Graphing Trigonometric Functions

in this video I&#39;m going to show you how to graph the sine the cosine the tangent the secant the cosecant the cotangent is designed to be a complete guide so let&#39;s start at the beginning it will start simple and we&#39;ll get a little bit more challenging as we go through this video I&#39;ll also have time stamps in the description below if you want to jump to a particular section of the video so the first thing we want to talk about are how to graph the basic sine graph and cosine graph and then we&#39;re gonna get into the transformation shifting stretching compressing etc we&#39;re also going to reference the unit circle and I&#39;m gonna show you where these graphs come from so starting with the sine graph we&#39;re gonna graph y equals sine of X and if we go to a unit circle remember the when you want to find the sine value you look at the y coordinate of the point on that unit circle so when we think about sine we say okay the sine of 0 degrees or in this case 0 radians either one you can see the sine value is going to be 0 so I&#39;m going to put a point right here at 0 now at 45 degrees okay it&#39;s root 2 over 2 and 90 degrees it&#39;s 0 comma 1 so at 90 degrees you can see the sine value is going to be 1 so I&#39;m just gonna write PI over 2 or 90 degrees the Y value of the output this is going to be 1 at 180 degrees or PI the Y value is 0 so I&#39;m going to put a point here at 0 at 3 PI over 2 ok our 270 the Y value is negative 1 so we put a point down here at negative 1 and then at 2 pi we&#39;re back where we started which the Y value is 0 so this is your basic sine graph and you want to memorize this basic shape this is going to help you as you go on to do more challenging problems so it&#39;s I think of it as starting at this midline it goes up to the maximum back to the midline down to the minimum and then back to the midline and then it starts over and it keeps repeating like that you&#39;ve got a Maxima at 1 a minimum at negative 1 now what happens if you wanted to graph y equals 2 times sine of X well this number that comes in front this is what we call our amplitude a and that&#39;s going to be a vertical stretch of 2 so what happens is instead of going up to 1 now you&#39;re gonna go up to 2 instead of going down to negative 1 you&#39;re going to go down - negative - and so again that&#39;s just a vertical stretch but it&#39;s still going to cross the x-axis at those same points so now see if we want to go to a slightly more challenging problem sailing number three we have y equals sine of 2x this number here okay this is B value the letter that comes or the variable or the constant I should say that comes before the variable ax that B value affects the period and we can use this formula period equals to PI over B to find out what that new period is so in this case the B value is 2 if we take 2 pi divided by 2 that&#39;s just going to be PI so our period is going to be PI what I like to do is I kind of like to mark the x-axis the scale first if the period is PI I&#39;m going to divide it up into four pieces just so we did here one two three four so pi divided by four gives us our scale that&#39;s gonna be PI over four to PI over 4 3 PI over 4 4 PI over 4 okay so 2 PI over 4 and 3 PI over 4 and 4 PI over 4 is PI we can see that the amplitude is 1 because 1 times anything is itself so it hasn&#39;t been stretched vertically at all and we haven&#39;t talked about the shifts yet that&#39;s the h and the K that&#39;s gonna pick up the graph move it left and right up and down so here we can see that the graph is just gonna have this basic shape again it starts at the midline it goes up to the maximum which is one back to the midline back down to the minimum which is negative one back to the midline the only thing that was different here is that the period changed it was two pi divided by two which is pi notice though that this number here it has the opposite effect on the graph instead of multiplying X&#39;s by two we actually divided the X&#39;s by two normally the period for sine and cosine of two pi here it would divided X&#39;s by two new period was just 1 pi so let&#39;s keep going so let&#39;s go to number four here now we&#39;re getting into y equals sine of X plus PI minus 2 so here you can see that the B value is 1 so which means that our period is 2 pi divided by 1 which is just 2 pi so let&#39;s go ahead and Mark our scale here at 2 pi divided up into four pieces one two three four if you take two pi divided by four that comes out to two PI over four which reduces to PI over 2 to PI over 2 which is PI 3 PI over 2 4 PI over 2 so I&#39;m just taking this PI over 2 by two or by 3 or by four and that&#39;s how I&#39;m getting the scale on the x-axis the amplitude is 1 so it&#39;s not being stretched vertically at all so that means it&#39;s gonna go up to a maximum of 1 and down to a minimum negative 1 but the number this group with the ax that&#39;s going to shift the graph in the X direction it has the opposite effect so plus pi is actually in a shift the graph left pi okay negative pi and then the minus 2 has the same effect that&#39;s gonna be a vertical shift the minus 2 is actually gonna shift it the same down to okay as the sine so what&#39;s gonna happen here is we can think of this graph is shifting left pi now notice how our scale is PI over 2 so it&#39;s a shift left PI that&#39;s actually like shifting left 2 steps 1 2 so here over here at negative PI and we&#39;re shifting down 2 so that&#39;s gonna be 1/2 so that we can think of this point right here is like our new origin like our new starting point okay so I kind of drew like a new y axis and a new x axis and so what I&#39;m going to do now is I&#39;m just gonna graph from here so I&#39;m starting at the midline I go up to the maximum which we said was gonna be you know 1 okay because it&#39;s not being stretched at all back to the midline back to the minimum and then back to the midline so that&#39;s our basic graph the only thing we did is we shifted everything left to and down to and we thought of that as our new starting point okay let&#39;s talk about cosine we&#39;re gonna get into some more challenging examples involving the amplitude changing the period changing as well as a phase shift left and right and a vertical shift up and down but let&#39;s talk a little bit about cosine first so when we talk about cosine we go to our unit circle the cosine we&#39;re talking about the x coordinates when we find that cosine value so cosine of 0 degrees or 0 radians is going to be 1 so with cosine at 0 we start at 1 at PI over 2 or 90 degrees the x coordinate is 0 so we go back down to 0 at PI which is here at 180 degrees the cosine is x coordinates negative 1 and then at 3 PI over 2 the x coordinate is 0 and then at 2 pi you&#39;re back to where you started the cosine is x coordinate you&#39;re back to one so you can see our basic cosine graph has that basic shape like that it starts at the maximum it goes to the midline down to the minimum back to the midline and then back up to the maximum and then it repeats okay so these are called sinusoidal graphs they have like that S shape so now what would the graph look like if you had y equals negative cosine of X well the negative what it does is it reflects the graph over the x axis so all the X values are going to be I&#39;m sorry all the Y the hell is gonna be the opposites the X values are gonna be the same so instead of being at 1 they&#39;re gonna be at negative 1 0 times negative 1 is still 0 negative 1 times negative 1 is positive 1 0 times negative 1 is 0 and 1 times negative 1 its negative 1 again we&#39;re just reflecting it so this graph is gonna look something like this dotted line here it&#39;s just been flipped again you want to memorize the basic shape so cosine has that basic shape sine has this basic shape and let&#39;s go to number 7 now y equals cosine of 1/2 X so just like sine we look at that B value that number that comes just in front of her to the left of the variable