College Algebra - Lecture 7 - Graphs

you now as we continue with this unit we're going to look at the graphs of equations and in this process we will actually use the graphing calculator that I hope you've had a little time to look at in the previous break so let's look at graphs of equations now first of all we need to decide what that means so let me give you a generic definition of graphs of equations so I'm going to repeat myself here to save a little writing graphs of equations what is that well what is the graph of an equation it is the set of points and remember in the coordinate plane in the plane that we have given coordinates and addresses in other words to every point all the points how are of the form X comma Y the set of points that and here's one of those mathematical words satisfy the equation so the graph of an equation is the set of points X Y that satisfy the equation and because these are points you can graph them and that's where the graph comes from now why do we see satisfy well the equation as you'll see is something that involves an equal sign of course that's the verb in the sentence sentence that is an equation and there will be X's and Y's and then what we put in the X's and Y's that we get is the set of points they satisfy the equation meaning the equation becomes true when they're in there so for example let me give you something to think about let's take an example here's an equation y equals 2x plus 5 and remember what I said about equations very early in the review this equals is the verb and the equation is a sentence and you can see there's a y and an x involved here now for the correct choices of y and X we will get true statements here not all values will work if I put in x equals 0 and I put y equal 100 that's not going to be true because 100 then would be 0 plus 5 and that isn't clearly not true however we can go ahead and list in the form of a table so I'll list X Y and then I'll list the point X comma Y so we're going to build a table here these are the kind of tables that you can see on your graphing calculator also and I will just pick a few points X say 0 1 minus 5 10 het cetera you have to realize there are an infinite number of real numbers to choose so my choice is an arbitrary choice and we'll talk about that later and how that will affect what the graph looks like and then we'll calculate what Y will be because up here this equation gives us a formula that says if X is given then Y is 2 times that number plus 5 so if x is 0 Y will be 2 times 0 plus 5 which means Y is 5 if X is 1 then Y is 2 times 1 which is 2 plus 5 that's 7 if X is minus 5 then Y is 2 times minus 5 which is minus 10 plus 5 is minus 5 and finally if we put 10 in for X we have 20 here plus 5 so Y is 25 and we could continue that with any number of points we liked having an x and a y creates a point that then we can then graph so this is the point 0 5 where the x value is in the first position the Y values in the second position that's what it ordered pair is and then the other points are 1 7 minus 5 minus 5 10 25 etcetera now I'm going to plot these points I'm going to actually draw them and then I'm going to try and produce a graph based on this really meager set of points there are only four points after all so here is an axis system and let me mark off in units of 5 because that seems to be about what I need so I will go up 5 here and then 10 15 20 and all the way up here to 25 because if you remember I had a point that requires that and then I will go down a few here and over a few to the left I'll mark the first stage as minus five here and minus five here and that may be all I need so let me plot the points that we're on the previous page if I show you what those are they are these four points if you recall zero five one seven minus five minus five ten and twenty five now go ahead and plot those let me start with zero five that's an easy one to remember zero is the x-coordinate so there is no direction in the X on the x axis and then I go up to five so there's a point on my graph the second point was one seven so one is about here and seven is a little above five so it's about here you can't do much better when you do this by hand so there's an approximation of where that point should be then there was a point minus five minus five so it's minus five and both of the coordinates that puts a point here and finally there was the point 10 25 which puts a point up here so there are the four points that I've graphed and if I done this carefully you can see that when I connect them up I have what appears to be seems to be a line now you cannot judge the picture strictly by looking at four points I warn you that this is dangerous okay and I will give you more information on this coming up but let's just take it for granted that four points is really not enough to determine a graph in general if I knew beforehand that this was a line then since I know Euclidean geometry only requires two points to determine a line any two of these points would be sufficient to draw the line but I don't know that that's a line yet we'll talk about that later so let me give you another example a little more complicated one let's raise the bar a bit and look at this particular equation y equals x squared now y equals x squared is a function that you may learn eventually it's called quadratic it has a picture that is a parabola we don't need to know that now we just need to know that in order to get Y we take X and square it and if you create a table once again I won't do too many of these but I think a another table at this point is appropriate and we just pick a few points like -3 minus 1 0 1 3 etc for X we know that what Y is is the square of this because Y is equal to x squared so I just Square these I get 9 1 0 1 9 etc there's a nice symmetry going on here and so this will be the points minus 3 9 minus 1 1 0 0 we know where that point is that's the origin point 1 1 point 3 9 and there's an infinite number of other points now if I plot these points as I did the other points on a coordinate system and on this one I will take the vertical scale to be in 5s again so 5 10 15 and 20 I think is as high as I need to go and I will take the X scale as 1 now you might say why am i doing that well I can choose whatever a scale I want on the x and the y axis it is always your choice and you choose a scale that is easy for you to graph what you need to graph so if that's 1 2 3 & 4 & going to the left of course minus 1 minus 2 minus 3 etc and I plot the points we have previously let me remind you what those were - 3 9 - 1 1 0 0 1 1 & 3 9 so minus 3 & 9 this is 10 so minus 3 9 is about there the next point was minus 1 1 that will be about there because this is about where 1 is say 0 0 is the origin and notice on the other side 1 1 mirrors the point minus 1 1 and then the point 3 9 mirrors the point minus 3 9 now when we connect these up we get a very nice symmetric curve we'll talk about symmetry later that looks something like that and that will become a very familiar curve to you as this course goes on but let me warn you once again this is dangerous plotting a curve by