 This video will talk about average rate of change. So a non-linear function has a rate of change that is not constant that we've been talking about quadratic. So let's look at a quadratic graph here real quick. And if we think about a quadratic, if I were to pick two points on this quadratic, say this one and maybe this one here, there would be a line I could draw between them. There would be a curve underneath that line and that makes it not constant. So we call this a secant line and if we could find the rate of change on that secant line then we would say that that's the average for the curve between those two points. So we have this function f of x and we have x1 and x2 is our domain and we have an arc between those two points. That's what we're talking about here. So there's my two really big points. So if you remember slope was y2 minus y1 over x2 minus x1 or the change in y over the change in x. Well we're doing the same thing here. We're changing the y's and changing the x's. We're just writing in a function notation since it's not a line. So we take the y value for whatever our x2 point was and subtract it from the x1 y value and then subtract the x's as long as they're not equal because if they're equal we're just at one point. So the average rate of change is used as function notation but it's basically still going to be our slope. You just have to understand that it's not slope because it has a curve and not just a straight line between the two points. So let's try it. Use the graph to find the average rate of change for part A. f of x is an element between negative 3 and 0. These are x values. So from negative 3 we're talking about this point right here and to x equals 0 we're talking about this point right here. So we want to find the average rate of change. So what's the y value at the 0? Negative 1.75 minus the y value at the negative 3 which looks like negative 4 and then we're going to divide that by this was the 0 minus the negative 3. Remember subtracting a negative is adding. So we have 4 plus a negative 0.75 divided by 0 plus 3 and we get, so this gives me 2.25 over 3 and 2.25 divided by 3 gives me 0.75. So the average rate of change is equal to 0.75 for this curve between x is negative 3 and x is 0. So now we need to do part B. We're going from 0 to 2.3. So again that's 0 over here to 2.3. So we start with the 2.3 y value, 0.02 minus the 0 value which is negative 1.75 so that will be plus 1.75. And then on the bottom we have the 2.3 minus 0 and 3.02 plus 1.75 ends up being 4.77. You can try it out with your calculator to double check and 2.3 minus 0 is going to be 2.3. If we divide those two we're going to end up with 2.07 and that is our average rate of change between those two points. So we have the secant line between them but the average on that curve would be a rate of 2.07. So how does the average rate of change in part A relate to the graph if the graph represented an object's velocity compared to the ARC or average rate of change in part B. Well this would be an average rate of change that is not as steep as the second part because we can see that it starts out slower but then it's going to increase faster and faster the bigger the x becomes. So the object in A would be slower velocity. So now we have a function that we're going to do this with and we want to know what it is over these intervals. Remember these are x's, x is an element of if that helps you out. And I'm going to come and use my calculator here and I'm going to go into my y equal and I'm going to put that equation in and the reason why I'm putting that equation in is because I want to go to my table to find out what the y values are so that I don't have to plug and chug a million times. So I'm going to go to my table and I'm going to table start at 0.25 and I'm going to go every 0.25 so that I can then just see what happens at 1 as well. So second graph at 0.25 I find out that the y values is 7 and then at 1 it's 16. So y is an element of 7 and 16. So my average rate of change then here is going to be 16 minus 7 over the 16 went with the 1 minus the 0.25. So we have 16 minus 7 which is 9 and 1.25 which is 0.75 and that's going to be equal to 12 and then if I look at 1 again the y value for that was 16 and the 1.5 I'm looking at my calculator even though you can't see it it tells me that it's 12. So the average rate of change is going to be the y value of 12 minus 16 and then I have the 12 went with the 1.5 minus my 1. 12 minus 16 is going to give me negative 4 and 1.5 minus 1 is going to give me 0.5 and negative 4 divided by 0.5 is going to be negative 8. Alright let's try another one. Although they have been around for decades water rockets continue to be a popular toy plus the graph is filled with water and pressurized using the handheld pump. It's released to go off. Here's the function and we want to answer the questions. It's not after h of t seconds but after t seconds. So find the rockets height at t1 and at t2. So when it's negative 16 times 1 squared plus 96 that's times 1 then we are going to have 80 and if I do negative 16 times 2 squared plus 96 times 2 I get 128. So this is 80 and it was 128 when I put in 2. We may need those numbers 80 and 128 remember those. Let's do t1 then it was 80 so that we don't forget later and t2 was 128 and then if we do negative 16 times 3 squared plus 96 times 3 we get 144. So would you expect the average rate of change to be greater between 1 and 2 seconds or between 2 and 3 seconds. Well let's I can sketch this graph for you right now. Starts on the ground it goes up and then it lands. I don't know exactly what the vertex is but I can guess that it's probably after 3. So it's steep it goes real fast at the beginning and then it levels off. So I would say between 1 and 2 should be the bigger average rate of change because it is starts fast and slows towards peak and then it says calculate the average rate of change and discuss your answer. Let's go back and do that over here. So t1 and t2 the y values were 128 minus 80 over 2 minus 1. 120 minus 80 over 2 minus 1 is going to give me 48 and then I'll do the other one down here and that is going to be 128 and 144. So 144 minus the 128 over 3 minus 2 and that gives me 16 and I was correct. So 1 to 2 was 48 and the 2 to 3 was only 16.