foreign hey welcome back this is Mr Kelly we're on 1.3 rates of change in linear functions and quadratic functions so let's review a little bit what the average rate of changes we know that that's the slope right of the secant line now secant line might be a new word for us so let's pretend like we're looking at a closed interval from A to B so maybe a is like right here it's my favorite a right there all right right let's label that so that would be the X values a what we call the Y value let's call it f of a okay F of a and then we'll just close that off as a coordinate point and then over here somewhere we have a value of B and so the point would be F of B so the x value would be B the Y value would be F of B all right so the secant line I don't know if you knew this you know what a secant line is I'm going to try my best here we go that was not bad for Mr Kelly the secant line goes through the two points and we learned in the last lesson that the slope of the line that goes through these two points or the secant line will give you the average rate of change so if we're using notation of a f of a and b f of B we can write that as what do we want to do we want to do Y2 minus y1 so let's do and get rid of that F of B minus F of a and then we have to do X2 minus X1 right so it's change in the dependent variables over the change in the independent variables that would be the formula for us so as I said today we're going to look at linear functions we're going to look at quadratic functions linear functions you know are straight lines right we want to find the average rate of change for each linear function so I'm going to look at number one right here the first thing I'm going to do is going to find two points and I love zero zero because that gives me some easy math and right here's the point three two so if I were looking at this interval it goes from zero to three so let's write that out in interval notation from 0 to 3 all right and then what is our average rate of change I know I have to go up to right and then over 3. so the up 2 is the Y right that would be on top and then we go over 3. that's interesting let's look at uh suppose you know the person next to you said well I didn't use those two points I looked at these two points right here it's a different interval but let's see what happens like maybe someone else will use different color here so we can look at it maybe someone else used I said different color someone used red maybe and they said all right from the interval of three to what do we go over to 6 here right what do you have to do it is still up two over three that's the average rate of change for number one all right well that's interesting I noticed I got two over three for both of them maybe that's a coincidence let's look at number two number two I'm going to use that point there that looks like a very nice point and this point right here is also a very nice point and when I say very nice look it goes through exactly the little crosshairs there so I know that the values are very uh they're very nice numbers they're very neat like negative two and then we have a seven and we're going to go all the way to negative one so what is the average rate of change from negative two to negative one well in this case we have to go down two so we're going to be very careful change in y is down two and we go over one still always going to the right one so that would give you with just negative two let's just simplify that everybody likes negative two now suppose you picked a different set of points look at this it's down two over one again so from negative one to zero we get the same thing negative two over one and guess what would happen if we picked two different points same thing you would get down two over one do number three by yourself it's a little bit tricky but it's a little bit easy you know what I mean pause the video and do number three all by yourself so how did you do I got a slope of zero for both of these the average rate of change is zero which makes sense because this line is just it's called flatlined right it's just going straight across and it does not change so the rate of change is zero the average rate of change is zero between these two points the average rate of change for a linear function did you notice how it changes here it doesn't it's constant so we're going to write down constant and that means that regardless of the input value interval length the average rate of change stays the same and that last part what that means is I could have used say number two the very first point in the very last point and what do we get here we have to go I'm going to do that let's do that just to prove my point real quick it doesn't matter if I'm going over by one or if I'm going over by let's start here and go all the way down to here how far over is that from negative 2 to 1 that would be that's over 3 but how far down do we go one two three four five six and then over three so it's down six over three notice I reduce it to negative two so it doesn't matter the length of the interval on the the average rate of change is going to stay the same for a linear function well what does that mean for numbers four five and six they want you to find the rate of change of the average rate of change for each linear function what does that mean that means that we have to find the slope so for number four it was 2 3 then he found the slope again and it was two-thirds and if I were to find it again guess what it would be constant two-thirds so what would the rate of change between these slopes be like how is the slope changing and you notice the slope for a linear function does not change it would be uh two-thirds each time so the change in that slope would be zero let's look at number five first we got negative two right and then for the next interval we got negative two and if we did another interval we would get negative two and we would find the difference in those average rates of change would be zero and lastly this last one was so easy right because we got zeros involved uh what do we get we get zero and then the next rate of change would be zero and then the next rate of change would be zero and then if we find the rate of change in those average rates of change it would be zero so that's you know if you're looking at the rate of change of the average rate of change or how is the slope changing for a linear function the slope doesn't change and that's why we get zeros when we're looking at those average rates of change it's always going to be zero easy enough well that's you know that's basically linear functions let's go to quadratic functions so with number seven uh we're given a quadratic function and it's up here we always want to know what that function is so look right here is our quadratic function equation y equals x squared minus 2X and we want to find the average rate of change on the given interval and so now we're using inequality notation for number seven we have X has to be between greater than one or equal to and then less than or equal to two so we're really talking about looking at this point right here and that's one right x equals one and then x equals two is at this point right here and we want to find the average rate of change which is the slope all right well the slope here would be up one okay so from one to two the slope is up one so it would be positive one and then we go to the right one which is positive one that's just one all right let's move over to number eight where we have the same function but a different interval goes from two to three so the interval from two to three if we notice it's a little different right so the interval from two to three uh we have to go up one two three and then over one so it's up three and then to the right one so we get an average rate of change of three on that one