Features of sinusoidal functions

we have a periodic function depicted here what I want you to do is think about what the midline of this function is the midline is a line a horizontal line where half of the function is above it and half of the function is below it then I want you to think about the amplitude how far does this function vary from that midline either how far above does it go or how far does it go below and it should be the same amount because the midline should be between the highest and the lowest point and then finally think about what the period of this function is how far how long do you have to how much do you have to have a change in X to get to the same point in the cycle of this periodic function so I encourage you to pause the video now and think about those questions so let's tackle the midline first so one way to think about it is well how high does this function go well the highest Y value for this function we see is 4 it keeps hitting 4 and a fairly regular basis and we'll talk about how regular that is when we talk about the period and what's the lowest value that this function gets to well it gets to y equals negative 2 so what's halfway between 4 and negative 2 well you could eyeball it or you could count or you could literally just take the average between 4 and negative 2 so 4 so we could the midline is going to be the horizontal line y is equal to 4 plus negative 2 over 2 just literally the mean the arithmetic mean between 4 and negative 2 the average of 4 and negative 2 which is just going to be equal to 1 so the line y equals 1 is the midline so that's the midline right over here and you see that you see that it's kind of cutting the function in it's where you have half of the function is above it and half of the function is below it so that's the midline mid line now let's think about the amplitude well the amplitude is how much this function varies from the midline either above the midline or below the midline and the middle ends in the middle so it's going to be the same amount whether you go above or below so if you one way to say it is well at this maximum point right over here how far above the midline is this how far above the midline is this is this well to get from one to four you have to go you're three above the midline or another way of thinking about this maximum point is y equals four minus y equals one well you had a you your Y can go as much as three above the midline or you could say your Y value could be as much as three below the midline that's this point right over here one minus three is negative one so your amplitude right over here is equal to three you can vary as much as three either above the midline or below the midline finally the period and when I think about the period I try to look for a a relatively convenient spot on the curve and I I'm calling this a convenient spot because it's at a nice when X is at negative two Y is at one it's at a nice integer value and so what I want to do is keep traveling along this curve until I get to the same y value but not just the same value Y value but the Y value I get the same y value and I'm also traveling in the same direction so for example let's travel along this curve so essentially our X is increasing our X keeps increasing now you might say hey have I completed a cycle here because once again Y is equal to one you haven't completed a cycle here because notice over here where our Y is increasing as x increases well here our Y is decreasing as x increases our slope is positive here our slope is negative here so this isn't the same point on the cycle we need to get to the point where Y once again equals one or we could say especially in this case we're at the midline again but our slope is increasing so let's just keep going let's just keep going so that gets us to right over there so notice now we have completed one cycle so this the change in X needed to complete one cycle that is your that is your period so to go from negative 2 to 0 your period is 2 so your period here is 2 and you could do it again so we're at that point let's see we want to get back to a point where we're at the midline and it you know I just happen to start right over here at the midline I could have started really at any point you want to get to the same point but also where the slope is the same we're at the same point in the cycle once again so I could go so if I travel one I'm at the midline again but I'm now going down so I have to go further now I am back at that same point in the cycle I'm at y equals 1 and the slope is positive and notice I traveled my change in X my change in X was the length of the period it was 2