A Level Maths Made Easy - Modulus Function

hey guys welcome back to another a Lev maths Made Easy video this topic is the modulus function and we're going to cover quite a bit in fact everything to do with the modulus function in my opinion this is one of the worst taught and also the topic that has the most misconceptions in it so I think the beginning part will start off quite nice so if you haven't done it before it' be a really good intro if you have done it before you might want to skip a bit further ahead but this topic here requires that you sketch functions all of the bloody time especially when you're solving equations so again there's a lot of misconceptions with this so I'm going to hopefully address all of them and that way you have a much better understanding of what the modulus function is and why it works I'll have another video on some of the really really hard modulus questions that can come up probably a bit later on but for now let's start with the actual beginning so as usual um most of the people that watch these videos are not subscribed so if you do find any of this useful please consider just subscribing really helps me out um if you look at the link in the description there is my website where I have tuition and all sorts of other things there and my email address in case you want to ask me any questions so of course the first question we should ask is what the hell is the modulus function well the modulus function is represented by these two vertical arrows arrows vertical lines here and essentially all they do is they make the value inside those straight lines positive so if you go to your calculator the button is actually called ABS on your calculator because it means absolute and if I put in a negative value let's say -3 I will get positive 3 out if I put in just normal positive three it stays as three so it doesn't flip the sign or anything like that it literally just makes the value positive and again it doesn't really matter how big the number is it will just make it positive so first things first we need to basically just Sub in values so F of 6 when the fun function has abs like this has the modulus part of it it will be 6 - 4 which means it's 2 which means the value is just equal to two and the same for B it would be the modulus of 3 - 4 then you would just make that the modulus of -1 which of course is just one so this is really really useful when we don't consider the negative values so when we deal things like speed distance and not acceleration but speed and distance and lots of other values in real life we usually only care about the positive values we don't care about negatives but as you sub in for all of these as you can see anytime it's a negative it just becomes positive and the same thing happens over here so with the last three questions we have G and what will happen here is the value subbed in will become positive first then then you're going to work it out so the answer can be negative the only thing that matters is that the value that you sub in initially becomes positive so again anything inside the modulus signs become positive and that's all you get so hopefully you can find that's nice and simple so let's see where this goes a bit later on the next thing that's really useful is when you have to sketch graphs so sketch each of the following graphs showing the coordinates of any points of intersection with the axes so for example if I just draw the regular Y = X + 4 graph here is what I get so I'm going to draw this in blue we'll get a line that goes through here Y X + 4 which means the Y intercept is 4 and this value over here is when yal 0 which is when so 0 = x + 4 so - 4 = x so that value be - 4 over here so what happens when you put the modulus sign around the expression the equation well all that happens is anytime this is negative so this thing inside the bracket is negative it becomes positive so let's let's kind of test this out for a second um let's say we have X is a number smaller than minus 4 cuz as I can see Y is negative when it goes below the xais right that's when y becomes negative so let's test it out right if we have X = -5 what's going to happen well we have the modulus of - 5 + 4 which means we get Y is equal to the modulus of -1 which means Y is equal to 1 so instead of going down constantly now the value is actually going up right because again instead of it being -15 it's going to be5 sorry instead of it being -51 instead it's going to be -5 + 1 so all of the values where Y is negative now become positive so the graph will actually look exactly the same as this it's just that it almost bounces off the xais and becomes positive hopefully that makes a little more sense okay so how else does this look well again if we take the graph of y = x^2 - 9 once again if I draw this out we're going to have an x s graph will look vaguely similar to that where these two points well what are they well remember we would have x^2 - 9 = 0 so that's x + 3 x - 3 = 0 you have x = - 3 x = + 3 by the way this topic is heavily heavily heavily based on being able to sketch a bunch of different functions even ones that you haven't done before before so you should definitely keep that in mind and the way that I'm doing this is exactly the way I'll do it in the exam where I'm drawing the original graph and then using a different color to show what the new graph would look like and oh yeah down here