now let's look at nine from three to four as we move along this function we get a point of three and a point of four uh what do we have one two three four five five up and then one over and if it helps you to draw that slope triangle that's okay but we're gonna document what we have here from three to four then we need to go up five and to the right one okay that's just five over one can we do this if we're looking at uh interval notation and kind of figure out the pattern here what do we think well let's use the function which is x squared minus two x I'm going to write that down here so I can look at it y equals x squared minus two x and so they've given us this interval but we don't have a fancy graph look these graphs don't go far enough so we're just going to use that equation equation is y equals x squared minus two x so let's just find y of four we can write it like that that means I'm going to plug a 4 into Y and we know the equation is x squared minus two x so that would equal 4 squared minus two times four and that'd give me what 16 minus eight and that's just going to be eight then we can figure out y5 so y of five is I'm just plugging a 5 in here as the input when you get 25 which is 5 squared minus 2 times 5. so 25 minus 10 25 minus 10. what does that even work that all out and then you could probably get 15 there okay so the difference between these two if I were to find these out this would be the 0.48 and the second one would give the point 5 15. so I can look at it and say the change in y is 7 right because it's going to go up 7 and then over one so we get seven you do the last one by yourself go ahead and do number 11 use the equation here and find the average rate of change between those points all right how'd you do what do we get y5 and Y of six we already did y5 so we find 24 and 15 subtract and we get an average rate change of nine so can I just let's look at the average rate of change across this quadratic function as we move on equal intervals look this is a distance of one this is a distance of one this is a distance of one and as we look at those average rate of changes I'm going to write those down the first for number seven we got an average rate of change of one and then for eight we got three and then five and then seven and then nine so when you're looking remember these are the average rates of change and how is that average rate of change changing well you have to go up to you have to go up two again and as we travel across these different intervals you notice that we get the exact same rate of change of the average rate of changes that's a little bit tricky to think about it's a lot of words so think about the average rate of change or the slope how is the slope changing for this function and the slope is increasing by the same amount which means we have a very small slope and you move over the slope has increased by a certain level and you move over and it's increased by the same level this is what happens for quadratics so we say that the average rate of change for a quadratic function does not stay the same we can see that it changed across these intervals right for consecutive equal length input value intervals which means that these distances are the same the rate of change of the average rates of change of the quadratic function is constant so if you look essentially look at the slopes and how are the slopes changing they remain the same for a quadratic for the above problem the constant is two okay so for number 12 they want us to find the rate of change of the average rates of change so first thing we need to do is find some output values right so they just give us a function so let's just pick some input values I just pick negative 2 negative one zero and one you could you know you could start at zero and go to ten or however far you want to go it doesn't matter what input values you pick but you do have to make sure that the distance between them is equal and so if you figure out all these input values we're going to get a 2 for negative 2 for negative 1 we get an 8 0 is 10 and then one is eight all right maybe some of you guys can do that math in your head that would be good but now we want to find the rate of change of these average rates of change so we need to find the average rate of change right here so from two to eight what is the average rate of change that is going what up six as we go over one so it'd be six over one I'm just going to call that a six and then as we go from 8 to 10 you go up two as you go over one that's the benefit just going over one makes easy slopes and then as you go from zero to one uh what do you do you go down two so let's look at the average rate of change and how that's changing when you go from six to two then you're going down four and as you go from two to negative two you were going down four and you guessed it if you kept going down this quadratic function as you go to the right on the x-axis as X increases the rate of change of the average rates of change would be constant at negative four okay so we did number 12 together you think you can do 13 by yourself I think you can go ahead and do number 13 pause the video you can do that one by yourself go find the rate of change of the average rates of change all right here's what I got I chose intervals starting at zero because I love zero math zero math is easy so I plugged in from F of zero all the way to F4 to find some intervals here and then I found the average rate of change now what is the rate of change of this average rate of change from one to seven that is up six so I'm going to say positive six and then positive six and then positive six so you can see the ab the rate of change of the average rate of change is positive six that's easy enough so for a quadratic the rate of change of those average rate of changes is constant whoo all right number 14 is our very last one it's different type it's talking about uh would they give us a table of values and they tell us that the function is either concave up which we remember kind of looks like this right or concave down which looks like this but it's not both which means it doesn't squiggle around like a polynomial function it says determine if the function is concave up or concave down now what we're going to do to do this is we're going to use the average rate of change here let's figure it out from 18 to 20. remember this is the Y value so we'll do 20 minus 18 which is 2 over 1 and then we'll go to the next equal length interval here so we go over and we get zero right over one and the next one you guessed it is negative two over one and then negative four over one so how is the average rate of change changing this is the average rate of change we get two over one and then zero the negative two and negative four notice how the average rate of change is decreasing which means that your slope of your secant line if I drew a secant line here first it's two right and then it goes to zero so then the line goes like this and then it goes to negative 2 which looks like this and then it goes to negative 4 which looks like this and if we were sketching that function it would first have a positive slope and then it would go oh marker here come on marker I have a positive slope and then the slope was zero and then it's a negative slope and more negative that's concave down this isn't too difficult and some students can look at this and just figure it out but if the average rate of change is decreasing then we'd say it is concave down and you guessed it if it's increasing it would be concave up here's how I can write that out the function is concave down because the rate of change is decreasing over equal length input intervals all right I think that's it good luck to you all on 1.3 this is Mr Kelly remember it's nice to be important but it's more important to be nice so