what's up everybody in this video dealing with function analysis we're going to talk about the rate of change in quadratic functions so let's explore the average rate of change over consecutive equal length intervals for quadratic functions so again the key thing here is consecutive equal length intervals in a quadratic function so they got to be consecutives they got to be going in order and they have to be the same length so it doesn't matter the length you pick the length right so it could be five so if we get negative five to zero that could be the first interval and then the next one would be zero to five and the next one would be five to ten and the next one would be 10 to 15. so that would be an example of consecutive I mean they're going consecutively in order there's no gaps in between them one ends at zero the next one starts at zero one ends at five the next one starts at five and they also have to be equal length Okay so we're going to do this by first looking at a graph and then we're going to examine a function to kind of see what happens when we're looking at the rate of change over these consecutive equal length intervals so first here is my beautiful quadratic function in red here and we're going to find the rate of change in each of these intervals the average rate of change over each interval so we're going to look from negative two to zero zero to two two to four four to six sixty eight so here they are consecutive every time an interval ends the next one begins and they're all equal length they're all length of two so we can actually see them down here so we're going from negative two to zero that interval then 0 to 2 then 2 to 4 then 4 to 6 then 68. so we're looking at those intervals there all right so the next graph here is me actually plugging those secant lines in so you can actually see the the blue the green the brown the purple and the and the turquoise secret lines that actually show the average rate of change between each of these intervals and I took the time to actually go ahead and do that just to try to save you the time but hopefully by now you guys know the process my goodness have you gotten this far you should have the process of finding the average rate change you simply subtract the Y's on tops track the X on the bottom from point to point so anyway here are the rates of change from negative two to zero we have negative 1.6 from 0 to 2 negative point eight from two to four it's actually a flat line you can actually see that Brown Line right there there's no rate of change rate change of zero then from four to six we're point eight and then from six to eight we're at 1.6 so what we want to do here is say all right well what do you notice what do you notice about these rates of change so I noticed a couple things that should all make sense to you so first I noticed that they are increasing right uh it started off as a negative 1.6 and then and then it increased to negative 0.8 and then increased to zero then it increased to 0.8 that it increased to 1.6 so as I looked at these equal length consecutive intervals the rate of change increase you can actually see it in the lines here so first it was pretty negative then it was still negative but not so much then it flattened out at zero and then it got more and more positive so that means we are concave up so that's one thing that we could identify here when we look at this um we learned that concave up is when your rates of change increase over that interval so since we see that here we could literally see it we could also see that we're concave up hopefully that all makes sense all right we also noticed that um as we went through the interval two to four we we were we were negative on the left side and then we were positive then in between we were zero so that means that somewhere in that interval from two to four somewhere down in this interval we switched from a negative rate of change to a positive rate of change and we know that when H switch is made from a negative to a positive rate of change that there must be an extrema and look you could see the extrema the the Min there in this graph so it's another thing that we notice now one more really cool thing that we notice is that the rates of change are not only increasing but they're increasing by the same value so they're they increase by 0.8 then they increase by another 0.8 and they increase by another 0.8 then they increased by another 0.8 so I could assuming that if I would go to the next equal length consecutive interval from 8 to 10 that I would increase by another 0.8 and that would be a very safe assumption so that would be 2.4 that'd be another 0.8 increase so remember we learned that with linear functions the rate of change is it's it's it never changes the difference between any consecutive rates of change is zero because it's always the same but a quadratic function the difference of your rates of change are constant meaning that they're a constant value the rate of change constantly increases by the same value with linear functions the rate of change don't change at all so there is no difference between them that's there's no difference whereas in a quadratic function there is a difference in consecutive equal length intervals rates of change and that difference is the same number it's a constant increase so that's something pretty cool that we notice here it's not always going to be 0.8 and it just so happens to be 0.8 for this function but it's always whatever the the change is it's going to be the same that's pretty cool all right let's do the exact same thing but this time no graph we're just looking at a function uh f of x equals negative 3x squared minus X Plus 4. here we're going to look at consecutive equal length intervals of length three so negative seven to negative four that's a length of three pick up right we left off negative four to negative one negative one to two two to five so every one of these intervals is first off consecutive and that they are all equal length three it doesn't matter what the length is last one was two this one is three just got to be consecutive and all equal all right so I actually took the liberty to go and do this for you I don't want to bore you with actually walking through because you should know how to find a rate of change between an interval right now but in this first interval we have negative 4 to negative 40. uh well that's terrible writing there sorry the point is let me start all over again here negative seven the Y value is negative 136 and at negative four the Y value is negative 40. and if you take the time to find the average rate of change between those two points you get an average rate of change we'll call that an Roc of 32. then from negative 4 to negative 1 at negative 4 the point is once again negative 40. and at negative one the point is that two and the rate of change between those two points is 14. from negative one to two so let's see here at negative one the point is at two and at two the point is that's uh negative 10. again please I'm just plugging it in I'm just trying to speed it up you could use your calculator if you want to check but the ROC the rate of change is negative four and then from two to five so at two I'm at negative ten at five if you plug it into the function you get Negative 76 the rate of change between those two points is negative 22. so once again what do we notice well we first notice that the rate of change is decreasing the rate of changes decreasing 32 14 negative 4 negative two so if your rate of change is decreasing As you move throughout the function that's the definition of concave down so this is going to be a parabola opening down it's going to be concave down but hey we knew this was a parabola opening down because that negative leading coefficient but that's proves through the definition that we are concave down because the rate of change is decreasing but the other thing that's really important to note is that the rate of change decreases by the same number it goes down 18. do you like that that color there it's like a psychedelic there uh 14 minus 4 so that's going to go down 18 down 18. so we notice that the rate of change is decreasing yes that's what makes it concave down but it's decreasing by the same value so it's decreasing by a constant that's pretty cool that's pretty awesome so we could assume that the next equal length interval which would go from 5 to 8 would have let's see all I got to do is take negative 22 and subtract 18. the rate of change for that one without me even doing the math the rate of change would be negative 40. so that's going to be down another 18. so here's what we know what we notice is that over equal length consecutive intervals the average rate of change changes at a constant rate this means that the rate of change of the rates is linear so when you are constantly going down by the same value down 18 down 18 down 18 down 18 you have a constant rate of change and a constant rate of change means it's linear so the rate of change within a quadratic function is linear what's the rate of change within a linear function it's it's absent it's zero there is no change of the rates of change in a linear function it's always the same but in a quadratic function we notice that there is a change in your rates of change and that change is in fact constant which means it's linear so for consecutive equal length input value intervals the average rate of change of a quadratic function can be given by a linear function that's pretty cool right so what is the rate of change of the rate of change negative 18. pretty cool go back to this example what is the rate of change of our rates of change positive 0.8 and that's a feature that's a characteristic of quadratic functions whatever the rate of change is the rate of change is changing that that is for a fact and the rate of change is changing by a constant value in this problem it's positive 0.8 and this problem was negative 18. so when you change by a constant value you are linear so the chain the quadratic is not linear my gosh it's a quadratic so the the output values of the quadratic is a quadratic right that's what a quadratic does but the rates of change for consecutive equal length intervals are changing by a constant value which makes that change linear a little confusing so I'm trying to explain a couple different ways as I can but I think that's a pretty cool feature that you will notice in any quadratic function so you're either going to be concave up or concave down depending if your rates of change are going up or going down and whatever those rates of change are they're going to increase or decrease by a constant value all right that is it for examine the rate of change of quadratic functions hopefully it was pretty short and pretty simple but some of those videos you might have to watch a couple times to fully comprehend