would be 9 so once again all of the values below the x axis so when Y is negative will now be positive so instead of crossing at Min - 99 it's going to actually cross at positive 9 and once again instead of going below the x-axis it's going to like bounce off and all of these values will be reflected up so anything below the x-axis now goes like this and it's going to look something like that and then it would still go up like this so it still follow this up this up so it's like a big W and that's what that look like and again that's because all of the negative values for y become positive which is really really key feature there once again we've got a few more that I want to show you yal the modulus of sin x now it does say between 0 and 2 pi if you haven't done the radians chapter this is the same as between 0 and 360° so once again if we draw out this graph and what I'm actually going to do to make it my life a little bit easier is I'm going to move it whoops to the right because we don't care about negative values of x and if you remember how the sign graph looks it goes like this that's almost perfect that's really nice so zero and then it goes you know 90° it goes between + one and -1 and then that's 180 that's 270 so every 90° something interesting happens so it goes between 0 and 1 1 and 0 uh 0 and -1 and min-1 and 0 now again have a think in fact pause the video and see what will happen to this by the way if you had this between 0 and 2 pi so if you had it in radians it would be piun / 2 pi 3 piun / 2 and 2 pi well again all of the negative values for y are reflected upwards so instead of it going through the x-axis again it bounces off the x-axis so this 270 minus 1 is now going to be up here it's going to be 270 and + one so it's going to look something like this and again it does follow this original curve and that would be Y is equal to the modulus of sin x so again my method is you draw the original graph and then anything below the xaxis you put to the well you put above the x-axis and that's basically the only difference so now once again we have yal 1X all of these graphs are graphs that you need to be able to do just on the Fly and if you remember the Y = 1 of X graph looks like it looks like this if you want to video on like how I memorize or how I know or learn how to kind of sketch a bunch of different functions then do let me know in the comments and again anything below the x-axis is now reflected up to the positive so it' be like this and then of course this one stays exactly the same cuz it's already positive remember the modulus function doesn't affect positive values it only affects negative values lastly but certainly not least we're going to do the Ln graph so and again this is just to kind of show you the types of graphs that you're expected to be able to sketch they can show you oh sorry give you pretty much anything they give you tan cos s whatever and normally your Ln graph looks like this where it goes through the position sorry the point one and zero now now the only difference here is that this is going to bounce off and come up this way because again anything below the x-axis becomes positive so I know we've kind of repeated that a hell of a lot but when else is this useful well it's also useful when you're solving equations and this is where I find students make mistakes all of the time they do not sketch graphs out so here we're this is what we're going to going to do with this equation here's how schools normally teach it is they just say oh you make it negative and you make it positive and that's it now that is not a good enough reason okay it's not good enough instead let's draw out the graph first and the reason why I'm going to draw out the graph is to illustrate to you why you can't just arbitrarily make things positive and negative okay you can't do it it's not it's just an idiotic thing to do the reason why I says otic is I really really really like this topic I think it's a very good topic I think who I think that you can gain a lot of marks for doing very little sometimes these questions can be worth two or three marks it's a linear equation now why am I getting into this if I say if I draw the graph of x - 5 quickly that's what it's going to look like roughly and if you wanted it equal to three all you would do is you would draw a line at y 3 and you can clearly see that it only crosses crosses the um equation crosses the graph once so there's one solution so there's one solution for x - 5 = 3 cool I fully agree with that but look what happens when we actually draw the X -5 modulus function well now everything that's below the x-axis now goes above the x-axis how many times does it cross it crosses twice so normally in our linear equation if we had x - 5 = whoops - 5 = 3 we have one solution because the graph of x - 5 crosses y = 3 once but now we have the modulus function there are two solutions because when we sketch it it will cross twice now how do you know what the values are well if you notice this graph here this blue part of the graph is just x- 5 that is the graph of y = x - 5 just this blue part here so one of my Solutions is when x - 5 equals 3 because again the blue part is just the graph of y = x - 5 which means one value is xal 8 boom now have a think about what the red line what the equation for that graph is well if you notice the Y intercept over here is minus 5 over here it's + 5 so this