now let's talk about graphing trigonometric functions let's start with the sine function sine x sine x is basically a sinusoidal function it's a sine wave and that's how it looks like at least that's one period this ends at two pi that's one cycle of the wave now let's say if you put a negative in front of the sine function it's going to flip over the x-axis so instead of going up initially it's going to start from the origin it's going to go down and then back up and then back down so that's the shape of sine and negative sign now keep in mind this wave keeps on going forever in both directions but for the course of this lesson i'm going to focus on graphing one period which is basically one cycle of the wave now what about the graphs of cosine x and negative cosine x cosine starts at the top whereas sine starts at the center so that's one period of the cosine wave let me do that a little bit better but it can continue going on forever negative cosine starts at the bottom it goes up to the middle and then goes up and then back down so that's the graph of one period of negative cosine so this is one cycle now let's go back to the sine graph let's draw two cycles of this graph so one cycle you need to break it up into four useful points one cycle is two pi you want to break that up into four points such as pi over two pi and three pi over two now if we want another period let's add two pi to it so we want to go to four pi in between two pi and four pi is three pi and between two pi and three pi it's five pi over two you add these two then divide by two if we add three pi and four pi that's seven pi and then divided by two we get seven over two sine starts at the center and then it's going to go up back to the middle down back to the middle so that's one cycle of the sine wave and then it's going to go back up back to the middle down and then back to the middle that's why it's helpful to plot the points first before putting everything else if you break up each cycle into five key points which equates to four intervals it's going to be easier to graph the sine wave let's do the same for cosine let's graph two periods of the cosine wave so one period is going to be two pi two periods four pi for each period or each cycle break it up into five points which is four intervals the first point by the way is the origin it's zero so these points will be the same as the graphics sign now we know that cosine starts at the top it's going to go back to the middle and then to the bottom back to the middle and then to the top and it's going to alternate so it's going to look something like this so that's one cycle and here's the second cycle so that's two cycles of the cosine wave now let's talk about the amplitude of the sine wave the generic formula is a sine bx plus c plus d now we're going to focus on a a the number in front of sine is the amplitude so in this case the amplitude is equal to 1. so when you graph the sine wave you plot your four points of interest for one full cycle the amplitude is going to be one so it's going to vary from one to negative one so we know sine starts at the center it's going to go to the top back to the middle and then to the bottom and then back to the middle so it's going to look like this and we know the period is 2 pi now what if we wanted to graph two sine x so if we increase the amplitude this graph is going to stretch vertically so it's going to vary from 2 to negative 2. by the way this is the amplitude it's the distance between the midline of the sine wave and the highest point now let's plot one period so this is going to be 2 pi so once again sine is going to start at the middle then it's going to go up back to the middle and then down and then back to the middle so it's going to look like that and so the amplitude tells you how much it's going to stretch or compress vertically consider the equation y is equal to negative 3 cosine x what is the amplitude of this function the amplitude is always a positive number so you ignore the negative sign and it's going to be 3. the amplitude is the absolute value of a the number in front of cosine now let's go ahead and graph it let's plot one period so let's break it up into five points or four intervals now the amplitude is three so we need to vary the sine graph or rather the cosine graph from negative three to three so cosine typically starts at the top but we have negative cosine so it's going to start from the bottom then it's going to go to the middle to the top back to the middle and then back to the bottom so that's how we can graph one cycle of negative three cosine x now keep in mind this graph can keep on going forever in both directions so let's say if we want to write the domain and range of this cosine graph the domain for sine and cosine graphs will always be the same it's all real numbers the range is based on the amplitude the lowest y value is negative three the highest y value is three so that's how you can write the domain and range of this particular cosine graph now let's talk about finding the period so given this sine function a sine b x we know a represents the amplitude now b is not the period itself but it's used to find the period the period is two pi divided by b so in the case of sine x b was equal to one so the period was two pi divided by one now let's go ahead and graph these two functions sine x and sine 2x let's see what effect b has on a graph now we know the general shape of sine x it has a period of 2 pi and for the most part it looks like this now if b is equal to 2 in this example the period is going to be 2 pi divided by b so the period is pi so therefore it's going to do one full cycle in less time so to speak so what happens is the graph it shrinks horizontally so one full cycle occurs in one pi two cycles occur in two pi here's another example go ahead and graph this function two sine one half x so first we need to find the amplitude the amplitude is the number in front of sine that's two the period is two pi over b where b is the number in front of x so in this case is one half two pi divided by one half is four pi so this one is going to stretch horizontally the amplitude is 2 and the period is 4 pi but we need to break it up into four intervals that's 1 pi 2 pi 3 pi and 4 pi sine starts at the center then it goes up back to the middle down and then back to the middle so we're going to have a graph that looks like that so if you have a fraction what's going to happen is it's going to stretch horizontally let's try another example let's graph 4 cosine pi x so first identify the amplitude and the period the amplitude is simply 4 in this example and a period is 2 pi over b in this case b is the number in front of x so b is pi 2 pi divided by pi is 2. so that's the period in this example so let's go ahead and make a graph so the amplitude is four it's going to vary from four and negative four the period is two so two should be about here and we need to break it into four parts so this is one one half and then