connecting points when you have so few points or even in fact if you had a million points it would still be dangerous you require mathematical knowledge to actually know that this is the graph and that there's not something else happening there since that's sometimes hard to believe let me now give you a classic student error which I hope will make the point so I'll even write that down classic student error now here here's what a student will do of course this won't be you but here's what a student might do here is an equation I give you Y is equal to X to the fourth minus 5x squared plus 4 now that's a little bit more complicated than the previous ones but it doesn't matter let's go ahead and plot a few points by first creating a table as I've done before and I'll pick some easy numbers for X say -2 - 1 1 & 2 ok those are easy numbers let me put them into Y and you might say well it's rather complicated but assuming that I've done these correctly you will see for example if I put -2 in that Y will be 0 for example - 2 to the 4th will be 16 - 2 squared is 4 so that's 16 minus 20 that's minus 4 plus 4 is 0 and it will turn out if you put minus one in you get zero one in you get zero and two in you get zero you can check those on your own but that leaves you with points minus 2 0 minus 1 0 1 0 & 2 0 dot dot dot you could say that this goes out to infinity but maybe you feel confident plotting these 4 points okay let's go ahead and look at what a plot would look like if you plotted those four points if you do it turns out to be a very easy plot if this is 1 2 and this is minus 1 minus 2 then the points on the x-axis that you plot are minus 2 0 which is that point minus 1 0 1 0 & 2 0 and so you would conclude that the graph if you connected these dots is the same thing as the x-axis now the fact is you have gotten this horribly wrong in fact the actual graph looks something like this true graph nothing at all like the graph that you thought it was based on these four points so there is always danger when you don't plot all possible points and since there are an infinite number of points there's always danger we need to have some other mathematical ideas that will tell us how the curve will appear even though we know we can't plot an infinite number of points even with a computer so that brings up the next question what kind of a graph do we really want and we're going to use a term for that that's become common now that we have graphing calculators the definition that I'm going to write down is the following a complete graph is the term we'll use a complete graph now what could that mean well in short it means a graph that contains within the window you're looking at all the interesting stuff about the curve that you're trying to graph so a complete graph reveals all the important features the important features of the graph for example Peaks valleys flats where the curve just goes horizontal and then other words you may know asymptotes etc in short there should be no hidden behavior now that is a tall order and as I said we cannot answer the question completely without some calculus we can however make some strong inroads into it using what we're going to learn in this course but that's exactly what we want to do complete graph reveals all the important features of the graph Peaks valleys flats asymptotes anything else of interest so since we're going to be using graphing calculators before I go ahead and show you some examples on the graphing calculator let me give you what one might call a general procedure for graphing on a calculator save this also applies of course to a computer algebra system first of all you want to write whatever your equation is as y equals something in only the variable X calculator is limited it must have a variable on the left usually Y but it doesn't have to be Y and only a single variable on the right but it has to be separated out this way then what you want to do is pick a window now as I said and you saw earlier window choice makes a big difference as to what you'll see once you've done that you will then enter your y equals something from above and I'll show you that by example with the calculator in a moment and then finally you'll graph it using the graph key on your calculator so that is the procedure the first part writing this has to be done by yourself beforehand so let me start an example and then we'll continue the example onto a calculator here's the example we're going to look at 6x squared plus 3y equals 24 now as I said I need to rewrite this so that the Y is alone on the left there's an equal sign and everything else on the right involves X or constants so this is just a matter of rearrangement I'll move the six x squared to the other side so I'll have 3y here I'll have 24 minus 6x squared and then I'll divide both sides by 3 so Y will become equal to 24 divided by 3 is 8 6 divided by 3 will give me 2x squared so this is the equation I am now going to graph 8 minus 2x squared so onto the calculators all right here's the calculator now I'm going to walk you through the entire process first I will turn it on you see I'm already in the graph mode let's hit the y equals key so I can type in the function which is the equation that we are currently looking at function is actually a term we'll use later eight minus two x and this calculator is unlike any other it allows you to type in powers the way you would write them notice that these superscript two is above just the way you would write that by hand so that's a nice feature that this calculator has alright I said the next thing I needed to do is pick a window so let's hit the window key now I've given some thought to this window so let me go ahead and type in first minus 5 to get X min 5 4 X max the scale is 1 I will leave it that way Y min I will make minus 10 and Y max I will make 20 and I'll leave the scale well no I think I'll change the scale to 2 for the Y scale so now I've chosen that I've entered my function and I picked a window now all I have to do is graph it well I just hit the graph key and lo and behold you will see that the curve appears on the calculator you may have noticed that it we had a little blinking spot up here in the upper right hand corner that indicates that the calculator is busy and it is busy graphing the the equation and so there's a picture now I might decide that the window I chose is not a good window I would like to improve this window a bit so I might say the Y is a little bit too high because I'd like to have the graph fill the screen a little bit more so maybe I could bring that down how would I do that this is called experimentation or fiddling if you like go back to the window come down to Y max which was the top I've gone up to 20 let me see what it would look like if I change that to 15 go back and hit graph of course it's the window has changed so the graph has to be reconstructed and you can see it seems to fit the screen a little better now this is the kind of thing you're going to do all the time you're going to look at the screen and you get a a new window based on what you see you'll also change the window based on