value on the end is going to be something + 5 now have a look at the gradient the gradient should be the same as the blue line but it's going down instead of up so it's minus X that's the other equation that we're solving so again the way that some teachers teach it and I would argue it's not even teaching it is they say well x - 5 = 3 and then brackets X x - 5 = 3 but they put a minus on the outside this is not always true and in some cases it makes it worse so I've got a couple of questions later on where you can't just do this arbitrarily you'll get the wrong answer instead I prefer my way where you actually look at the graph and you actually understand what the hell is going on but that's where your teacher got it from it's where again this is technically true but only because when you expand it you get the actual equation of the line but anyways if we solve this we then get - x = -2 and therefore X is equal to 2 so hopefully that makes a little more sense but again you have to be very careful with where you're dealing with this now if you're wondering why X is not a negative number it's because on my sketch the intersections on the negative side in actuality it seems like it should be more steep like this so it crosses on the right hand side of the graph but again that doesn't really matter the sketch is mainly there to tell you how many times it intersects so let's see if we can do this now so once again I've given you basically the same type of question I would recommend pausing the video and just quickly having it Go and seeing what would happen but for those of you that have remained again draw the graph out get into this habit because I guarantee you in your exam you may get screwed over so once again we're looking at plus one and it's we're looking for the intersection when it equals 15 so like so and again when we actually do this calculation it's going to intersect quite far away but there's two points of intersection the first one is when the normal graph x + 1 = 15 so x = 14 but in the other one the Y intercept is now somewhere below this line it's at minus one right if we continue this red line along if we wanted to maybe I'll continue in green it would cross at minus one and it's obviously Negative X because the gradient is negative once again in this case you can just put the negative on the outside of the bracket I'm not saying that you can't do that I'm saying don't do it without reason okay so that is the way how you find the opposite part of the graph and you getx = 16 so X is equal to -16 and those are the two answers and again for three marks it's worth doing the slightly longer method to make sure you're not making a mistake but let me show you two questions now where you cannot just arbitrarily make things negative so so here is exactly where people make mistakes because here if you do it the normal way well first of all you might say x - 2 = x + 4 okay but then which one do you make negative so do you do x - 2 = - x + 4 or would it be - x - 2 = x + 4 or would it be both of them negative and the most common question I get from students is in the mark scheme they'll only do two of these because these graphs will only intersect twice I'll prove it to you in a second but they might also only intercept once by the way but the most that they'll intercept is twice at minimum well actually no at minimum zero technically but they'll intercept once or twice so here we have four different equations we'll get four different X values and the worst thing is and this is the really key thing is for two of them you'll get an answer that's nonsense so you might get like xal X or something like that which means you know 0 equals 0 and people start to panic so if you did this all four of them you would get the right answer but you have to be prepared that for at least two of them maybe three of them again you will get an answer like this which it it's a null statement right you can't derive anything from that because they don't intercept each other you will not get x equals z okay for me that's confusing well a it's twice as much work right sometime it might even be four times as much work if it only intercepts once and two getting this in your exam would scare the life out of you so instead of doing this I'm going to get rid of all this and we're going to be smart about it we are going to draw the graphs so let's have a look again draw a decently large axis if we do x- 2 first that's going to look like this um maybe I'll bring it up a little bit just to make it look at who careful wait hello just a little bit just to make it like this and then of course when we actually draw the uh modulus function it probably should be a little less steep than that but that's okay so that is my graph of y = x - 2 and then the graph of y = mod x - 2 and then I'm going to use two different colors so I'm use black for this one so we have X+ 4 so that'll be I mean it should actually in be a little higher because that's a two so something like that and then uh let's use green for the rest of it so if you have a look at this two things I want to say they intercept once once right they intercept right here now here's the question I get given is and this is a really good point how do we know that those two graphs if they go off infinitely don't eventually intercept outside of my sketch because I've done it before where when I've drawn the two graphs on a single set