X and we can see that that B value is 1/2 if we want to find the period we just say Q B divided by B which is 1/2 when you divide by a fraction it&#39;s really like multiplying by the reciprocal so instead of multiplying dividing by 1/2 I&#39;m multiply by 2 over 1 which gives us a period of 4 pi so what I like to do is I like to mark my x-axis with the period 4 pi and I divide it up into 4 pieces just like the four quadrants 1 2 3 4 so our scale is going to be pi then times 2 is 2 pi and then times 3 is 3 PI 4 PI so just took 4 pi divided by 4 and that&#39;s what I&#39;m counting by to get to each successive point on the x-axis our amplitude is 1 because 1 times anything is itself and we know cosine has that basic shape where it starts at the maximum back to the midline down to the minimum back to the midline and then back to the maximum and that&#39;s your basic graph the only thing that changed here was instead of dividing X&#39;s by 2 we actually did the opposite we multiply the XS by 2 that&#39;s why our period is 4 pi instead of the normal period of 2 pi so let&#39;s go to number 8 now we&#39;ve got y equals cosine of X - PI over 2 plus 1 you can see the amplitude is 1 the period is going to be the same 2 pi because 2 pi divided by the B value here is 1 it&#39;s just going to be 2 pi but what&#39;s happening is a graph is shifting right PI over 2 remember the one group of the X has the opposite effect - PI over 2 you&#39;re actually going to write PI over 2 and then plus 1 is gonna shift up 1 so what I&#39;m gonna do is I&#39;m just going to label my x-axis to PI divided up into 4 pieces that&#39;s going to be PI over 2 pi 3 PI over 2 and 2 pi and we can see we haven&#39;t it&#39;s not being stretched at all in the vertical direction but what&#39;s happening is we&#39;re shifting right PI over 2 up 1 we can think of that as a new starting point or like our new origin okay it&#39;s a cosine graph remember the cosine starts at the maximum so you&#39;re gonna start up here you&#39;re gonna go back to the midline down to the minimum back to the midline and then back to the maximum so that&#39;s your basic graph right there now notice the amplitude is 1 so from that starting point that&#39;s why it went up 1 and then the minimum I went down 1 from the midline and again you have that basic shape it had a normal period of 2 pi it was just everything was being shifted right PI over 2 and up 1 let me erase the whiteboard I&#39;ve got some more challenging problems involving all of the different components a B H and K let&#39;s look at those next and then we&#39;ll jump into graphing the cosecant the secant and then the tangent the cotangent okay number nine let&#39;s look at y equals 3 sine of 1/2 X minus PI minus 2 so we&#39;ve got a little bit of everything in this problem the first thing I like to do though is I like to look at the scale on the x axis the horizontal stretch or compress so let&#39;s figure out what the period is we&#39;ve got the B value of 1/2 so 2 pi divided by 1/2 looks like let&#39;s see so 2 pi divided by 1/2 we know when we divide by a fraction it&#39;s like multiplying by the reciprocal so that&#39;s going to be like 2 pi times 2 over 1 which is 4 pi so we&#39;re gonna go ahead and Mark our scale we&#39;ve got 4 pi divide that by 4 we get 1 pi 2 pi 3 PI 4 PI but you can see that the graph is shifting left pi I&#39;m sorry - pi is actually having the opposite effect it&#39;s shifting right pi and then down to so we can think of this as our new starting point right PI down to and that&#39;s that&#39;s your point right there now you can see the amplitude is 3 so that&#39;s a vertical stretch of 3 and so and we know we have a sine graph so we can see that from this point it&#39;s going to start here it&#39;s going to go up 3 so let&#39;s see when you&#39;re over here you&#39;re gonna go up 3 1 2 3 that&#39;s the maximum back to the midline then down to the minimum 1 2 3 which is gonna be right down here and then back to the midline and you&#39;ve got one period of your sine graph again notice that the one that&#39;s grouped with the ax that&#39;s going to be the phase shift it&#39;s gonna have the opposite effect on minus pi shifts right pi this is your vertical shift ur came out it&#39;s shifting down to think of that as your starting point pay attention to the whether it&#39;s a sine graph or a cosine graph and then the amplitude is a vertical stretch so think of it as your new starting point science starts at that that origin goes up to the maximum but that maximum is going to be 3 units from that midline same thing with the minimum it&#39;s going to be 3 units from the midline and you&#39;ve got it so let&#39;s look at another example of involving all the different components here number 10 y equals 2 times cosine of the quantity 4x plus pi plus 1 now what&#39;s different about this problem is see how there&#39;s a 4 okay this B value it&#39;s inside of the parenthesis it&#39;s grouped with the ax what we need to do is we need to factor it out and when you factor it at the 4 out you&#39;re gonna have to factor it out of both of these terms so we&#39;re dividing out 4 so our equation is going to look like this now y equals 2 times cosine of 4x plus PI over 4 notice if you distribute that 4 back in you get 4x plus 4 PI over 4 which is 1 pi so that&#39;s the key you definitely want to have just one X inside of the parentheses so you have to factor out that coefficient so you can see here that the B value is 4 so if we take 2 pi divided by 4 that reduces to PI over 2 so that&#39;s our hour period PI over 2 if we divide that by 4 or multiply by 1/4 either way you&#39;re going to get PI over 8 that&#39;s our scale so 1 PI over 2 PI over 8 3 PI over 8 4 PI over 8 we can see that our graph the shifting left pi over 4 because remember this has the opposite effect on the graph so it&#39;s going to go left PI over 4 so PI over 4 is actually two steps so left PI over 4 because PI over 8 times 2 is 2 PI over 8 which is PI over 4 and then plus 1 it&#39;s going to shift up 1 so that&#39;s our new starting point we can think of that as our new y-axis our new x-axis or new origin it&#39;s a cosine graph with an amplitude of 2 so from here it&#39;s going to start at the maximum so I&#39;m going up to back to the midline down to the minimum so that&#39;s going to be down to back to the midline and then back to the maximum up to so that&#39;s going to be our basic cosine shape the only difference is that it&#39;s being stretched it&#39;s also being shifted and the period is changing as well so that was a good example so now what we&#39;re gonna do is we&#39;re going to transition over to working with the secant and the cosecant graphs so just a quick refresher when you think about the cosecant it&#39;s really the reciprocal of the sine graph see 1 over sine and when you think of the secant a function it&#39;s the reciprocal of the cosine ok function so 1 over cosine so we&#39;re going to use that information to graph these graphs so when we look at y equals 2 secant of X we can think of this as 2 over cosine of X now let&#39;s start a little bit simpler let&#39;s just say we wanted to graph y equals not cosine of X but 1 over cosine of X which is equal to secant of X so let&#39;s look at our basic values on the unit circle so when we think of cosine we said cosine starts at the maximum it goes to the midline then to the minimum dent in the midline back to the maximum that&#39;s the basic shape right this is 1 this is negative 1 this is PI over 2 pi 3 PI over 2 and 2 pi right but here&#39;s the thing when you&#39;re at 0 see how the cosine value is 1 so if this is equal to 1 right here 1 divided by 1 is still 1 right when you&#39;re over here at 0 PI over 2 see this is 0 what&#39;s 1 divided by 0 well that&#39;s undefined you can&#39;t divide by 0 so what happens is we get a vertical asymptote here when you&#39;re at PI cosine is equal to negative one what&#39;s one divided by negative one that&#39;s still negative one and then when you&#39;re over here