equals 0 Y intercept so that's pretty damn good and now we need to factorize what's inside the bracket so we're looking for something that times is to minus five and adds to plus four so you're looking for five and minus 1. so we have X X plus 5 doesn't really matter the order x minus one equals zero so now my value for the X intercepts are either x equals zero x equals minus five x equals one you just flip the sign of what's inside the um bracket so all of the points we have now are zero zero again so we don't need to put both of them if we don't want to we also have minus five zero and we have one zero so if we're sketching this now we have thanks now sketching cubics and also cortex and things like that it can be a bit tricky okay um but it's not too bad at the moment if you want to do it so what we're going to do here is we're going to put the -5 here the zero is here and the one is over here and again it's a positive cubic so it starts from the bottom just like that and that is all you need to do for two sorry how many marks is that that was should have been three yeah good okay then for Part B Part I show showing the coordinates of the point of intersection with the corner axes sketched on separate diagrams and X plus one and two x okay so F of so again let's think about what we're doing here we're subtracting one from X chords so if we have a look at all our values all we're going to do is move everything to the left so that's not really too tricky to do let's just draw our X and Y values and all we're going to do is subtract one from all of the X values so minus five we're going to move that to minus six now remember this is also a root which has coordinates 0 0 so we need to subtract one there so it's minus one zero minus one and the last one is still at zero so we're doing the exact same thing we're just shifting everything along and drawing a beautiful looking graph like that okay perfect and then for 2x remember it's inside the bracket so we do the opposite and effect X so we're going to be doing is dividing X chords by two so again it's going to the same thing is going to be squished in so once again let's draw our axes so instead of minus five it's now minus 2.5 and I like to actually make the shape a bit thinner as well zero zero obviously stays the same and one is going to now become 0.5 so we can actually make the shape really thin now to so technically the actual height should be the same so it should actually still go like this so it should still reach the same height as the previous one it's just compressed inwards it gets there faster hopefully that makes sense okay since again we actually went through the shape of this graph at the beginning so y equals 1 over X so if we think about what's Happening Here is if we say that y equals 1 over X or rather if we say this is f of x then what they're doing here is asking us to sketch y equals f of x plus two so everything is going to be shifted upwards now you need to be really careful because this equation does actually have intercepts of its own so if you remember with the one over X graph you have something that looks along the lines of this goodness gracious right and the asymptotes are along here now if I were to drag the graph upwards it is going to meet the x-axis at a point over here somewhere so we need to be really careful of that it also means the asymptote is going to move upwards which we have to label so giving the equation of any asymptotes as it says in the question but this asymptote here along the y-axis is going to actually remain identical to what it looks like now so what we could do is is it going to cross the y-axis the answer is no but what we can because if we Sub in X is zero we get something hideous and disgusting is if we want to find the x-intercept right if you remember rightly all we need to do is find when y equals zero so we have zero equals one over X plus two move the minus uh make it minus two equals one of x it we have minus two x equals one so x equals minus half so let's draw it we're going to have something along the lines of so it's going to look similar to this but I would like to draw the asymptotes first so this asymptote here we would call it y equals zero we move it up by two so we're going to have a dotted line going along here at y equals plus two now none of our lines should meet that we also have it says giving the equation so what we do is write that as x equals zero and now we draw our graph it's going to look the same it's just going to and again be careful make sure it doesn't look like it's crossing the x-axis uh the asymptotes because if it looks like it crosses if it touches at all you lose the mark for shape and this coordinate here is minus a half zero and that's it so you should be able to transform pretty much any graph even if you're not familiar with it just by thinking of what this shape is and then what I'm doing here now what some students ask me is what would it look like if same question but it was f of x plus two then you'd have something along the lines of 1 over x minus two and that's what you're plotting okay so for it to be inside the bracket it needs to be in the same part of the function as X right so if you look this is the main function here and then we're adding 2 to it if that plus 2 was here in the same function which is a divide then it would be inside the bracket I believe this is the last question for today again not a super long lesson um but it doesn't look like any of you have any questions