this is lesson 7 4 which is graphing sine and cosine functions our essential question is how can you identify key features of sine and cosine functions this is an overview of what we're going to talk about so we have y equals a sine of b x we can also write y equals a cosine of b x so we're going to focus on the a and the b and how they affect your equation so the amplitude of our graph is the absolute value of |a| so we can see down here that our graph is oscillating around our midline and it's going between so this is 1 so this is going up to a 1/5 and down to negative 1/5 so our amplitude is one half so you can see that in the equation up here our a value is one half our b value is going to tell us the period of the graph so if we look at the graph here we start at 0 we go up down and back to that same starting point so the period is the amount of time it takes to get or the amount of angle it takes to get back to the original starting point so with this graph we can see it's right here this right here is pi so this would be half a pi or pi over 2. so our period is pi over 2 and our amplitude is a half so the period comes from we're going to take 2 pi because that's the period of our parent function in this case divided by our b values so our b value was 4 so that's where the pi over 2 came from and we're going to look more at that when we do some specific examples okay so the first question is what is the period of the graph of y equals sine of x so sine of x remember is talking about the y values so if we think about on our unit circle we start out at zero degrees and that our y value would be zero so then as we go around the unit circle our y value gets bigger it reaches our top point at pi over two then it starts to get smaller now it's zero right here at pi then it gets negative in the third quadrant down to 270 degrees or three pi over two is at its lowest point and then it goes back to zero and that is one period of pi or of okay so 2 pi radians gets us around the unit circle we can keep going so this kind of shows us how that graph is developed so it's taking from the unit circle we're going our y values are going to give us the graph of sine okay so if we go back to our notes here what is the period of the graph of y equals sine of x so if we think about that illustration we just looked at it's we're going to reach the same point as soon as we've gone around the unit circle so the period of the graph of y equals sine of x is just 2 pi okay and then it says what are other key features of the graph of y equals sine of x so if we go over here so key features so we have a midline at zero okay our graph is going because again we're talking about why when we're talking about sine so it's going between a positive one value and a negative one is our y values um and we can see so and the period again we talked about is two pi amplitude is one so those are our key features that we're talking about so our a value is one our b value is also one so the a value tells us our amplitude is one and the period is 2 pi so that's how we get so this is our equation for the parent function of sine okay so example 2 asks us to identify the amplitude and period of y equals 3 cosine of x so you can see the blue graph is just y equals cosine of x and the orange graph is y equals 3 cosine of x so the only difference between those two graphs is the 3 and that 3 is going to change the amplitude so you can see on our graph that it's going to change that it goes up to positive 3 and down to negative 3. so the amplitude would be 3 and the period is still 2 pi does not change the period by putting that 3 in front of cosine okay and then part b says what are the amplitude and period of y equals negative sine of 2x so this is where we have to be careful the amplitude is if we look at our a value here we have a negative number out there that does not change remember the amplitude is the absolute value of a so the absolute value of negative one is still one so the amplitude in this case is still one that negative is going to tell us that we have a reflection over the x-axis okay and then the period the period will be different on this one so to find the period we take 2 by 2 pi divided by b so our b value in this case is 2. so 2 pi divided by 2 is just going to be pi so that means that the period in this case is going to be shorter so you're going to have a higher frequency of the function and we'll talk about frequency here in the next example okay so example three is to graph um a sine of bx and y equals a cosine of dx and we're going to talk about what is the frequency so the frequency is the reciprocal of the period so if we're finding the period we're going to take 2 pi over b and in this case our b is one half so be careful here because sometimes we see two pi two divided by a half and we think oh that's one but it's not think about how many one half pieces fit into two and it's four so this would be 4 pi so that means that the frequency would be 1 over 4 pi it's the reciprocal of the period so that means that you have one cycle of your graph in four pi angles so this graph would be more spread out okay you would have a horizontal stretch instead of a horizontal shrink