at 3 PI over 2 that&#39;s 0 again 1 divided by 0 is undefined you get another vertical asymptotes wherever the graph is crossing the x-axis that&#39;s where your vertical asymptotes are going to be now notice when I&#39;m at this point say for example right here say for example this is like a cosine value of 1/2 well 1 divided by 1/2 when you divide by a fraction slike multiplied by the reciprocal so 1 divided by 1/2 is really going to be like two you&#39;re up here so you can see this graph is looking like this now when you&#39;re over here let&#39;s say like that when cosine is negative 1/2 1 divided by negative 1/2 is going to be like negative 2 right so this graph is looking something like this etc so what you can see is the coasts cosine graph if you graph that and you look at these key points where it reaches a maximum or a minimum that&#39;s going to be where the secant graph bends or turns okay so I&#39;ll show you another example so say for example this one y equals 2 secant of X what you would do is you would start by graphing y equals 2 cosine of X so cosine of X looks like this it has an amplitude of 2 okay has a period of 2 pi like that wherever it crosses the x-axis those are where our asymptotes are going to be and then the secant graph is going to have the shape where it goes the opposite direction like that okay and again the reason it works is because see how when this value is let&#39;s say for example 1 well then you have 2/1 which is actually going to be 2 right and then same thing over here when this is like 1/2 2 divided by 1/2 that&#39;s going to be 4 so you can see that as this gets smaller and smaller the cosine graph the seeking graph section here it&#39;s going to be larger and larger because when you divide by a small and smaller fraction you&#39;re multiplied by the reciprocal this is going to go up and up and up so that&#39;s the same idea when you&#39;re looking with the cosecant graph remember cosecant is the reciprocal of sine so if we wanted to graph this graph let&#39;s just graph y equals 3 sine of PI over 4 times X the sine graph we know looks like this ok the amplitude is 3 so it&#39;s gonna reach a maximum here at negative three and I&#39;m positive three and down to negative three wherever it crosses the x-axis that&#39;s where our vertical asymptotes are okay sometimes when you&#39;re doing this secant and cosecant graphs that helps to draw you know a little bit more cycles okay more periods and then what you can do now is you can say our graph is going to look like this now the one thing that I haven&#39;t done yet is look at the period so let me do that on last year usually I do it before I graph this but I would take the PI over 4 remember the period is 2 pi divided by PI over 4 okay 2 pi divided by B which is like multiplying by the reciprocal so the PI&#39;s cancel and it&#39;s actually going to have a period of 8 so you can see this is going to be right here it&#39;s going to be 8 divided up into 4 pieces 2 4 6 8 that&#39;s our scale and you&#39;ve got your cosecant graph okay let&#39;s do a little more challenging one number 13 we&#39;ve got y equals 4 secant 1/4 X plus 2 pi minus 1 so that&#39;s a lot right but what we&#39;re gonna do first is we&#39;re gonna graph the same equation here except for instead of secant we&#39;re gonna think of it as cosine okay if we take the period 2pi divided by b which is 2pi divided by 1/4 when you divide by 1/4 it&#39;s like multiplied by the reciprocal that&#39;s going to be 8 pi so our period is going to be 8 pi here divide it into 4 pieces that&#39;s gonna be 2 pi 4 PI 6 pi 8 pi okay but the graph is shifting left 2 pi down 1 so we&#39;re gonna think of this as our like our new origin like right like that okay and the amplitude is 4 it&#39;s a cosine graph so we&#39;re going to go up 4 so this is going to be 1 2 3 4 so it&#39;s going to start right here it&#39;s going to go back down to the midline okay and then it&#39;s going to go down to the minimum which is going to be right here at down 4 from the midline then back to the midline and then back to the maximum okay which is 4 above the midline because the amplitude here is 4 so that&#39;s going to be your basic C your basic graph something like that okay and it&#39;s going to continue okay but what we need to pay attention to is this cosine graph crosses the midline okay so it&#39;s crossing basically like this so it&#39;s crossing right here it&#39;s crossing right here okay it&#39;s crossing right here etc so our secant graphs actually going to look like this it&#39;s going to approach the asymptotes but it&#39;s not going to cross them or touch them so that&#39;s the basic idea and you&#39;ve got it so let me erase the whiteboard we&#39;re going to talk about tangent and cotangent next okay now switching gears over to tangent and cotangent when you graph tangent we&#39;re going to be using the values from negative PI over 2 to positive PI over 2 and remember from the unit circle the tangents the Y value divided by the x value so let&#39;s just go ahead and graph the basic parent graph first and what I&#39;m going to do is I&#39;m going to start over here at negative PI over 2 and tangent is Y divided by X now negative 1 divided by 0 we can&#39;t divide by zero that&#39;s undefined so we&#39;re gonna get a vertical asymptote right here at negative PI over 2 now at negative PI over 4 tangents Y over X anything divided by itself is 1 but we have a negative divided by positive which is negative 1 so over here at negative PI over 4 this is going to be right here at negative 1 at 0 Y divided by X is 0 divided by 1 is going to be 0 so that&#39;s gonna be another key point at PI over 4 which is 45 degrees tangent is y over X of course anything divided by itself is going to be 1 and then at PI over 2 tangent is y over X 1 over 0 we can&#39;t divide by zero that&#39;s undefined we get another vertical asymptote so that&#39;s one period and what&#39;s interesting about tangent as opposed to sine and cosine is that the period is 1 pi divided by B naught 2 pi divided by B so you can see that from negative PI over 2 to positive PI over 2 this is actually a period of pi and then what happens is this graphs going to repeat like that same thing over on this side as well and just going to keep repeating like that but just 1 period one cycle from negative PI over 2 to positive PI over 2 but for number 15 what do you think this 2 does to the graph well you can see the two is this coefficient in front here and it&#39;s going to be like a vertical stretch if it&#39;s greater than one or a vertical shrink if it&#39;s between zero and one of course if it&#39;s negative that&#39;s going to reflect it over the x-axis so if I was to graph a number 15 on the same graph what would happen is two times zero is still zero one times two is going to give us two and negative 1 times 2 is going to give us negative two so what would happen here is the graph would actually be something like that where it&#39;s being stretched it&#39;s going up faster two times faster okay for number 16 we&#39;ve got y equals tangent of 1/2 X what do you think the 1/2 dust of the graph well you can see that this number here is in front of our variable ax that&#39;s our B value pi divided by 1/2 remember when you divide by a fraction it&#39;s like multiplying by the reciprocal so that means that this is period is actually going to be not 1/2 but actually doubled it&#39;s actually gonna be 2 pi now when you graph a tangent an important thing to realize is that see how half the graph is to the left of the y-axis and half the graph is to the right so when you figure out what the period is like in this case we&#39;ve figured out the period is 2 pi I&#39;m gonna cut that period in half so 1/2 of 2 pi is going to be 1 pi and I&#39;m gonna put negative pi so from negative pi to pi that&#39;s actually a period of 2 pi right so I&#39;m going to go ahead and sketch in our asymptotes okay just as dotted lines and then what I like to do is I like to cut this in half again so half of