the kind of calculations that you do and we'll learn about that later in the course I wanted to show you a couple of other things while we're here and we'll see this with other pictures later there's a key here called trace watch what happens when I hit trace several things appear on the screen in the upper left corner you see the function appears maybe a little hard to read it says y1 equals 8 minus 2x squared now the newer calculators actually put the function up there and on the bottom you see x equals 0 and y equals 0 now if I use the arrow keys and I move away from that look what's happening actually that wasn't y equals 0 was it that was the top there so that was why you call another number but you see what's happening as the cursor is moving along the x and y values below are changing because they are giving the coordinates of the cursor approximately those are decimal approximations but this is a nice way to see and let's see what happens if we come on down to the x-axis we like at one point for the x value to be some number and the Y value to actually go to zero I see it didn't happen let me go back at this point the Y value if you can read it is positive it's point one two something the next stage as I trace further down I have minus point five so it's clear that I've crossed the axis but you won't always get the point that you're looking for we'll have to talk about that later and those are limitations of the machine and there are ways to get more appropriate approximations one other thing I wanted to show you while we're here is the table feature which a lot of calculators now have if I hit the table key a table appears and on the left there are X values and you can set this table up any way you like it is currently set up with integer values 4 5 6 7 8 etc and look what's on the right the Y values associated with them the top of this column is y1 and that's the function we entered 8 minus 2 x squared and so if you put 4 in there you get the minus 24 value there I believe that's what that says yeah and then you get the other values and you can also see those numbers appearing below as I change this directly below the table y1 minus one fifty four minus one twentieth setter is appearing below and the other columns are for what well if we have more functions we can have a table in which all of those functional values are listed we'll get back to this later I wanted to give you some indication of what this would look like now while we're here let's go ahead and change the function to something more complicated and fiddle a little bit and I'll show you something about zooming so go back to y equals well let's first go back to the graph and then let's go to y equals and I'm going to hit my clear key and wipe out that function now I'm going to type in one that's more complicated X cubed so you can see how it looks when you do this minus 11x squared see this calculator is so nice because the powers appear above minus 190 X and then plus 200 now that's a function with coefficients you probably wouldn't have seen before we had calculators to look at things because those are big numbers and that makes it hard to calculate with by hand but with a calculator there's no problem now I thought about this a little bit and we're going to have to experiment but here's a starting point for the window we'll go from minus 12 enter to 1 to 12 and then the scale I will make equal to 4 for the X scale and then I'll go from minus two hundred to 200 in the y-direction and then the scale I will make 100 so let's see what the graph looks like when we do this now I purposely chose this so that the graph would not be a complete graph that parts of it are clearly outside the window it seems to go up on the Left come down in the middle and it seems to disappear at that point now it looks like something's missing and what you need to do is fiddle with this experiment now you can simply experiment by changing the window directly as we've seen before but there are some other features you need to know about one is zooming there's a key called zoom I hit the zoom key now I have a variety of options if you look to the right of zoom I have an automatic zooming which I won't do I have zoom box which I'll show you later zoom in which I'll show you later but right now I want to look at zoom out because I think I'm too close and if i zoom back if I move backward perhaps I can see more of the graph so I've got it set it zoom out if I hit the enter key it will now take the curve and zoom out which means pull yourself backwards from the screen and you see what's happening you can see much more of what's going on and you can begin to say aha think I know what the shape of this is it's not mathematically precise just yet but we're beginning to get a feel for what this picture ought to look like it looks like I ought to extend the whyme in the lower part so that I can actually see more of the bottom here that seems to be another valley down here so let me go ahead and just change that window and come down here to Y min I'm currently it's at minus 800 let me go down to say minus 1000 and go back to the graph and see if that gives me the valley I was looking for all of this is experimentation the more mathematics you know the better you are at it because you know where the valley will occur and calculus helps you do that but we will learn ways of doing that also now see I didn't go far enough so let's go back to the window come on down to Y min and it'll be a little bolder go to minus 1500 and let's see I want to get 1500 there we go and then go back to the graph and now let's see if I picked up that valley it goes up comes down well I'm doing better let's go back one more time and see if I can pick that up I'm going to be even bolder I'm going to go to minus 2500 this time and go back to the graph and I think this time I will probably have the valley I was looking for this is the kind of experimentation you'll be doing with more mathematical knowledge you won't have to go through so many stages but now I have what I think is a complete graph and since I know something about this graph the fact that it's a cubic I know that there's one and one down and that's all there is to it that is again something we'll learn later all right well I think that's enough for now - looking for looking at the graphing calculator [Music] I had talked about graphs of equations let's now look at intercepts now that is the word that's used when we cross the x or y axis with the graph they're often called X intercepts and y intercepts so let's go ahead and look at the two of these intercepts now these are useful to give you points on the curve that are easy to find what am I talking about well here are the axes x and y and here's a curve coming along here like so now there's some nice points if I'm trying to pick points out the graph that are very easy to see there's this one this one this one and this one those are all places where the curve crosses the x axis and here's another one there's one point up here where it crosses the y axis these are all particularly nice points why are they nice well all of these points