of axes I actually get two two solutions because they actually intercept further outside of the scope of my sketch and students then say to me well how do you know I can tell by looking how can I tell well here I know for a fact they do not intercept again why because their gradients are the same they are parallel which means these two lines going off will stay equally distant from each other forever and the red and green line going off will stay equally distant like train tracks forever so the black and blue parallel red and green parallel because no matter what they will have the same gradient the only reason why the black and red one intercept is because the red one has a negative gradient and the black one has a positive gradient that's the only reason why they intercept so the only solution to this equation is the red one where you have - x - 2 so you basically have - x + 2 and you have the positive black line where in fact I didn't actually write what the black line is that's honestly bit sloppy x + 4 and Y = mod x + 4 so that's where you would have equals x + 4 that is the only solution so I'm going to add X to both sides take away four from both sides so X = -1 that is the only solution to this equation and again the reason why I know it's the only one is because those lines are parallel they will not cross again if we had something like 2x + 4 and x - 2 then they would intercept again somewhere to the left and that's how you could tell okay so let's try it again using one that's um a little different right so we have y = x - 5 and Y = 7 - x and once again do not do that thing where you just shove negatives in front of all of them it is stupid it's just not a good idea it's a really dumb idea let's draw x - 5 first it's going to be down here somewhere whoops so y = x - 5 so then when it becomes positive it's going to go quite shallowly shallowly shallowly sure that's an adverb right that's five - 5 and if you draw the black one that's 7 - x so it's a graph that goes down it crosses at seven at seven of course and then B back up and I also know by the way that these should have the same gradient because they should so y = 7 - x and then the green part is going to go off with the exact same gradient as this one that's actually a pretty good sketch I'm actually relatively happy with myself for once um that's sounded depressing but hopefully you get my point so how many times do they intercept well they have the same gradient it well the magnitude of their gradient is the same it's one which means they're parallel which means it crosses once so it crosses right here so that is when we use the black line 7 - x equal to the Blue Line x - 5 have you noticed what I've done I didn't even use the whole putting a negative outside right because I've drawn all the graphs I've drawn all four graphs so you don't even need to do the minus x - 5 or whatever because guess what I've drawn all the graphs I can just look I can see well the black and the Blue Cross well black is 7 - x blue is x - 5 how easy is that remember I something that I say to a lot of students is keep it simple right the the simpler you keep it the less likely you are to make mistakes X so add X to both sides 2x - 5 add 5 to both sides um we get 12 and again x = 6 and this does make sense because it's on the positive X AIS the very last thing is again one of the most annoying Parts because what we're going to do is we're going to look at the difference between the mod of f ofx being over the whole of the function so basically making y positive so make if if you remember your Transformations this might help and F of mod x where essentially you're making X positive and this is a really really really key skill so honestly with this first one I'm sure you guys already know what this one will look like so you have 3x - 1 so normally that would look like a a straight line going down and diagonal but in this case it's going to go down down quite Steep and then go up it should be equally steep where that is plus one and it hits the ground at a third comma 0 how do you know that just make y equal to 0 you get 3x = 1 x = 1/3 but how does it look when you have mod f ofx so if I write that out for you this is the equation that we're basically sketching now here is where I'm actually going to draw it separately okay and what I mean by that is I'm going to draw it in the classic way that I did before where I draw both diagrams on one thing so if I do 3x - 1 first you get something along those lines right that's minus one it's y = 3x - 1 now when I make X positive what does it mean well it means that and I I've I've thought of so many ways to explain this because it's quite tricky but essentially if I look at my if I look at a plotting table right look at a plotting table the positive values will be the exact same as the ne as the negative so if we do 0 1 2 and 3 0 would be Min - one 1 would be 3 - 1 which is 2 uh that' be 6 - 1 which is 5 and 8 now if I sub in a negative value you would expect the values to get Negative right so normally you know 3 * -1 is -3 takeway 1 is -4 so normally you'd expect -4 - 7 - 10 but we're making the x values positive so what does that mean it means that subbing in minus one gives the same result as subbing in one because we're making the x values positive which means instead of having that we have 2 5 8 because again I put in a negative value it gets made positive because