Pi is going to be PI over 2 some people call these like the halfway points or the midway points and negative PI over 2 ok so now what we&#39;re gonna do is you can see that this coefficient in front of the tangents just 1 so we&#39;re just going to be going at these halfway points up to 1 and then down to negative 1 and it goes right here through the origin and we&#39;ve got a good sketch of our graph so remember the one that&#39;s grouped with the ax actually has the opposite effect it looks like we&#39;re dividing by two we&#39;re actually multiplying the X values by two and that&#39;s stretching it horizontally let&#39;s look at another another example number 17 this one has more going on here the first thing I like to do is figure out what the period is so I would look at that coefficient that B value I&#39;d say well let&#39;s see pi divided by 2 that&#39;s PI over 2 but again because half&#39;s on the left and half on the right side of the y-axis what we&#39;re gonna do is we&#39;re going to cut it in half again making this PI over 4 and negative PI over 4 which gives us a period of PI over 2 you can cut it in half again PI over 8 and negative PI over 8 and then now let&#39;s look at the phase shift okay the left and right and the vertical shift up and down so the number of this group with the ax remember this has the opposite effect the minus PI over 8 is actually going to shift it to the right PI over 8 okay pause and PI over 8 and this one has the same effect the pulse one&#39;s gonna shift it up 1 but what am i doing this one a little bit different than some of the other problems that I&#39;ve done in this video is let&#39;s just graph what the original graph would look like without the shift so we would have asymptotes right here okay like this and like this but now what&#39;s going to happen is it&#39;s going to go up here to one it&#39;s going to go down here to negative one and through the origin but what the PI over 8 is going to do is going to shift it right PI over 8 so each of these points is going to move right PI over 8 and then up one so this point would go right here and right here this point is gonna go right an up one and this point is gonna go right and up one like that so now you can see our graph is looking something like this okay and where our asymptote was here at negative PI over 4 is now I&#39;m going to shift right PI over 8 so it&#39;s going to be right here and the one that was over here at PI over 4 is now going to be one step over here at 3 PI over 8 and so now you&#39;ve got your your tangent graph so it shifted right PI over 8 up one okay now let&#39;s take a look at cotangent so when you think about cotangent you&#39;re thinking on the unit circle that&#39;s actually the x-coordinate divided by the y-coordinate and the unit circle values we&#39;re going to use for cotangent are from zero to pi okay zero to 180 so cotangent of zero is going to be 1 over 0 which is undefined so right here at zero we&#39;re gonna have an asymptote okay vertical asymptote at PI over 4 X divided by Y is going to be 1 because anything divided by itself is 1 and then over here at PI over 2 X divided by y 0 divided by 1 is going to be 0 okay and then over at 3 PI over 4 X divided by Y is going to be negative 1 and then over here at PI X divided by Y negative 1 divided by 0 is undefined so we get another vertical asymptote like that so you can see the graph is going down to the right ok whereas with tangent it was going up to the right and the other interesting thing again is that the tangent half of the graph is to the left of the y-axis 1/2 is to the right unless it&#39;s been shifted whereas with cotangent you can see that that whole graph is to the right of the y-axis okay but again the graph repeats so there&#39;s going to be another branch over here ok another branch over here like that and it&#39;s going to keep repeating but for one period one cycle from 0 to pi to the right of the y axis going down to the right is your basic graph also get a couple more challenging cotangent graphs ok for number 19 we have y equals 3 cotangent PI over 2 times X so first thing I like to do is to figure out what the period is using this formula so for tangent and cotangent we&#39;re going to be using the period equals pi divided by B for sine cosine secant and cosecant it&#39;s two pi divided by B so if we do PI divided by PI over 2 that&#39;s really the same as pi times 2 over pi because when you divide by a fraction you&#39;re multiplying by the reciprocal notice the PI&#39;s cancel and we get a period of two so there&#39;s no pi in the period they&#39;re just two so what I&#39;m gonna do is I&#39;m gonna label this as two and if I divide this by four cutting it up into four pieces that&#39;s going to be one half one and one and a half okay and so now you can see our asymptotes are gonna be right here at two and zero okay just like our parent function right here okay hasn&#39;t been shifted at all but what does the three do to the graph well the three is going to be a vertical stretch so this point here which is normally at one is now going to be at three so right there crosses at the midline here and then this point here which is only a negative one is going to be multiplied by three it can be a negative 3 1 2 negative 3 so you can see this was just a vertical stretch and then the period had changed using the PI over 2 so let&#39;s look at number 20 now a little bit more challenging still we&#39;ve got y negative cotangent 1/4 the quantity X minus pi minus 1 okay so there&#39;s a lot involved in this one the negative is going to reflect it over the x axis that&#39;s going to make the Y value is the opposite sign right but let&#39;s figure out what our period is I always do that first so I do pi divided by B now the B is 1/4 but when you divide by a 1/4 it&#39;s like multiplied by the reciprocal so it&#39;s actually gonna be 4 pi so let&#39;s label this 4 pi let&#39;s divide it up into 4 pieces 1 pi 2 pi 3 PI 4 PI and then now let&#39;s see so normally cotangent is going down to the right right but the negative is going to reflect it so now it&#39;s going to be going up to the right so let&#39;s kind of just draw our basic graph here we&#39;re to do it in two steps so normally a cotangent would be here it&#39;s gonna reflect it&#39;s gonna be down here at negative 1 crosses at the midline and then it goes up to positive 1 and then you have your other asymptotes like that so this would be your graph like that right but now what happens here the minus Pyatt and the minus 1 well this is actually had the opposite effect it&#39;s actually going to go right pi then the minus 1 is going to go down 1 so we go right pi at this point and down 1 okay that&#39;s going to be right here this point is going to be right pi and down 1 right here and right pi and down 1 right here now the up and down isn&#39;t gonna affect the vertical asymptotes but the right pi is so normally the asymptotes here it&#39;s gonna go right pi so now it&#39;s going to be right here and then the one that was at 4 pi is now gonna be here at 5 pi and so now you can see there&#39;s your cotangent graph so if you&#39;re enjoying these videos subscribe to the channel my channel is all about making learning math less stressful so you can raise your grade pass your class and go on to pursue your dreams I look forward to helping in the future videos I&#39;ll talk to you soon