down here all of these X intercepting points have something in common if you look at them as an ordered pair they'll have an x value of some kind depending on where they are but what is their Y value neither up nor down the Y value is 0 so this is X comma 0 they all have that form the X's are different but they all have the same form that makes them particularly nice likewise if I'm looking at this one y intercept point up here what form will it have well it has a particular height some Y value height whatever that is but notice that it is not in either of the directions for X so the x coordinate of this point is 0 that makes this point nice because one of the two coordinates is 0 that's the reason people look for intercepts because there's it a 0 involved and 0 is nice to deal with now if I have a picture it's easy to find the intercepts what I'd like to do is look at the equation and find the intercepts from the equation so let's examine that procedure finding intercepts and those are either the X or the y intercepts finding intercepts from an equation well the key to doing this is something I mentioned but when I talked about the coordinates x-intercepts what do they look like they have the form x comma 0 in other words the y-value is 0 and then the x value is the number we need to find so that's what we do we let y be 0 in the equation and we solve for x and that's all there is to it to find the x-intercepts we let y be 0 and solve for x and if that works for X if you're looking for y-intercepts then you do the a similar operation you're looking now for points where the Y value is some unknown number but the x value is 0 and the same procedure works let X be 0 this time and solve for y and that's all there is to it so let me show you that with an example here's an example here is an equation y equals x squared minus 3 let's first look for x-intercepts now I'm doing this without looking at the picture eventually I would like to look at the picture but right now I'm not going to look at the picture this is a calculation for points of the form X comma 0 that I can do strictly from the equation so what was the procedure again I let Y be 0 and I solve for X well here's my expression if I let Y be 0 I have 0 equals x squared minus 3 so I just need to solve for X out of this that's fairly easy I move the 3 to the other side so I 3 equals x squared and then I can reverse that if I like to have my X's on the left but then what I need to do is take the square root of both sides and so I end up with X being plus the square root of 3 or minus the square root of 3 because either one of those squared gives me three and so I have my two x-values and so what are the points that are the x-intercepts that I was looking for well one is plus square root of three zero and the other is minus square root of three zero now we'll look at a picture on the graphing calculator of this in a moment but let's go ahead and look at the y intercept version so this is continued let me write the equation down again y equals x squared minus three and now let's look for the y intercepts and you know what the procedure is now y intercepts we are looking for points of the form 0 Y so we will let X be 0 and solve for y so y equals 0 squared minus 3 well this is much easier than the other one y equals -3 there you go and so what is the one point here that is the y-intercept point it is the point 0 comma minus 3 so there we have it we have found the X and y intercepts now let's go back to the calculator and graph this and see what things look like so here's the calculator I first have to type in the expression as before so I have x squared + -3 is the expression I need to pick a window now my window is something that again you just try a few things I've done a little bit of preliminary work here so I'll try something fairly simple - 5 - 5 in the x-axis and I'll let the X scale be 1 and then minus 4 - 4 and the Y and I'll let the Y scale be 1 and let's see what we look what this looks like so there's a nice simple graph that we've got out of this now remember what we found we found that the x-intercepts were to occur at plus or minus square root of 3 comma 0 so let's see what happens when we trace this I will hit trace now and we come down to the curve C hit trace I'm sorry let me hit the trace button there there I'm on the curve let me come up here to the right to the axis and what number is appearing there around X it's at 1.8 - right now if I go backwards a bit it's one point seven four go backwards a little bit more that's where I've got one point six six somewhere in there is square root of three if you remember square root of three is approximately one point seven three so it looks like that's really true that is at square root of three if I come on down to the bottom that's where it is crossing the y axis let's see if I can get that lined up directly there oh look that works out perfectly X is zero Y is minus three so the trace happens to give me the exact value this time so sometimes that will be that will occur sometimes it won't now I wanted to show you something else while I'm here we've looked at trace let's go back to zoom now there are some other features of zoom if we want to get a better approximation for square root of three for example let me look at soom in now let me go back to the graph before I do that and turn the trace on and let me put my cursor up here toward the right see the cursor is rising up there to about where I need it to be now that's about where square root of 3 is let me go back to the zoom key go over to the right come on down to zoom in and I will hit enter to make this happen so if I do that what has happened now I have zoomed in on the curve and the zoom in screen has centered on where the trace left the cursor is trace again and there's the cursor you see and if I now look at the cursor I've got numbers that are much closer look at that one point seven to one point seven oh if I go that way I go far to the right one point seven four all of those are closer than I did then those numbers were before if you continue to zoom in things will get better now let me show you an alternative way to zoom in zoom and come over to the right and go to zoom box this is one of my favorite aspects of the calculator I hit enter and notice nothing has happened well I have to set the box up this this feature allows me to set the box create an actual box that the curve will zoom in on now what I want to do is box the intersection of the curve with the x-axis so I have to choose the corners of the box I've put the cursor up there that's about where I want the corner to be a 1-1 corner of the box now to make that corner stay I hit enter now you don't see anything happening but watch what happens when I move the cursor when I go over to the right it draws a line behind it see that and if I come down it's creating a box now I create the box around the point I'm interested in and if I hit enter now the screen will zoom in on that box and give me perhaps a better approximation to that intersection that I had before so let's do that I hit enter and I have zoomed in on that box of