today's lesson is from one point three advanced functions properties of graphs of functions so it's going to be a little bit different lesson today because there's a lot of terminology and I didn't want to write it all out so you can stop the video anytime you want have a look at the notes so I'm going to be discussing with you so key terminology first of all we have symmetry now you know what symmetry means is something balanced is it the same on each side so you have different types of symmetry you have line symmetry if there is a line x equals a that divides a graph into two parts such that each part is a reflection of the other in the line x equals a then you have line symmetry and obviously the parabola would be the perfect example you learn that in grade 10 the axis of symmetry everything was the same distance between the axis of symmetry even if I drew it over here right so this has line symmetry the other symmetry is point symmetry so a cubic function y equals x cubed if you spin it around about this point right here if you spin this around then you would have the same graph right if I just brought this up here would look the very same as this part here so a cubic function is an example of point symmetry point symmetry the graph is symmetrical about a point a B if each part of the graph on one side of a B can be rotated 180 degrees to coincide with point on the other side so yes this is point symmetry now end behavior is something that you will have to describe many times in the grade 12 curriculum so it's important that you understand this and I have another little exercise we're going to do together a little bit later on and behavior what is happening to the graph as X approaches positive and negative infinity so I'm talking about what happens as I go out this way and as I go out this way so quite obviously for parabola as X approaches infinity so as they go this way Y or f of X this part is also approaching positive infinity right it goes on to remember these arrows mean it continues as X approaches negative infinity f of X also approaches positive infinity so this is still going up right so the Y's are getting higher or I've got a little shorter version here you could write as X approaches plus or minus infinity f of X approaches infinity okay so intervals of increase and decrease now so we have an interval within a functions domain where Y values get larger moving from left to right is increasing opposite for decreasing ie the Y values get smaller so in other words what you're doing is as I start here say I start here I'm coming up from the bottom and going I'm increasing increasing increasing increase then I hit a maximum value here and after that it's decreasing to here so my increasing intervals would be from negative infinity to 0 it's neither increasing nor decreasing at that point and then it decreases between 0 and whatever this value is here I think I wrote it as 4 so it decreases from 0 to 4 and that increases again from 4 to infinity now the tricky part about writing these brackets this is called interval notation and that is that if you're using a round bracket and I'll go into much more detail in a minute with that if you're using round brackets it means you're not including the endpoints of your interval so I can't include negative infinity because negative infinity it isn't a value right a negative infinity means the the graph is taking arbitrarily large values so because you have an arbitrary large value can't say it's called infinity infinity doesn't exist ok it just means it's getting really really really big negatively so between negative and friendly until I get to zero but not at zero because at zero the function is neither increasing nor decreasing it has reached a maximum in this case and the U here means Union so this just means Union which means I'm also including this interval which is from 4 to positive infinity the decreasing interval here is from 0 X is an element of remember this little isse imbel means element of X is an element of the values between 0 & 4 but not including 0 and not including 4 ok so on that theme I'm going to go over some of the bracket interval notation that you will be using to describe intervals of increase and decrease now in this case it was printed out in black and white which means we're going to have to color in what it actually means on the number line so this said the bracket interval is from A to B the inequality would be X is between a and B and the number line drawing would be remember if it's increase or lessen or greater than or not equal to right if it's equal to it's a closed circle so this is an open circle this is an open circle and I'm between these two so X is greater than a and less than B in words now if I have a round bracket and a square Rockit the square bracket again means lessen or equal to in this case it could be greater than or equal to if it was on the other end like in this one so now I have an open circle on a it's not including a but we go all the way down here to B and we include B so that's why it's solid there so that would be how you would draw it X is greater than a and less than or equal to B so let's go through this a little quicker now so we've got a square bracket on this end and round bracket that B and X is between these two values again so like that now you can pause the video and take a look at this handout if you want to spend a little more time but I'm just going to go through it quickly so this one we have two square brackets which means we're including both the endpoints and all the points in