in this video I&amp;#39;m going to show you how to graph the sine the cosine the tangent the secant the cosecant the cotangent is designed to be a complete guide so let&amp;#39;s start at the beginning it will start simple and we&amp;#39;ll get a little bit more challenging as we go through this video I&amp;#39;ll also have time stamps in the description below if you want to jump to a particular section of the video so the first thing we want to talk about are how to graph the basic sine graph and cosine graph and then we&amp;#39;re gonna get into the transformation shifting stretching compressing etc we&amp;#39;re also going to reference the unit circle and I&amp;#39;m gonna show you where these graphs come from so starting with the sine graph we&amp;#39;re gonna graph y equals sine of X and if we go to a unit circle remember the when you want to find the sine value you look at the y coordinate of the point on that unit circle so when we think about sine we say okay the sine of 0 degrees or in this case 0 radians either one you can see the sine value is going to be 0 so I&amp;#39;m going to put a point right here at 0 now at 45 degrees okay it&amp;#39;s root 2 over 2 and 90 degrees it&amp;#39;s 0 comma 1 so at 90 degrees you can see the sine value is going to be 1 so I&amp;#39;m just gonna write PI over 2 or 90 degrees the Y value of the output this is going to be 1 at 180 degrees or PI the Y value is 0 so I&amp;#39;m going to put a point here at 0 at 3 PI over 2 ok our 270 the Y value is negative 1 so we put a point down here at negative 1 and then at 2 pi we&amp;#39;re back where we started which the Y value is 0 so this is your basic sine graph and you want to memorize this basic shape this is going to help you as you go on to do more challenging problems so it&amp;#39;s I think of it as starting at this midline it goes up to the maximum back to the midline down to the minimum and then back to the midline and then it starts over and it keeps repeating like that you&amp;#39;ve got a Maxima at 1 a minimum at negative 1 now what happens if you wanted to graph y equals 2 times sine of X well this number that comes in front this is what we call our amplitude a and that&amp;#39;s going to be a vertical stretch of 2 so what happens is instead of going up to 1 now you&amp;#39;re gonna go up to 2 instead of going down to negative 1 you&amp;#39;re going to go down - negative - and so again that&amp;#39;s just a vertical stretch but it&amp;#39;s still going to cross the x-axis at those same points so now see if we want to go to a slightly more challenging problem sailing number three we have y equals sine of 2x this number here okay this is B value the letter that comes or the variable or the constant I should say that comes before the variable ax that B value affects the period and we can use this formula period equals to PI over B to find out what that new period is so in this case the B value is 2 if we take 2 pi divided by 2 that&amp;#39;s just going to be PI so our period is going to be PI what I like to do is I kind of like to mark the x-axis the scale first if the period is PI I&amp;#39;m going to divide it up into four pieces just so we did here one two three four so pi divided by four gives us our scale that&amp;#39;s gonna be PI over four to PI over 4 3 PI over 4 4 PI over 4 okay so 2 PI over 4 and 3 PI over 4 and 4 PI over 4 is PI we can see that the amplitude is 1 because 1 times anything is itself so it hasn&amp;#39;t been stretched vertically at all and we haven&amp;#39;t talked about the shifts yet that&amp;#39;s the h and the K that&amp;#39;s gonna pick up the graph move it left and right up and down so here we can see that the graph is just gonna have this basic shape again it starts at the midline it goes up to the maximum which is one back to the midline back down to the minimum which is negative one back to the midline the only thing that was different here is that the period changed it was two pi divided by two which is pi notice though that this number here it has the opposite effect on the graph instead of multiplying X&amp;#39;s by two we actually divided the X&amp;#39;s by two normally the period for sine and cosine of two pi here it would divided X&amp;#39;s by two new period was just 1 pi so let&amp;#39;s keep going so let&amp;#39;s go to number four here now we&amp;#39;re getting into y equals sine of X plus PI minus 2 so here you can see that the B value is 1 so which means that our period is 2 pi divided by 1 which is just 2 pi so let&amp;#39;s go ahead and Mark our scale here at 2 pi divided up into four pieces one two three four if you take two pi divided by four that comes out to two PI over four which reduces to PI over 2 to PI over 2 which is PI 3 PI over 2 4 PI over 2 so I&amp;#39;m just taking this PI over 2 by two or by 3 or by four and that&amp;#39;s how I&amp;#39;m getting the scale on the x-axis the amplitude is 1 so it&amp;#39;s not being stretched vertically at all so that means it&amp;#39;s gonna go up to a maximum of 1 and down to a minimum negative 1 but the number this group with the ax that&amp;#39;s going to shift the graph in the X direction it has the opposite effect so plus pi is actually in a shift the graph left pi okay negative pi and then the minus 2 has the same effect that&amp;#39;s gonna be a vertical shift the minus 2 is actually gonna shift it the same down to okay as the sine so what&amp;#39;s gonna happen here is we can think of this graph is shifting left pi now notice how our scale is PI over 2 so it&amp;#39;s a shift left PI that&amp;#39;s actually like shifting left 2 steps 1 2 so here over here at negative PI and we&amp;#39;re shifting down 2 so that&amp;#39;s gonna be 1/2 so that we can think of this point right here is like our new origin like our new starting point okay so I kind of drew like a new y axis and a new x axis and so what I&amp;#39;m going to do now is I&amp;#39;m just gonna graph from here so I&amp;#39;m starting at the midline I go up to the maximum which we said was gonna be you know 1 okay because it&amp;#39;s not being stretched at all back to the midline back to the minimum and then back to the midline so that&amp;#39;s our basic graph the only thing we did is we shifted everything left to and down to and we thought of that as our new starting point okay let&amp;#39;s talk about cosine we&amp;#39;re gonna get into some more challenging examples involving the amplitude changing the period changing as well as a phase shift left and right and a vertical shift up and down but let&amp;#39;s talk a little bit about cosine first so when we talk about cosine we go to our unit circle the cosine we&amp;#39;re talking about the x coordinates when we find that cosine value so cosine of 0 degrees or 0 radians is going to be 1 so with cosine at 0 we start at 1 at PI over 2 or 90 degrees the x coordinate is 0 so we go back down to 0 at PI which is here at 180 degrees the cosine is x coordinates negative 1 and then at 3 PI over 2 the x coordinate is 0 and then at 2 pi you&amp;#39;re back to where you started the cosine is x coordinate you&amp;#39;re back to one so you can see our basic cosine graph has that basic shape like that it starts at the maximum it goes to the midline down to the minimum back to the midline and then back up to the maximum and then it repeats okay so these are called sinusoidal graphs they have like that S shape so now what would the graph look like if you had y equals negative cosine of X well the negative what it does is it reflects the graph over the x axis so all the X values are going to be I&amp;#39;m sorry all the Y the hell is gonna be the opposites the X values are gonna be the same so instead of being at 1 they&amp;#39;re gonna be at negative 1 0 times negative 1 is still 0 negative 1 times negative 1 is positive 1 0 times negative 1 is 0 and 1 times negative 1 its negative 1 again we&amp;#39;re just reflecting it so this graph is gonna look something like this dotted line here it&amp;#39;s just been flipped again you want to memorize the basic shape so cosine has that basic shape sine has this basic shape and let&amp;#39;s go to number 7 now y equals cosine of 1/2 X so just like sine we look at that B value that number that comes just in front of her to the left of the variable X and we can see that that B value is 1/2 if we want to find the period we just say Q B divided by B which is 1/2 when you divide by a fraction it&amp;#39;s really like multiplying by the reciprocal so instead of multiplying dividing by 1/2 I&amp;#39;m multiply by 2 over 1 which gives us a period of 4 pi so what I like to do is I like to mark my x-axis with the period 4 pi and I divide it up into 4 pieces just