course the picture looks more or less the same but if I hit trace now and I look around that point look at that stays around 1.73 even when I go to the left and the right because the scale is so fine at this point and of course you can zoom in as much as you want to get whatever scale you want so these are features of cow calculators that I wanted to point out to you at this point okay let's go back to the pad and I'll show you something else I have to warn you that I do abbreviations so let me go ahead and give you a couple of abbreviations so that you will understand what I'm saying my abbreviations I don't know how standard they are there are a couple that I'm you're going to see again and again here is one wrt what could that possibly mean I write wrt with the line under it what it is going to mean it means with here's the W respect there's the R - so I will say with respect to something and I'll use this abbreviation and I will also use eqn of course for equation we'll be using this word a lot and of course you've seen other abbreviations I use already but I wanted to point out these two because I'm going to start using those a little bit more [Music] we've looked at graphs of equations and intercepts crossing the axes let's go on to symmetry of graphs the idea behind symmetry of graphs before I draw anything is that if the graph is symmetric then you only have to draw one part of it and you realize that the other part or parts are identical to the first part so you save yourself a lot of work so let's go ahead and look at symmetry and there are three standard kinds of symmetry that we look at all of these will be a matter of noticing something and then writing down a definition that corresponds to what we notice so let me show you visually what's going to go on here and then we'll see if we can get an algebraic representation of that here is a graph I'm going to draw and if I'm lucky I will get this to look very symmetric there we go there's a graph now if you think of this graph as having symmetry around the x axis this is the x axis you can think of the x axis as a mirror where the graph above is reflected in the mirror to the graph below and so this is all one graph that has two halves and there is symmetry involved now how can we write down algebraically that there's symmetry it's nice to see that symmetry but I'd like to write it down algebraically so I can recognize it from the equation and so that I will then be able to draw only the top and reflect that across the bottom and save myself half the work well let's see what would a point on this graph look like well an arbitrary point is going to have coordinates X Y so this is X and this is a height of Y those are the coordinates for the point now because of the symmetry if I come down here below I know that this point is going to have coordinates first of all X has the same x-coordinate as the previous point but since I have symmetry across the x-axis if I've drawn this correctly then it goes down the same distance it went up which means the point here is X minus y so the point over here is minus y well that seems to be the key to recognizing this kind of symmetry XY lies on the curve if and only if X and minus y lie on the curve so when we describe this we say in English that this is a curve that is symmetric with respect to there's my little abbreviation the x-axis and you just need to think of the x-axis as a mirror now let me show you how the fact that X Y and X minus y must lie on the curve for this kind of symmetry to be present will allow you to look at the equation and then see that there's symmetry without ever drawing the picture so I'll even write that down so X comma Y and X comma minus y both satisfy the equation and there's that abbreviation I promised I'd start using so what is another way to say that another way to say that and this becomes the practical way of using this replacing Y by - why does not alter the equation that's the practical test that you will use so this is the actual test that you can apply to an equation you can look at the equation and everywhere Y occurs replace it by - Y if the equation as a result does not change you have an equation that is symmetric with respect to the x axis so the example I'm going to use here and carry on to the next sheet is X equal Y squared so let me carry that on X equal Y squared and let me see if this is symmetric with respect to the x axis I will replace Y by minus y now you have to be careful that you do this correctly the X doesn't change equals the Y up here is squared so when I replace the Y by minus y the minus y is also squared so you have to understand where the minus is going to go it simply is replacing the Y and the Y up here could have been thought of as Y squared like that so I'm now replacing the Y by minus y and I have to ask myself is this the same equation as the one above well of course it is this is X equal whether you square minus y or Y it's still Y squared because the minus part is minus 1 square that you get 1 so yes it is the same equation as the original so we have symmetry with respect to the x axis and if you look at the picture of this I'll draw it up here in small the actual graph of this looks like this and you can see the symmetry with respect to the x-axis there but this is a technique for checking symmetry without actually drawing the picture okay good that's symmetry with respect to the x-axis well there's another axis the y axis so let's likewise outline that procedure perhaps a little more quickly suppose we have a curve here's the x axis here's the y axis and now the Y axis will be the mirror and let me draw one if I can draw something that looks reasonably symmetric here there we go that's fairly symmetric and we ask ourselves what is it that makes this symmetric with respect to the y axis well let's take a point here X Y here's X here's the height Y and what is it that makes this symmetric well the point corresponding to this X Y across the mirror across the Y axis which would be over here has what coordinates well let's see it has the same y coordinate cuz it's the same height as the one to the right and the x coordinate is the same distance as it is to the right except that is now to the left so this is a coordinate minus X so this is minus X comma Y so the mirror makes symmetry from left to right and we described this of course as a curve that is symmetric with respect to the y axis and to give you an example well first of all I'll go ahead and outline the same procedure as before so so you'll have this down so X Y and minus X Y both both points satisfy satisfy the equation so here's the test replacing X by minus X doesn't doesn't alter the equation so there is the test for symmetry with respect to the y-axis and if we want to look at an example of this sort here's one take the equation X equal Y sorry y equals x to the fourth minus x squared plus one and I'll carry that to the next page and if you're thinking ahead of me here I will copy it again y equals x to the fourth minus x squared plus one I'm going to replace the X everywhere by minus X so here is the replacing replace X by minus