between this one shows a is included and then X is greater than a which means it's going this way so to infinity so that's why there's a round bracket remember infinity you cannot put a square bracket on infinity from negative infinity and including a so this is just going the other direction so we have here and we have it going this way oops put that in there okay in this one two round brackets so not including the a value but greater than this parents not working so well for me and this one it's going from it's less than a so lesson means you leave an open circle and you go this way oh it's really fading okay and finally negative infinity to positive infinity so that's including an element of real numbers means everything right so this is the entire number line in all directions okay so that's all the possible intervals for real numbers between an A and a B value okay so let's go back to we finish this properties and we've done that and now we're going to look at this kind of here which is talking about odd functions a function that has rotational symmetry about the origin okay so we just talked about the cubic function and my pencil and so for a cubic function you have it's was hiding if I take the cubic function and I cube negative two of course that's going to give me negative eight this is going to give me negative one zero one eight so if I drew for you a cubic function it would be 0 0 1 & 1 2 & 8 so about like that so it's coming down like this that's going to do the same thing on the other side so this would be y equals x cubed now in order for it to be odd now notice this line here about the origin and that is because if I said what would be F at negative x is equal to the negative of f of X now that sounds a little confusing but not if you watch carefully here so let's say I was going to say let's say X is 2 is f at negative 2 equal to the negative of F at 2 okay so don't get confused this is really quite simple it just sounds weird F at negative x is equal to the negative of f of X so f at negative 2 is negative 8 right so we have negative 8 on this side and on this side we have the negative of f of 2 and F at 2 is 8 so negative 8 equals negative 8 so that means it has odd symmetry okay so what's odd an odd function okay so this means it's odd actually it's kind of cool I think that F at negative 2 is negative of effort too now this does not hold true if I have my cubic function over here for instance oops that's a really bad cubic function so let's say I I'm going all the wrong ways alright guys it's been a long hot summer day here so I have it coming down like this and then like this okay so you can see right away that F at let's say this was 3 if I moved the cubic function over to the right 3 this would be the same thing as y equals x minus 3 cubed you know that because you know your transformation it shifted to the right three units so if I plugged in let's say four so f at four would be equal to one right because that would be one cubed but F at negative four would be equal to minus seven cubed which I don't know off the top of my head but it's not one so this is not odd okay so be careful with that it has to be symmetry about the origin which you know is zero zero right so if unless it's here this isn't going to be odd okay even functions functions whose graph are symmetrical about the y-axis now we talked about lime symmetry so you do know that a parabola has line symmetry so if I do a problem like this this would be an even function now it would only be even if the only transformation we can apply to this would be to shift it up or down so a vertical shift so I move my problem here it's still an even function or if I put it down here it would still be an even function but if I moved it over here that would not be because it's not about the y axis okay so even an odd has to be the origin has to be the y axis continuous functions are functions you can draw without lifting your pencil that's the easiest way to describe it so something like if I had a graph like this and I drew a square root function is that continuous yes I drew it without lifting my pencil doesn't matter that I started here and there's nothing over there it's continuous for X is greater than or equal to zero now if I had a function that wasn't continuous you did learn one in grade 11 and that was when you had a hole in the graph do you remember that so if I have five a function that's going like this and for some reason it wasn't defined right here I had a hole this would not be considered not to be continuous okay so any holes and graphs you don't see that too often but sometimes you have seen them in another one that wouldn't be continuous would be something like 1 over X I I really have to lift my pencil in order to drop both parts of that graph this is not continuous so most polynomial functions are continuous the radical functions exponential functions so you have to get used to the idea of what is what is continuous what isn't continuous and you also will see on your test and in your homework exercises where they give you these different characteristics of functions and ask you which functions have the following characteristics so you need to understand your your parent functions really well to that end there is a homework assignment and the homework it's number 12 on one point three properties of graphs I'm going to show it to you in two parts here so that you can freeze frame and have a look at it so this would be something your teacher would probably ask you to fill in it's a very important set of descriptors