like the four quadrants 1 2 3 4 so our scale is going to be pi then times 2 is 2 pi and then times 3 is 3 PI 4 PI so just took 4 pi divided by 4 and that&amp;#39;s what I&amp;#39;m counting by to get to each successive point on the x-axis our amplitude is 1 because 1 times anything is itself and we know cosine has that basic shape where it starts at the maximum back to the midline down to the minimum back to the midline and then back to the maximum and that&amp;#39;s your basic graph the only thing that changed here was instead of dividing X&amp;#39;s by 2 we actually did the opposite we multiply the XS by 2 that&amp;#39;s why our period is 4 pi instead of the normal period of 2 pi so let&amp;#39;s go to number 8 now we&amp;#39;ve got y equals cosine of X - PI over 2 plus 1 you can see the amplitude is 1 the period is going to be the same 2 pi because 2 pi divided by the B value here is 1 it&amp;#39;s just going to be 2 pi but what&amp;#39;s happening is a graph is shifting right PI over 2 remember the one group of the X has the opposite effect - PI over 2 you&amp;#39;re actually going to write PI over 2 and then plus 1 is gonna shift up 1 so what I&amp;#39;m gonna do is I&amp;#39;m just going to label my x-axis to PI divided up into 4 pieces that&amp;#39;s going to be PI over 2 pi 3 PI over 2 and 2 pi and we can see we haven&amp;#39;t it&amp;#39;s not being stretched at all in the vertical direction but what&amp;#39;s happening is we&amp;#39;re shifting right PI over 2 up 1 we can think of that as a new starting point or like our new origin okay it&amp;#39;s a cosine graph remember the cosine starts at the maximum so you&amp;#39;re gonna start up here you&amp;#39;re gonna go back to the midline down to the minimum back to the midline and then back to the maximum so that&amp;#39;s your basic graph right there now notice the amplitude is 1 so from that starting point that&amp;#39;s why it went up 1 and then the minimum I went down 1 from the midline and again you have that basic shape it had a normal period of 2 pi it was just everything was being shifted right PI over 2 and up 1 let me erase the whiteboard I&amp;#39;ve got some more challenging problems involving all of the different components a B H and K let&amp;#39;s look at those next and then we&amp;#39;ll jump into graphing the cosecant the secant and then the tangent the cotangent okay number nine let&amp;#39;s look at y equals 3 sine of 1/2 X minus PI minus 2 so we&amp;#39;ve got a little bit of everything in this problem the first thing I like to do though is I like to look at the scale on the x axis the horizontal stretch or compress so let&amp;#39;s figure out what the period is we&amp;#39;ve got the B value of 1/2 so 2 pi divided by 1/2 looks like let&amp;#39;s see so 2 pi divided by 1/2 we know when we divide by a fraction it&amp;#39;s like multiplying by the reciprocal so that&amp;#39;s going to be like 2 pi times 2 over 1 which is 4 pi so we&amp;#39;re gonna go ahead and Mark our scale we&amp;#39;ve got 4 pi divide that by 4 we get 1 pi 2 pi 3 PI 4 PI but you can see that the graph is shifting left pi I&amp;#39;m sorry - pi is actually having the opposite effect it&amp;#39;s shifting right pi and then down to so we can think of this as our new starting point right PI down to and that&amp;#39;s that&amp;#39;s your point right there now you can see the amplitude is 3 so that&amp;#39;s a vertical stretch of 3 and so and we know we have a sine graph so we can see that from this point it&amp;#39;s going to start here it&amp;#39;s going to go up 3 so let&amp;#39;s see when you&amp;#39;re over here you&amp;#39;re gonna go up 3 1 2 3 that&amp;#39;s the maximum back to the midline then down to the minimum 1 2 3 which is gonna be right down here and then back to the midline and you&amp;#39;ve got one period of your sine graph again notice that the one that&amp;#39;s grouped with the ax that&amp;#39;s going to be the phase shift it&amp;#39;s gonna have the opposite effect on minus pi shifts right pi this is your vertical shift ur came out it&amp;#39;s shifting down to think of that as your starting point pay attention to the whether it&amp;#39;s a sine graph or a cosine graph and then the amplitude is a vertical stretch so think of it as your new starting point science starts at that that origin goes up to the maximum but that maximum is going to be 3 units from that midline same thing with the minimum it&amp;#39;s going to be 3 units from the midline and you&amp;#39;ve got it so let&amp;#39;s look at another example of involving all the different components here number 10 y equals 2 times cosine of the quantity 4x plus pi plus 1 now what&amp;#39;s different about this problem is see how there&amp;#39;s a 4 okay this B value it&amp;#39;s inside of the parenthesis it&amp;#39;s grouped with the ax what we need to do is we need to factor it out and when you factor it at the 4 out you&amp;#39;re gonna have to factor it out of both of these terms so we&amp;#39;re dividing out 4 so our equation is going to look like this now y equals 2 times cosine of 4x plus PI over 4 notice if you distribute that 4 back in you get 4x plus 4 PI over 4 which is 1 pi so that&amp;#39;s the key you definitely want to have just one X inside of the parentheses so you have to factor out that coefficient so you can see here that the B value is 4 so if we take 2 pi divided by 4 that reduces to PI over 2 so that&amp;#39;s our hour period PI over 2 if we divide that by 4 or multiply by 1/4 either way you&amp;#39;re going to get PI over 8 that&amp;#39;s our scale so 1 PI over 2 PI over 8 3 PI over 8 4 PI over 8 we can see that our graph the shifting left pi over 4 because remember this has the opposite effect on the graph so it&amp;#39;s going to go left PI over 4 so PI over 4 is actually two steps so left PI over 4 because PI over 8 times 2 is 2 PI over 8 which is PI over 4 and then plus 1 it&amp;#39;s going to shift up 1 so that&amp;#39;s our new starting point we can think of that as our new y-axis our new x-axis or new origin it&amp;#39;s a cosine graph with an amplitude of 2 so from here it&amp;#39;s going to start at the maximum so I&amp;#39;m going up to back to the midline down to the minimum so that&amp;#39;s going to be down to back to the midline and then back to the maximum up to so that&amp;#39;s going to be our basic cosine shape the only difference is that it&amp;#39;s being stretched it&amp;#39;s also being shifted and the period is changing as well so that was a good example so now what we&amp;#39;re gonna do is we&amp;#39;re going to transition over to working with the secant and the cosecant graphs so just a quick refresher when you think about the cosecant it&amp;#39;s really the reciprocal of the sine graph see 1 over sine and when you think of the secant a function it&amp;#39;s the reciprocal of the cosine ok function so 1 over cosine so we&amp;#39;re going to use that information to graph these graphs so when we look at y equals 2 secant of X we can think of this as 2 over cosine of X now let&amp;#39;s start a little bit simpler let&amp;#39;s just say we wanted to graph y equals not cosine of X but 1 over cosine of X which is equal to secant of X so let&amp;#39;s look at our basic values on the unit circle so when we think of cosine we said cosine starts at the maximum it goes to the midline then to the minimum dent in the midline back to the maximum that&amp;#39;s the basic shape right this is 1 this is negative 1 this is PI over 2 pi 3 PI over 2 and 2 pi right but here&amp;#39;s the thing when you&amp;#39;re at 0 see how the cosine value is 1 so if this is equal to 1 right here 1 divided by 1 is still 1 right when you&amp;#39;re over here at 0 PI over 2 see this is 0 what&amp;#39;s 1 divided by 0 well that&amp;#39;s undefined you can&amp;#39;t divide by 0 so what happens is we get a vertical asymptote here when you&amp;#39;re at PI cosine is equal to negative one what&amp;#39;s one divided by negative one that&amp;#39;s still negative one and then when you&amp;#39;re over here at 3 PI over 2 that&amp;#39;s 0 again 1 