X what do I get Y doesn't change X is replaced by minus X so that's minus X to the fourth now - this is where you be careful this is x squared I'm replacing the X by minus X and squaring it plus one and you see that minus X to the fourth minus x squared because those are even powers will be the same thing as X to the fourth and x squared so this is y equals x to the fourth minus x squared plus one so yes we do have symmetry with respect to the y axis and here's the picture you can check this out in your graphing calculator if you're looking for things to practice with the picture looks something like this it comes down here goes up here comes down here and goes up like that and you can clearly see the symmetry from side to side so there there are two types of symmetry there is a third and you may say well there are only two axes where is this third symmetry going to come from lastly let me give you an example of the symmetry I'm talking about this is a little bit different and perhaps unusual you're used to thinking of mirrors as a line or the mirror in your room that you look in the same thing with this y-axis you're thinking of a mirror from side to side now I'm going to talk about a mirror that consists of a single point in this case the origin and let me give you an example of a curve that one perhaps can see is symmetric in that sense there's a symmetry across the point zero zero here because on this curve if I take a point XY here and I follow it through the origin to where it has its corresponding point on the other part of the curve if this is X and this is y you can see that this point must be at minus y distance downward and minus X to the left so that it's coordinates are minus X and minus y and now you see why we've done this previously we replaced Y by minus y to get symmetry with respect to the x axis X by minus X to get symmetry with respect to the Y but if you replace both and you still have symmetry this is symmetry with respect to the origin this point in the middle so this is symmetry or symmetric with respect to the origin and the origin remember is the point zero zero so let me show you the procedure then for checking you know where this is headed so I will go ahead and just write it out we want to replace X by minus X and simultaneously Y by minus y to see if the equation is not altered if it is not altered then it is symmetric with respect to the origin and a classic example of that is this particular function y equals 1 over X now of course this only makes sense let me put in parentheses where X is not 0 because if X were 0 we'd have division by 0 and that doesn't make any sense that's undefined this curve will occur many times throughout this course it has lots of features we're going to discuss but let me go ahead and discuss its symmetry with respect to the origin y equal 1 over X I want to do the replacement now and the replacement is to replace both Y and X by minus y and minus X respectively so I'll put a minus y here equals 1 over minus X I just put the parentheses in to make sure you're clear that I'm replacing the X and y by their negatives so I have minus y equals 1 over minus X but you know that that's the same equation as the original because if I multiply both sides by minus 1 I have y equal 1 over X and so I haven't changed the equation so this equation is symmetric with respect to the origin its picture is so important I'm going to put it on its own sheet we will see this time and again so it's worth memorizing this graph the graph looks more or less like this it never touches the x or the y axis there are no X or Y intercepts neither here nor here here and are here these lines simply approach the x axis from both directions infinitely close but never touch the same thing with the y axis here this is the curve y equal 1 over X clearly undefined when x is 0 there's no point on the graph here and this is symmetric with respect to the origin because if you take any point over here you follow through that you can see there's a corresponding point there you take a point up here and follow it through there's a corresponding point there so the symmetry is clear and the fact that this has all of these interesting properties will be something that we deal with later [Music] let's look at another topic that deals with curves that are a little bit simpler let's go back to the list here we've looked at graphs of equations we looked at intercepts crossing the axes and symmetry of graphs now those are both techniques for producing graphs fairly easily let's now look at lines the simplest possible graph you call them straight lines perhaps in Euclid when you were studying plain geometry in your high school years but we'll just call them lines and what we are going to do is define a property of lines that you could not have in plain geometry this is a property that will be known as slope but let's go ahead and get started on this lines now in a plane like the plane and you studied in Euclidean geometry in a plane there are there is no measure of slanted Nisour a line because there are no references there are no reference points there's no horizontal there's no vertical however in a plane with a coordinate system like our plane with the rectangular coordinate system and in our case that's x and y axis lines have what you might refer to say if you were beginning to think about this as slanted nests and the slanted nests is with respect to the axes the idea is that if you have a line then the only way that you know it is slanted is if you have some reference line and you can somehow see that it is slanted with respect to the reference line and that's what we're going to discuss now slanted Ness is too long a term for one thing and it's not the term that is currently used but there are a variety of words that convey this idea that you've probably run into in your life here are some of them great if you've ever driven on a road you might have seen that the grade is often marked and that is simply the degree of slanted nosov the road you might have seen the word pitch carpenters use this when they talk about the roof a pitch of a roof for example or the pitch of a staircase that they're building that is simply a slanted nasaan ttle in europe slanted this is often referred to as gradient but the word that we will use and is most commonly used for this is slope and so since we have this idea let's go ahead and see if we can come up with a definition that will make us feel satisfied an actual mathematical definition for slope so now since I said we needed the references the x and y-axes here they are and let me draw a line here of some kind so this is a line and now I want to know how slanted is it with respect to these axes well the first thing is we're only going to compare it with the x axis the horizontal we can choose either axis we like but we're going to choose the x axis because that's conventional and to measure the slant to this I'm going to use a fact about lines that you may remember from geometry everything about a line is determined if you know two points on the line one point is not enough because the line could twist around