between one and two you add them up one plus two is three then you average it or you divide it by two so it's three over two so those are the four points of interest cosine starts at the top then it's going to go to the middle and then back to the bottom to the middle and to the top so we're going to have a graph that looks like this that's one cycle and if we wish to extend it to draw another cycle this is going to be three uh next one is 2.5 or five over two and then three plus four is seven but then divided by two so three point five is seven over two the next point is going to be at the middle and then back to the bottom back to the middle and then to the top and that's it so that's how you can graph 4 cosine pi x so when you find your period make sure you put that first on the x axis and then break it into four intervals now what is the domain and range of this function as you recall the domain for any sine or cosine wave is all real numbers the range is from negative four to four it's from the lowest y value to the highest y value now let's talk about what to do when there's a vertical shift let's say if we wish to graph sine x plus three so the vertical shift is three the amplitude is one so what you want to do first is you want to plot the vertical shift so at 3 i'm going to draw a horizontal line that's going to be the new center of the graph the amplitude is 1 so sine is going to vary one unit higher than the midline and one unit lower than it so it's going to vary between two and four now we're still going to plot just one period so let's write our four key points sine starts at the top and then it goes to the middle actually i take that back sign starts at the middle and then it goes to the top and then back to the middle to the bottom and then back to the middle so this would be one sine wave so that's how you can graph sine x plus string let's try another example let's graph two periods of two cosine x minus one so this is going to be one cycle and two cycle but let's start with the first cycle so the midline is at negative one now the amplitude is two so we got to go up two units and down two units now cosine will start at the top and then it's going to go to the middle back to the bottom and vice versa now we need to plot one more cycle so this is pi and this is three pi so it's going to go back to the middle and then to the bottom back to the middle and to the top so that's how we can graph two cosine periods now what is the range for this graph notice the lowest y values at negative three but the highest is at one so the range is from negative three to one let's go ahead and graph this one negative three sine x plus four so feel free to pause the video actually let's also let's change it a bit let's make it one-third x plus four the majority of the graph will be above the x-axis so let's draw the center line at four first the amplitude is three so we're gonna have to go up three four plus three is seven and then down three starting from four four minus three is one so the range is going to be from one to seven now let's find the period we know the period is two pi divided by b and b is one third so it's two pi divided by one third so it's equal to six pi and let's break it into four points half of six pi is three pi half of three pi is three pi over two and if you multiply this number by three it will give us to this point which is nine pi over two now we know that sign starts at the center positive sign will go up initially but negative sign will go down and then it's going to go back to the middle and then to the top at 7 and then back to the middle so that's how you can plot negative three sine one third x plus four now let's talk about how to graph this function sine x minus pi divided by two how can we do so so considering the generic formula a sine bx plus c plus d anytime there's a c value there's a phase shift which means that the graph is going to shift either to the right or to the left and so you want to find the phase shift because sine won't start at the origin in this case so to find the phase shift set the inside equal to zero and solve for x so when you set b x plus c equals to zero x is going to equal negative c divided by b and this is your phase shift that's where it starts on the x-axis so let's set x minus pi over two equal to zero so we can see x is at pi over two so that's where the sine wave is going to start now let's go ahead and graph it the amplitude is one and the period is two pi over one so it's two pi but first plot pi over two because that's where the phase shift is and then what you want to do is add one period to the phase shift so you're adding two pi to pi over two two pi is the same as four pi over two so this will give you five pi over two so this is going to be three pi over two and you want to break it into five key points this is one pi over two in between one and three is two two pi over two is pi in between three pi over two and five pi over two we have four pi over two which reduces to two pi now the amplitude is 1 so it's going to vary from 1 and negative 1. now sine starts at the middle but we're not going to start at the origin in this example we're going to start at the phase shift which is pi over 2. positive sign is going to go up negative sign is going to go down first so negative sign will be like this positive sign will have that shape and then at 2 pi it's going to have a y value of negative 1 and that 5 pi over 2 is going to be back on the x axis so that's how you can plot this particular sine wave with a phase shift now let's try another example let's say if you want to plot 2 sine x minus pi over 4 plus three so we have a vertical shift of three an amplitude of two the number in front of x is one so two pi over one is two pi the period is still two pi but we do have a phase shift so if we set the inside equal to zero the phase shift is positive pi over four the majority of the graph will be above the x axis so we're going to plot it up there so let's plot the midline first or the center line which is at three the amplitude is two so we need to travel through two units above the center line which will take us to five three plus two is five and then two units down three minus two is one so the graph is going to vary from one to five and that's the range of this sine function now the phase shift is going to start at pi over four that's where the sine wave is going to start and if we add one period to that the period is two pi two pi over one is the same as a pi over four we need to get common denominators so if we add these two numbers this will give us nine pi over four so that's where the first period will end the midpoint between one and nine is five and the midpoint between one and five is three and between five and nine is seven now we can graph it so let's start with the phase shift sine is going to start at the middle and then it's going to go up back to the middle and then down and then back to the middle so this is the graph of just one period you