divided by 0 is undefined you get another vertical asymptotes wherever the graph is crossing the x-axis that&amp;#39;s where your vertical asymptotes are going to be now notice when I&amp;#39;m at this point say for example right here say for example this is like a cosine value of 1/2 well 1 divided by 1/2 when you divide by a fraction slike multiplied by the reciprocal so 1 divided by 1/2 is really going to be like two you&amp;#39;re up here so you can see this graph is looking like this now when you&amp;#39;re over here let&amp;#39;s say like that when cosine is negative 1/2 1 divided by negative 1/2 is going to be like negative 2 right so this graph is looking something like this etc so what you can see is the coasts cosine graph if you graph that and you look at these key points where it reaches a maximum or a minimum that&amp;#39;s going to be where the secant graph bends or turns okay so I&amp;#39;ll show you another example so say for example this one y equals 2 secant of X what you would do is you would start by graphing y equals 2 cosine of X so cosine of X looks like this it has an amplitude of 2 okay has a period of 2 pi like that wherever it crosses the x-axis those are where our asymptotes are going to be and then the secant graph is going to have the shape where it goes the opposite direction like that okay and again the reason it works is because see how when this value is let&amp;#39;s say for example 1 well then you have 2/1 which is actually going to be 2 right and then same thing over here when this is like 1/2 2 divided by 1/2 that&amp;#39;s going to be 4 so you can see that as this gets smaller and smaller the cosine graph the seeking graph section here it&amp;#39;s going to be larger and larger because when you divide by a small and smaller fraction you&amp;#39;re multiplied by the reciprocal this is going to go up and up and up so that&amp;#39;s the same idea when you&amp;#39;re looking with the cosecant graph remember cosecant is the reciprocal of sine so if we wanted to graph this graph let&amp;#39;s just graph y equals 3 sine of PI over 4 times X the sine graph we know looks like this ok the amplitude is 3 so it&amp;#39;s gonna reach a maximum here at negative three and I&amp;#39;m positive three and down to negative three wherever it crosses the x-axis that&amp;#39;s where our vertical asymptotes are okay sometimes when you&amp;#39;re doing this secant and cosecant graphs that helps to draw you know a little bit more cycles okay more periods and then what you can do now is you can say our graph is going to look like this now the one thing that I haven&amp;#39;t done yet is look at the period so let me do that on last year usually I do it before I graph this but I would take the PI over 4 remember the period is 2 pi divided by PI over 4 okay 2 pi divided by B which is like multiplying by the reciprocal so the PI&amp;#39;s cancel and it&amp;#39;s actually going to have a period of 8 so you can see this is going to be right here it&amp;#39;s going to be 8 divided up into 4 pieces 2 4 6 8 that&amp;#39;s our scale and you&amp;#39;ve got your cosecant graph okay let&amp;#39;s do a little more challenging one number 13 we&amp;#39;ve got y equals 4 secant 1/4 X plus 2 pi minus 1 so that&amp;#39;s a lot right but what we&amp;#39;re gonna do first is we&amp;#39;re gonna graph the same equation here except for instead of secant we&amp;#39;re gonna think of it as cosine okay if we take the period 2pi divided by b which is 2pi divided by 1/4 when you divide by 1/4 it&amp;#39;s like multiplied by the reciprocal that&amp;#39;s going to be 8 pi so our period is going to be 8 pi here divide it into 4 pieces that&amp;#39;s gonna be 2 pi 4 PI 6 pi 8 pi okay but the graph is shifting left 2 pi down 1 so we&amp;#39;re gonna think of this as our like our new origin like right like that okay and the amplitude is 4 it&amp;#39;s a cosine graph so we&amp;#39;re going to go up 4 so this is going to be 1 2 3 4 so it&amp;#39;s going to start right here it&amp;#39;s going to go back down to the midline okay and then it&amp;#39;s going to go down to the minimum which is going to be right here at down 4 from the midline then back to the midline and then back to the maximum okay which is 4 above the midline because the amplitude here is 4 so that&amp;#39;s going to be your basic C your basic graph something like that okay and it&amp;#39;s going to continue okay but what we need to pay attention to is this cosine graph crosses the midline okay so it&amp;#39;s crossing basically like this so it&amp;#39;s crossing right here it&amp;#39;s crossing right here okay it&amp;#39;s crossing right here etc so our secant graphs actually going to look like this it&amp;#39;s going to approach the asymptotes but it&amp;#39;s not going to cross them or touch them so that&amp;#39;s the basic idea and you&amp;#39;ve got it so let me erase the whiteboard we&amp;#39;re going to talk about tangent and cotangent next okay now switching gears over to tangent and cotangent when you graph tangent we&amp;#39;re going to be using the values from negative PI over 2 to positive PI over 2 and remember from the unit circle the tangents the Y value divided by the x value so let&amp;#39;s just go ahead and graph the basic parent graph first and what I&amp;#39;m going to do is I&amp;#39;m going to start over here at negative PI over 2 and tangent is Y divided by X now negative 1 divided by 0 we can&amp;#39;t divide by zero that&amp;#39;s undefined so we&amp;#39;re gonna get a vertical asymptote right here at negative PI over 2 now at negative PI over 4 tangents Y over X anything divided by itself is 1 but we have a negative divided by positive which is negative 1 so over here at negative PI over 4 this is going to be right here at negative 1 at 0 Y divided by X is 0 divided by 1 is going to be 0 so that&amp;#39;s gonna be another key point at PI over 4 which is 45 degrees tangent is y over X of course anything divided by itself is going to be 1 and then at PI over 2 tangent is y over X 1 over 0 we can&amp;#39;t divide by zero that&amp;#39;s undefined we get another vertical asymptote so that&amp;#39;s one period and what&amp;#39;s interesting about tangent as opposed to sine and cosine is that the period is 1 pi divided by B naught 2 pi divided by B so you can see that from negative PI over 2 to positive PI over 2 this is actually a period of pi and then what happens is this graphs going to repeat like that same thing over on this side as well and just going to keep repeating like that but just 1 period one cycle from negative PI over 2 to positive PI over 2 but for number 15 what do you think this 2 does to the graph well you can see the two is this coefficient in front here and it&amp;#39;s going to be like a vertical stretch if it&amp;#39;s greater than one or a vertical shrink if it&amp;#39;s between zero and one of course if it&amp;#39;s negative that&amp;#39;s going to reflect it over the x-axis so if I was to graph a number 15 on the same graph what would happen is two times zero is still zero one times two is going to give us two and negative 1 times 2 is going to give us negative two so what would happen here is the graph would actually be something like that where it&amp;#39;s being stretched it&amp;#39;s going up faster two times faster okay for number 16 we&amp;#39;ve got y equals tangent of 1/2 X what do you think the 1/2 dust of the graph well you can see that this number here is in front of our variable ax that&amp;#39;s our B value pi divided by 1/2 remember when you divide by a fraction it&amp;#39;s like multiplying by the reciprocal so that means that this is period is actually going to be not 1/2 but actually doubled it&amp;#39;s actually gonna be 2 pi now when you graph a tangent an important thing to realize is that see how half the graph is to the left of the y-axis and half the graph is to the right so when you figure out what the period is like in this case we&amp;#39;ve figured out the period is 2 pi I&amp;#39;m gonna cut that period in half so 1/2 of 2 pi is going to be 1 