that point but once you've picked out a second point the line is fixed that's the nature of lines so let's suppose we have two points on the line x1 y1 and let's call this one x2 y2 now those are the coordinates of those points then we've done this before when we talked about midpoint and the distance between points but the same kind of argument will now bear different fruit if we go horizontally here until we are right below the point x2 y2 and then we go vertically up to this point so say horizontally I'll put an arrow that way and then vertically up this way you can describe our path as follows you can say that this was a run horizontal distance traveled will be referred to as a run and this could be referred - as a rise now these are words that are very commonly used and you'll see them used again what exactly is the numerical value of the rise in this case and the numerical value of the run well that's easy to decide because we've done this argument before the coordinates of the corner point we know what is the x-coordinate here that corresponds to this point well it's the same as the x-coordinate of the point above so that's X sub 2 what is the y-coordinate at this point well it's the same height as this point over here so that must be Y sub 1 and then as we did before in the review or in the beginning of this units excuse me the distance between these two points is the x2 minus x1 values the rise is is similarly calculated it is the distance between these two points which refers to the Y values and that will be y2 minus y1 so now we've quantified the rise in the run and the slope the slanted nough swill be described in terms of these two numbers as you move along how high do you rise and that will determine how slanted you are now let me give you one warning rise is use interchangeably with what we might call fall if we had a line coming down this way and we went over to the right and then went down you'd probably want to call that fall but it is referred to also as rise so let me do this plus or minus rise is either a plus or a minus a number and that would take into account a fall because this is a line where there is a positive run and a negative rise okay getting back to what I've discovered up here I'm going to use these two numbers now and I'm going to describe what I call the slope of this line and here is the definition and I'll give you several different forms of it so that you'll see what it actually means I hope the slope of the line and of course I'm referring to the line I drew on the previous page is well first of all the commonly used letter for slope is M that seems to have occurred for historical reasons no matter it is a letter that's used so we will use it now I will describe it to be the vertical rise which remember could be plus or minus over the horizontal run some people just say rise over run but I wanted to emphasize the vertical and horizontal here now there's another way to say that the rise if you remember from the previous picture the rise was the difference between these two y-values you might say that in going from here to here the Y value has changed there was a change in Y and in going from here to here there was a change in X so I can describe the run as a change in X and the rise as a change in Y so I can write that down change in Y over change in X now there is a symbolism for that if you haven't seen this before here it is the Greek letter Delta is used for change in and we write Delta Y over Delta X so the words change in are translated into this symbol Delta that's a delta so change in Y over change in X this will lead you on to some of what we do later in college algebra but I'll also lead you on to calculus but there's one notation and finally to use the letters I used on the previous page what is the change in Y exactly it's y2 minus y1 over x2 minus x1 it is the difference of the Y's over the difference of the X's and let me point out an obvious algebraic fact if I reverse these two that's the same as multiplying this through by negative 1 isn't it because then this would become neck - why - positive y1 and I could write that is y1 minus y2 now if I multiply the bottom also by negative 1 I'd have x1 minus x2 but then I've multiplied the top on the bottom by negative 1 that's like multiplying the whole expression by 1 negative 1 over 1 is just 1 so this expression is the same as this one so all that matters is x2 and y2 are the first two numbers used and x1 y1 out of the second or the x1 y1 of the first and the x2 y2 or second as long as you're consistent you can calculate the slope either way now there's something else here that's hidden until you take a pause and look at this the slope here is defi defined as a quotient now there's a perennial problem with quotients the bottom of the quotient is always a problem because if the bottom of the quotient is ever zero the quotient is undefined so if we want to be careful here we have to say one other thing we have to say that x1 cannot be the same as x2 because if x1 and x2 are the same the bottoms become 0 now what would that mean graphically well let's see note when x1 equals x2 what happens graphically well let's see if we have a picture like this and this is that point x1 which is the same as x2 so that's the same x-value then if we have two points that have coordinates x1 y1 and x2 y2 and x1 and x2 are the same then we have a line that's vertical that's a vertical line that's what happens when X 1 and X 2 are equal now coming back to the definition of slope that means the slope is undefined if we have a vertical line why would that make sense well ask yourself if this is the x axis and this is the y axis just how slanted is this line in a sense it's infinitely slanted but we will say that it has no slope no slope because the slope requires that we have this difference on the bottom and that's 0 another way to say that is to say that it has an undefined slope or rather that the slope is undefined so no slope is indicated for vertical lines so they are a separate category of lines but apart from vertical lines every other line has a slope and just to repeat what we saw in the previous page any two points determine the slope of a line because if you remember we had slope was equal to the difference of the Y's over the difference of the X's and that only requires two points now just to give you a sense of what slope means in the picture remember when we drew the picture originally that we had a run that was positive because the x2 was greater than the x1 so this distance was positive and then going vertically we had a rise that was positive because the y2 and the y1 y2 is larger than the y1 so if you took the rise over the run here you take a positive number over a positive number and get a positive slope now since we move from left to right the X values are increasing as we go to the right remember we have 1 & 2 & 3 etcetera as we go from left to right it appears that when the slope is positive the line is rising to the right and that gives us the clue as to what the number slope will mean to align so let me go ahead and give you the summary of what that will