for the different functions and we'll go over the list of maybe two or three of these and then I'll let you take a look at it on your own time just pause take a look and make sure you understand the different characteristics so let's do let's do this one here first one over X people always hate that one because it's got so many complicated parts to it like asymptotes right so what is the domain X is an element of real numbers X is not equal to zero right can be zero the range Y is an element of real numbers Y is not equal to zero another asymptote here right that function has two asymptotes a vertical and horizontal the intervals of increase none why does that happen remember when you're talking about intervals of increase it's as you move from left to right you say oh no it's increasing right there see it's going up but no you're reading from left to right so this is going down this is going down as we move from x-values getting larger or more to the left or this way always down down so there's no intervals of increase ya know and intervals of decrease it's decreasing from negative infinity notice around bracket negative infinity to zero and you could put you here zero to infinity now we have to break it at zero because it's neither increasing nor decreasing at the point X is zero because the graph doesn't exist there location of discontinuities and asymptotes okay we've got that X is equal to zero Y is equal to zero those are simply the equations y equals zero x equals zero the equations of the asymptotes there are no zeros remember zeros where you cross the x-axis does not happen the y intercept does not cross the y axis the symmetry is point symmetry oh but you wouldn't have guessed that but if you look at this graph if I this is the point here about the origin if I flip it all the way around this would go over there this would go over there right so point symmetry and it would also be considered an odd function okay and behaviors so now I'm looking at what happens as X approaches infinity so as X approaches infinity Y approaches zero right because I go this way I'm approaching zero as X approaches zero from the right now this is something you probably didn't see before either don't know if you can read this tiny little writing a point zero from the right when you get your textbook in the fall if you're there already or not the solutions to this will also be in the back of the book but this will give you a little head start as X approaches zero from the right so as I approach zero here from the right that means I'm coming this way so as I approach from the right what's happening well H of X approaches infinity so that's what I've got here as X approaches zero from the right y approaches infinity as X approaches zero from the left now we're on this side here y approaches negative infinity and I think I covered both of them here as X approaches plus or minus infinity Y approaches zero turning points that's another thing that wasn't on this chart I added it to my classes notes because turning-point simply means where does a graph reach a Max or min and change direction so for instance for this graph here there would be your parabola there's a turning point right here at 0 0 so it goes down and then it goes back up so that's this one doesn't have a turning point this one has one this one doesn't have any this has a turning point here this doesn't have a turning point this doesn't have a turning point but the sine function it has a turning point every 90 degrees you reach the max you go to a min you go to max and min right okay so maybe we'll look at let's look at the exponential function so P at x equals two to the X here's my graph goes through zero one you should know that has an asymptote here right there's your exponential graph you can plug in some values if you want remember that to to a negative 1 is 1/2 to the negative 2 is 1/4 and on the other side of course you squared are u cubed squared you raise it to the fourth power what is the domain X is an element of real numbers I can plug in any value I want what is the range well Y is greater than 0 not equal to right greater than it never crosses the Y the X so was greater than zero intervals of increase remember read from left to right so as I go this way it's going up so it's increasing for all values of X so negative infinity to positive infinity and again we just have these round brackets intervals of decrease none doesn't go down it's always increasing that's why it's called an exponential growth function location of discontinuities and asymptotes it has an asymptote here of y equals 0 you know that y equals 0 if you can't remember if it's X or Y remember that on this axis all Y values so if I said 0 0 1 0 2 0 3 0 everything is 0 for y does it have any zeros no doesn't cross the x-axis does it have a y-intercept yes one in this case it would be 0 1 symmetry none how sad the end behaviors as X approaches infinity let's look at the graph again as X approaches infinity Y approaches infinity as X approaches negative infinity can you see it ok as X approaches negative infinity y approaches 0 so there we go like this and there are no turning points okay so I'll just scroll this really slowly back up to the top so if you want to have a look at it and freeze-frame and take a look at all the different properties for each of the graphs and you can stop it on your own and that's today's lesson I hope that helps you out and gets you prepared for your unit 1 test make sure you understand this very completely when I do a test I'll try to put in some examples