for those of you that have turned up uh so we can remember if you are unsure of a topic it's good to show up to these to make sure that you're kind of okay with it but then you can also ask questions because I can't exactly answer questions I mean I can't answer questions afterwards obviously but it's easier when I'm actually at a whiteboard so to speak so once again sketch this curve three marks we should be fairly okay with doing this now if I want the wine to set which again is the easiest with subbing in x equals zero right so we have plus four times minus one is minus four times 2 is -8 so our y-intercept is zero minus eight x intercept which is when f of x equals zero or y equals zero whichever one you want to call it uh all we're looking at is the fact that X plus four x minus one two minus X so be careful they've kind of written this backwards equals zero so we have either X plus four equals zero so x equals minus four we have x equals one now with this one you have to be careful you have 2 minus x equals zero so x equals two okay so be careful with that the shape of this is also going to be a bit tricky because one of the X's is minuses is a minus the overall X cubed is going to be minus X cubed right x times x is x squared times minus X is minus X cubed remember when I say it's a negative cubic I only care about the X cubed bit everything else can be negative positive it doesn't really affect if it starts from the top or bottom but since this is negative X cubed it starts from the top okay because again if you expanded that cool cool cool let's keep going X y I like to put the um these values first so we have minus four over here so one would be something like here and two would be something like here it's not actually it's not bad and zero minus eight so it crosses somewhere like over here somewhere and it starts from the top so it's going to look interesting to say the least so down and then it has to kind of so basically when it goes through -8 that doesn't have to be the bottom of the curve it's not necessarily the minimum right the minimum could be over here so since what I liked is our draw like this and it looks a bit more natural and that would be your full three marks a mark for the Curve and again remember the next two marks come from having all the right points there are four coordinates they give you two points but it's not two marks which it's not half a mark each you drop one for everyone you get wrong yeah sounds a bit crappy but is what it is Part B sketch on separate diagrams cool so if we look at Part B the two equations are f of x plus two and minus f of x minus f of x I think I'm going C now my memory is poor okay so sketching this nice and easy the hang on there is actually one thing I forgot to do plus two on the inside it's minus two from X chords and minus f of x means I'm timesing y by -1 so in other words I'm just flipping the sign so flip sign of Y which is the same as saying a reflection in the x-axis because keep in mind one thing that some students kind of get wrong is that these this transformation affects every coordinate along this line every single one so the reason why I say students get it wrong is because we like to focus on these points here and that's a good thing to do right we only focus on the important points because there's an infinite number of coordinates so we focus on the important bits but keep in mind that this minus f of x it doesn't mean the shape will be the same but this is like on the other side it means it's actually starting from the bottom so times this y coordinate by minus one you're starting from down here times this one by minus one you're ending up up here the whole thing is changing it's minus 2 from X uh this actually isn't too bad we have minus four here so that becomes minus six then we have one which would become minus one which I'm going to put over here two becomes zero and if you want you can put this as minus eight uh minus 2 minus eight but it's not necessary and like I said just now is instead of it coming from the top it's going to come from the bottom uh whoa hang on a minute no one yeah that's fine I scared myself sorry um you go down and then up and then down easy and for this one if you notice all of our coordinates except one is actually going to stay the same anyway and that's again why I caution I caution caution I encourage caution because the only coordinates changing is this minus eight everything else is actually the same but the shape is completely different so that minus eight now becomes a plus eight and this still stays as a one and a two okay so that's a really key thing to remember with that and again it's starting from the bottom and a perfect reflection again you don't have to be like Michelangelo it's okay to kind of make you know it'd not be perfectly the same and it's a sketch right so it the main thing is the coordinates and the general shape right like if I join these points up like a like a zigzag that's wrong or if I did something like I went up and then straight line and then straight line or something you know like something weird like that you need a general shape okay uh I think that's pretty much it for everyone I think everyone's perfectly fine with this so I'm going to call it quits here so again remember you can join these live streams uh 6 P.M every single Thursday you can ask questions you can do all of this fun stuff and also make suggestions right to make it as useful as possible but I'll see you in the next one