in this video we're going to focus on how to graph s and cosine functions uh particularly with horizontal face shifts but we're going to start with some basic structures some basic equations and then we're going to make it progressively harder so let's say if we want to graph let's start with the very basic y equals sin x the amplitude is one and for this graph the period is 2 pi sign starts at the center it goes up back to the middle down and then back to the middle and here's the period 2 pi the amplitude varies from one and negative 1 now let's go over the four types so let's say if we want to graph negative sinx negative sinx is going to look like this instead of going up it goes down uh first so negative sign goes down positive sign goes up now let's say if you want to graph and by the way this is going to be 2 pi and the amplitude is 1 let's say if we wish to graph cosine x cosine doesn't start at the center positive cosine starts at the top then it goes down and then back up and then you can keep uh continueing the graph if you want same thing for this one you can continue it cosine also varies between one and negative 1 and a period is from here Peak to Peak so this would be 2 pi at that point that's one full cycle negative cosine if you wish to graph it it starts over here instead and goes like this that's negative cosine it starts at the bottom instead of the top so those are just some generic graphs that you just have to know so now let's say if we want to graph y = 2 s let's say 2 sin x let's just keep it simple so we still have the regular sine wave the period is 2 pi but the amplitude now is two so it's going to vary from Two and Ne -2 instead of one and negative 1 but now let's say if we want to graph y = 2 actually just sin 2x by the way this is known as a vertical stretch the graph will stretch two units um it doubled in the y direction if we have 2X it's going to be a horizontal shrink it's going to vary between negative 1 and one but it's going to be much shorter the period is going to be pi and the way you calculate that period is equal to 2 pi over b b is the number in front of X so it's uh 2 pi over 2 and so we get Pi for the period so a shrink from 2 pi to Pi the generic equation for a s or cosine wave is y equal to a sin BX + C plus d as we've talked about already a is the amplitude B helps you to find a period um C tells you the presence of any horizontal or phas shift and D is the vertical shift so let's try another example let's say if we wish to graph y is equal to negative s of 12x so because it's a negative sign it's going to instead of going up it's going to go down now because we have a half it's going to be a horizontal stretch by a factor of two if we calculate the period it's 2 pi over B and B is2 2 Pi / 12 is the same as 2 pi * 2 so the period has doubled into to 4 Pi the amplitude is still one so uh that's going to be the same so for the sake of practice let's try another example let's say if we want to graph 3 cine 1/3 x so the amplitude is three the period is 3 Pi / 1/3 so that's 3 Pi * 3 that's 9 pi so we have a vertical stretch of three and a horizontal stretch of three as well at the same time so we know cosine starts at the top so let's say this is three and that's neg3 and that's one cycle if we wish to continue it that's another the cycle so from here to here is one cycle so this is 9 pi and that's how you would graph 3 cosine 1/3x so now let's introduce the vertical shift and then we're going to introduce the phas shift so now let's say if we have y = sinx + 2 the two is the vertical shift so what you should do is first plot two which is going to be your Center Line your amplitude is one so you need to go up one from two which is three and down one which is one so sign starts at the center it's going to go up back to the center and then down back to the center and that's how you graph it so let's say if you want to graph 3 sin x + 4 so the vertical shift is four so let's plot that first and the amplitude is three so 4 + 3 is 7 and 4 - 3 is 7 is 1 so we need to go up to seven and stop at one or the lowest we can go is one so plot the midline which is where the vertical shift is and let's make this 3 sin + 4x it won't change these numbers that's going to be the same but instead of going up we're going to go down back to the center and then up okay so let's say if we want to graph y is equal to 2 sin 4x - 3 so let's start with the midline the vertical shift is -3 and amplitude is 2 so -3 + 2 is1 that's our maximum and -3 - 2 is5 that is our minimum and we have a positive sign graph the period is 2 pi over b or 2 pi over 4 which is pi over 2 so we're going to start here and let's plot this the period so let's say pi/ 2 is somewhere over here what we need is four points half of that is pi over 4 and half of that is pi over 8 and 3 * pi over 8 is 3 p pi over 8 so those are the four point points that we need so here's the first one it's going to positive sign is going to go up back to the middle and then down back to the center and so that's our graph that's one cycle for it okay let's try one with cosine so let's say Y is -3 cine 12x + 5 so let's plot the midline first let's go up three to eight and down three to two let's calculate the period it's 2 pi over b or 2 pi/ by half so it's 4 Pi so let's say four Pi is over here half of that is 2 pi half of that is 1 pi times that by 3 you get 3 Pi so those are the four important points that we need cosine starts at the bottom so which is at at two then it's going to go back to the center and then to the top back to the center Center and at the bottom so it's going to look like this so that's how we can graph that function so now let's introduce the vertical Shi if Y is equal to 2 s x - Pi / 2 + 