be and then we'll look at some examples later M the slope if the slope is greater than 0 the line is going up and up for us means from right from left to right if M is equal to 0 you might say that this is flat or you might say horizontal either one of those is inaccurate description the line has no slope because think about it how can a slope be zero slope was determined to be y2 minus y1 over x2 minus x1 that was defined to be slope well the only way a fraction can be 0 is if the top is 0 and the only way the top can be 0 is if the y1 and the y2 are the same and if they're the same heights the line must be horizontal finally if M is less than zero this indicates a line that is going down from left to right and then let me put in the final case M undefined that was the case where we had division by zero here were x-two and x-one are the same and we've already decided that that meant you had a vertical line which I suppose I can draw like this so here are all the possibilities for slope positive slope indicates a line that grows toward the right slope equals zero is a line that is flat or horizontal less than zero means the slope means the line goes down and if the slope is undefined it's a vertical line all right let me do an example of a problem here just to give you a taste of how one might calculate with such things for example you might be asked the following graph the line through a particular point say the point 3 2 with slope say M is equal to minus 4/5 so you're given a point on which to anchor your line and then you're told a given direction since it's negative at some sort of direction going downward and with a point and a slope like that you think you ought to be able to draw the line now eventually we'll have an equation for all of this but right now we actually have to draw the picture so let's go ahead and do that in our solution the point we're going through is 3/2 so we need to have at least 3 and 2 marked 3 is the x value 2 is the Y value so there's the point 3 2 marked on the coordinate system this is the x-axis that's the y-axis and now how am I going to use the slope to get another point on the line because I need two points on the line to graph it well the slope is rise over run now let me point out something here that's algebraically simple but gives you options I've written it with the minus on the top I could also have written it as 4 over minus 5 so the minus could be on the bottom it is the same fraction of course this way I have a run of 5 and a rise of minus 4 which would be a fall here I have a rise of 4 and a run of -5 now my personal preferences to keep my runs positive so I'm going to go five to the right so I need to go one two three four five that will take me to eight from the point three two I go a distance five that's the run the rise is minus four so I'm going to drop down for now I'm two above the axis I will have to go to below to drop down four so this is minus four and this is the rise this is a negative rise now I have a point what point is that well let's see the x-coordinate is clearly eight the y-coordinate is to below that's minus two well now I have this point and I have this point and of course graphically I just need to connect the two and I have the line drawn so there's the graph of the line eventually we'll talk about the equation of this line now before we get to equations of lines is one last technical mode I'd like to make and I then will show you something on the calculator so let's call this a technical note you will find in your textbook and elsewhere reference to what are call square screens now what does that mean remember what the screen of a calculator looks like it's not unlike a television screen it is longer horizontally than it is vertically so if I do have an axis system and I mark off the same number of units on each direction I might find that my X units all of indicating a space of 1 look like that whereas in order to get the Y units in here the same number I have to squash them together a bit so maybe a Y unit of 1 looks like this now that it's a little hard to tell that's a little bit smaller than this that's not a square screen square means that it looks like a unit here and a unit here are identical now how would you create such a screen well it really requires that you know the ratio of the two sides the horizontal and the vertical sides now that's not always clear some calculators have a key that allow you to set the screen to either a square screen so a square screen setting or sometimes it's referred to as a default screen setting but not all calculators have that option but now you have to ask yourself is that really that important to me does it matter that the units on the x and the y axes look alike to me well sometimes it does if you're trying to draw a circle for example you like the circle to look like a circle and not be squashed in one direction but that really is an aesthetic preference it doesn't change the nature of the equation or the fact that this is a circle and all the points are equidistant from the center it just looks funny so four circles perhaps that's worthwhile for everything else that doesn't matter so much let me show you something on the calculator now that will illustrate how different the same function the same equation will look so let me go to the calculator now and turn this on you'll see I'm in the area where I enter the equation I have 3x entered that will turn out to be the equation of a line I've already picked the window and let me show you what the graph looks like now here's the equation of that line I am NOT going to change the equation of the line all I'm going to do now is change the window now remember how that looks it's a line that slants up it's a line that goes up to the right so the slope is a positive number of some kind let me go to the window and change one thing I will change the the X and the y value the x-min and x-max values excuse me so minus 10 for X min to 10 for X max notice again that I did not change the equation let's go back to the graph it looks like it's much steeper than it was before it's not it just looks that way because I've changed the window let me change the window one more time to make another dramatic point change the window I will change the X min to minus 1 and the X max to 1 and now let's look at the graph this graph is much shallower it looks like it is not nearly as steep as it was before but in all cases it's exactly the same equation so the moral is if your window is different your equation may appear to be different if there's a certain way you'd like this equation to look what you need to do is to pick a window that makes you feel comfortable but mathematically it doesn't make any difference when we come back from this little break we'll go and look at the equations of lines so for the moment let's take a pause [Music] you you

Channel: UMKC

Views: 53,626

Rating: 4.9061031 out of 5

Keywords: Graphs of equations, graphing on calculator, Intercepts, x-intercept, y-intercept, symmetry, Lines, slope, square screens.

Id: s437HDMl-ok

Length: 74min 15sec (4455 seconds)

Published: Mon May 04 2009