the modulus then the same thing happens again so think about it if I put- 3 in make it positive that's 3 * 3 is 9 - 1 is 8 3 mod - 2 well you take the modulus of minus 2 it becomes 2 3 * 2 is 6 - 1 is 5 and so on so what does that mean it means our graph can be negative y values it's allowed to be be negative y value but all of the X values basically just become a reflection so the way I like to put it is you mirror along the Y AIS so whatever's on the right of the Y AIS you draw the same thing on the left hand side so you draw it like this now that might not seem too impressive right now but have a look at when we do something the later ones so the next one I'm going to do is I'm going to do a quadratic quadratic graph for you and you're going to see why things get a little bit weird because again the right side and the left side become perfect mirrors of each other when you take the modulus of X and the reason for that is every negative x value gets changed into its positive version so let's have a look here we have a classic quad itic however um there there's something else going on here so if I draw the graph normally here's what it would look like X and Y so first of all we want the Y intercepts so sorry the X intercepts which would be x = -1 and x equals 5 so if we label those boom and boom but if you expanded this you would notice it's a minus x squared so in our case it's actually going to look like a big sad face like so and where does it cross well if we Sub in xal 0 we get 1 * 5 which is 5 um I should actually label those two points as well so what happens when I take the modulus of this graph well remember everything below the y- axis becomes positive so that's going to be like this and like this which is you know fantastic right so now you're probably thinking you're probably stressing a little bit what the hell does this look like when you have F of mod of X so for the sake of Simplicity I'm actually just going to draw that all on one graph so I'm not going to draw two graphs so the right hand side will stay exactly the same but remember what I said about the left side so imagine I've just copied this part of the graph here moved it over it's honestly it's not I'm not happy with that drawing come on there we are that's gone slightly better well it becomes an exact mirror so I'm going to draw the same thing on the other side but mirrored around where that is well it's also five but it's negative five you know what I mean it's also five though so it looks like a giant N I will admit this isn't the best looking drawing uh I might even copy over um one that I've done on the um so basically I use Mathematica to also plot these so if I show you again hopefully you can see where I'm coming from with this um yeah this is speed this is all right so again if you look we have those flicks at the end for either side and again with this it comes up like a giant m shape now again I did this by plotting it on a software that's why it looks better now there is something I should mention with plotting graphs and things like that is they don't care about the shape you're not meant to be Pablo Picasso okay we're not da Vinci over here all we care about is the general shape so the fact that it's curved the fact that it's in the positive y quadrant will get you the marks so I would get full marks even though this is absolutely hideous why because I have the correct coordinates - 555 - 555 and because I have the general shape the idea that it is kind of like a a duplicate copy on both sides but I do admit that it's a little bit sloppy is a little bit sloppy I'll admit that to you and coming on to the last one which is uh fun now let's draw the one on the left so again remember this is going to be yal F ofx oh jeez um and you have an asmt yal 0 so you might notice that the yal F ofx graph is actually the same as mod f ofx because remember the e to the X graph can never be negative because when you put something to a power it can never be negative assuming it's a positive base but have a think about what could happen when we do and maybe I do it on the same set of axes F of mod x well again the left and right side will be complete mirrors of each other so instead we're going to have something that looks like this almost looks like a quadratic and again for the sake of me being very um I I knew I wouldn't be able to draw it super super well but just to show you guys so the original function uh and the mod f ofx function is the exact same and then here again it kind of just looks like it should be more of a tip which I think I've got the tip f well but again remember we're not public Picasso all we need to do is make sure that we label those coordinates and we're done so honestly here's the main misconceptions a with the drawing it's pretty hard but B the biggest misconception I see is with solving the equations I promise you right now if you sort this out in your head and you understand how to do this your life will become a lot easier I have a ton of extremely hard questions harder than any exam question ever that I can go through so do let me know if you want me to do that but generally speaking sketch the graph see how it overlaps because they can also give you quadratics things like that and it gets really fun but again if you found it useful please consider subscribing as it really does help me out but I hope to see you in the next one and have a great day