3 okay so let's make the graph we can see that the vertical shift is up three and the amplitude is two so 3 + 2 is 5 and 3 minus 2 is 1 so let's draw the horizontal line here now our phe shift to find it set the inside equal to zero and solve for x so it's going to be Pi / 2 next let's calculate the period the period is 2 pi over B and the number in front of X which there is no number and it's an invisible one so it's 2 pi over 1 which is just 2 pi now what you need to do is add 2 pi um to the where you start from to get your end point so pi over 2 + 2 pi 2 pi is the same as 4 pi/ 2 and 4 pi over 2 + pi over 2 gives you 5 pi over 2 so this graph is going to end let's uh extend it it's going to end at 5 Pi / 2 now what you need is Five Points you want to break into four intervals the midpoint between 1 pi over 2 and 5 pi over 2 between 1 and 5 is three and between 1 and three is two 2 pi over 2 is the same as Pi between three and five is four 4 pi over 2 is the same as 2 pi if you reduce it so now we have our Five Points so now you want to plot it sign starts at the center and because it's positive2 it's going to go up to five then it's going to go back to the middle down to one and then back to the middle and so that's how you can graph one cycle and if you want to you can continue it if you want let's try another example so let's say if we wish to graph y = 3 cosine 12x + Pi - 5 so I'm going to draw the graph like this so the vertical shift is atg5 and the amplitude is three so -5 + 3 that's -2 that's my maximum 5 - 3 is8 eight that's my minimum by the way if you want to find the range of this graph as you can see it's from uh from low to high8 to -2 for the last one that we did it varied from 1 to 5 so the range would be 1 to five the domain for S graphs and cosine graphs is negative Infinity to Infinity because they can keep going on forever unless you restrict it so I just want to mention that in case you had questions on it so we talked about the amplitude right it's three we consider the vertical shift now let's calculate the uh the phase shift where it starts to find the phase shift set the inside equal to zero and solve for x so if we move pi to the other side a half X is equal to Pi and if we multiply both sides by 2 x is equal to -2 Pi which I should have extended the graph here so let me redraw it okay so here's five -28 so here's -2 pi and here is our midline and I'm just going to rewrite the equation so now that we have the the phase shift this is where the graph is going to start let's calculate the period the period is 2 pi over b or just 2 Pi / half and we're going to get 4 Pi now what you should do is add your starting point your face shift to your period -2 Pi + 4 Pi is equal to 2 pi so that's where it's going to stop so over here so that's going to be just one cycle and we can continue it if we uh choose to so cosine starts at the top so at -2 Pi -2 which is right there and it's going to go back to the middle well first we got to break it into four intervals the midpoint between -2 pi and 2 pi is 0 and between -2 pi and 0 it's pi and between Z and 2 Pi is Pi so then cosine is going to go back to the middle which is going to be here at negative pi and at zero is going to go to all the way to the bottom at8 and that Pi back to the center and at 2 pi over it's going to be right there it's going to look like that now be careful don't draw a vshape now let's try one more example so let's say if Y is -2 sin 1/3 x - < / 2 plus three okay so most of the graph will be on the first quadrant so I'm going to draw it like this so let's start with the vertical shift it's at three and the amplitude is 2 so 3 + 2 is 5 3 - 2 is one that means that the range is from 1 to five and a domain goes on from negative Infinity to Infinity so this is the that's the midline and next we need to find the phas shift so set the inside = to 0 and solve for x so 1/3 x is = Tok / 2 and if we multiply both sides by 3 x is 3 pi/ 2 so that's where the graph is going to start so now we need to calculate the period the period is 2 Pi / 1/3 which is the same as 2 pi * 3 so that's 6 Pi so we need to add 3 Pi / 2 the pH shift plus the period which is 6 pi 6 Pi if we get common denominators is the same as 12 Pi / 2 so 3 + 12 is 15 so we get 15 pi over two so I'm going to extend this graph so here's 15 pi over two the midpoint between 3 and 15 is uh let's see 3 + 15 is 18 if you divide by two you get n so it's 9 Pi / 2 the midpoint between 3 and 9 is 6 3 + 9 is 12 / 2 is 6 so we get 6 pi/ 2 which reduces to 3 pi and the midpoint between 9 and 15 and this doesn't look like a 15 but it's supposed to be 15 the midpoint between 9 and 15 is um 12 so 12 Pi / 2 reduces to 6 Pi so now we have our five points and so now we can graph it sign starts at the center and because it's negative sign it's going to go down before it goes up then it's back at the center then it goes to the top at five and then back to the middle now let's say if you only want to draw one cycle notice that your domain is restricted from 3 pi/ 2 to 15 pi over 2 based on the graph that we drew because it starts here and it stops here however if you wish to continue the graph it can go on forever and the domain therefore 4 can be negative Infinity to Infinity so it really depends on whether if you choose to restrict your graph or if you choose to expand it so just know that for sine waves typically the domain is this negative Infinity to Infinity unless you put a restriction on it and here's the range so that's it for this video that's how you can graph sign and cosine functions um when you're given amplitude vertical shift uh phas shift and if you're just given you equation you know how to find everything now so that's it and thanks for watching