pi and I&amp;#39;m gonna put negative pi so from negative pi to pi that&amp;#39;s actually a period of 2 pi right so I&amp;#39;m going to go ahead and sketch in our asymptotes okay just as dotted lines and then what I like to do is I like to cut this in half again so half of Pi is going to be PI over 2 some people call these like the halfway points or the midway points and negative PI over 2 ok so now what we&amp;#39;re gonna do is you can see that this coefficient in front of the tangents just 1 so we&amp;#39;re just going to be going at these halfway points up to 1 and then down to negative 1 and it goes right here through the origin and we&amp;#39;ve got a good sketch of our graph so remember the one that&amp;#39;s grouped with the ax actually has the opposite effect it looks like we&amp;#39;re dividing by two we&amp;#39;re actually multiplying the X values by two and that&amp;#39;s stretching it horizontally let&amp;#39;s look at another another example number 17 this one has more going on here the first thing I like to do is figure out what the period is so I would look at that coefficient that B value I&amp;#39;d say well let&amp;#39;s see pi divided by 2 that&amp;#39;s PI over 2 but again because half&amp;#39;s on the left and half on the right side of the y-axis what we&amp;#39;re gonna do is we&amp;#39;re going to cut it in half again making this PI over 4 and negative PI over 4 which gives us a period of PI over 2 you can cut it in half again PI over 8 and negative PI over 8 and then now let&amp;#39;s look at the phase shift okay the left and right and the vertical shift up and down so the number of this group with the ax remember this has the opposite effect the minus PI over 8 is actually going to shift it to the right PI over 8 okay pause and PI over 8 and this one has the same effect the pulse one&amp;#39;s gonna shift it up 1 but what am i doing this one a little bit different than some of the other problems that I&amp;#39;ve done in this video is let&amp;#39;s just graph what the original graph would look like without the shift so we would have asymptotes right here okay like this and like this but now what&amp;#39;s going to happen is it&amp;#39;s going to go up here to one it&amp;#39;s going to go down here to negative one and through the origin but what the PI over 8 is going to do is going to shift it right PI over 8 so each of these points is going to move right PI over 8 and then up one so this point would go right here and right here this point is gonna go right an up one and this point is gonna go right and up one like that so now you can see our graph is looking something like this okay and where our asymptote was here at negative PI over 4 is now I&amp;#39;m going to shift right PI over 8 so it&amp;#39;s going to be right here and the one that was over here at PI over 4 is now going to be one step over here at 3 PI over 8 and so now you&amp;#39;ve got your your tangent graph so it shifted right PI over 8 up one okay now let&amp;#39;s take a look at cotangent so when you think about cotangent you&amp;#39;re thinking on the unit circle that&amp;#39;s actually the x-coordinate divided by the y-coordinate and the unit circle values we&amp;#39;re going to use for cotangent are from zero to pi okay zero to 180 so cotangent of zero is going to be 1 over 0 which is undefined so right here at zero we&amp;#39;re gonna have an asymptote okay vertical asymptote at PI over 4 X divided by Y is going to be 1 because anything divided by itself is 1 and then over here at PI over 2 X divided by y 0 divided by 1 is going to be 0 okay and then over at 3 PI over 4 X divided by Y is going to be negative 1 and then over here at PI X divided by Y negative 1 divided by 0 is undefined so we get another vertical asymptote like that so you can see the graph is going down to the right ok whereas with tangent it was going up to the right and the other interesting thing again is that the tangent half of the graph is to the left of the y-axis 1/2 is to the right unless it&amp;#39;s been shifted whereas with cotangent you can see that that whole graph is to the right of the y-axis okay but again the graph repeats so there&amp;#39;s going to be another branch over here ok another branch over here like that and it&amp;#39;s going to keep repeating but for one period one cycle from 0 to pi to the right of the y axis going down to the right is your basic graph also get a couple more challenging cotangent graphs ok for number 19 we have y equals 3 cotangent PI over 2 times X so first thing I like to do is to figure out what the period is using this formula so for tangent and cotangent we&amp;#39;re going to be using the period equals pi divided by B for sine cosine secant and cosecant it&amp;#39;s two pi divided by B so if we do PI divided by PI over 2 that&amp;#39;s really the same as pi times 2 over pi because when you divide by a fraction you&amp;#39;re multiplying by the reciprocal notice the PI&amp;#39;s cancel and we get a period of two so there&amp;#39;s no pi in the period they&amp;#39;re just two so what I&amp;#39;m gonna do is I&amp;#39;m gonna label this as two and if I divide this by four cutting it up into four pieces that&amp;#39;s going to be one half one and one and a half okay and so now you can see our asymptotes are gonna be right here at two and zero okay just like our parent function right here okay hasn&amp;#39;t been shifted at all but what does the three do to the graph well the three is going to be a vertical stretch so this point here which is normally at one is now going to be at three so right there crosses at the midline here and then this point here which is only a negative one is going to be multiplied by three it can be a negative 3 1 2 negative 3 so you can see this was just a vertical stretch and then the period had changed using the PI over 2 so let&amp;#39;s look at number 20 now a little bit more challenging still we&amp;#39;ve got y negative cotangent 1/4 the quantity X minus pi minus 1 okay so there&amp;#39;s a lot involved in this one the negative is going to reflect it over the x axis that&amp;#39;s going to make the Y value is the opposite sign right but let&amp;#39;s figure out what our period is I always do that first so I do pi divided by B now the B is 1/4 but when you divide by a 1/4 it&amp;#39;s like multiplied by the reciprocal so it&amp;#39;s actually gonna be 4 pi so let&amp;#39;s label this 4 pi let&amp;#39;s divide it up into 4 pieces 1 pi 2 pi 3 PI 4 PI and then now let&amp;#39;s see so normally cotangent is going down to the right right but the negative is going to reflect it so now it&amp;#39;s going to be going up to the right so let&amp;#39;s kind of just draw our basic graph here we&amp;#39;re to do it in two steps so normally a cotangent would be here it&amp;#39;s gonna reflect it&amp;#39;s gonna be down here at negative 1 crosses at the midline and then it goes up to positive 1 and then you have your other asymptotes like that so this would be your graph like that right but now what happens here the minus Pyatt and the minus 1 well this is actually had the opposite effect it&amp;#39;s actually going to go right pi then the minus 1 is going to go down 1 so we go right pi at this point and down 1 okay that&amp;#39;s going to be right here this point is going to be right pi and down 1 right here and right pi and down 1 right here now the up and down isn&amp;#39;t gonna affect the vertical asymptotes but the right pi is so normally the asymptotes here it&amp;#39;s gonna go right pi so now it&amp;#39;s going to be right here and then the one that was at 4 pi is now gonna be here at 5 pi and so now you can see there&amp;#39;s your cotangent graph so if you&amp;#39;re enjoying these videos subscribe to the channel my channel is all about making learning math less stressful so you can raise your grade pass your class and go on to pursue your dreams I look forward to helping in the future videos I&amp;#39;ll talk to you soon

Transcript for:Graphing Trigonometric Functions

Transcript for:
Graphing Trigonometric Functions