okay in this video we're going to talk about increasing decreasing and concavity from an AP pre-calculus perspective largely through looking at a couple of graphs so uh this is all stuff that maybe you would make note cards for and flip through a thousand times and just be able to like reel off really quickly let's get started uh here's a a picture of our function and first thing we're going to talk about is is it increasing or decreasing well f is definitely increasing in this case because the Y values are just getting bigger As you move from left to right always As you move from left to right um now we want to talk about the rate of change of f so rate of change is another word for slope you're always going to want to in your mind replace rate of change with slope every time you see it they're interchangeable so in this case the slope is positive um now the next thing we want to do is talk about concavity now in my experience when you're first learning concavity you basically just memorize what it looks like and then as you work with it more it kind of becomes intuitive but this curve that we're looking at here is concave up now what does that mean that means that F's rate of change or F's slope is increasing now to give you an idea of why that's happening what I'm going to do I'm just going to put a little line segment on a bunch of places uh and mark down what I think the slope would be at those kind of intervals or at those points so I think we get a slope of like three halves and then two and then 10 and you can see that those values are getting bigger as we move from left to right so the rate of change the slope of the curve is increasing now we're going to do that again all right we have this so again first thing we're going to talk about increasing or decreasing you can see the y- values are getting bigger As you move from left to right so f is definitely increasing now we want to talk about uh what's the rate of change so f is increasing that means that the rate of change the slope is positive okay so far so good uh the next thing is concavity so we want to say here f is and again initially you just memorize shapes uh F here is concave down now what is concave down tell us about the rate of change or the slope so F's rate of change or F's slope is decreasing now sometimes people have trouble dealing with this not so much for the increasing functions um for the increasing functions not the increasing rate of change when a function is increasing people tend to believe this but here are some segments so you go from a slope of maybe 10 to 1 to 1110th those are definitely decreasing As you move from left to right write those down on some note cards memorize them uh it's pretty easy to to get this down with just a little bit of practice let's look at the next one all right here we go step one we want to talk about is this function that we're looking at is this graph increasing or decreasing this is definitely decreasing because as you move from left to right the yv values are getting smaller if a function is decreasing what can we say about its rate of change or its slope well F's rate of change its slope is definitely negative because f is decreasing so those are kind of like saying the same thing you'd say um you know describe a function where F's rate of change is negative that would just be saying describe a function that is decreasing now we want to talk about concavity so what is the rate of change of the rate of change so f is concave in this case concave up and again uh you know you memorize the shapes you put them on no cards you flip through them a couple times you'll be totally fine with that so F here is concave up which means we want to talk about F's rate of change so F's rate of change because you are concave up every time you find yourself saying concave up you're going to say that the rate of change is increasing so interchangeable things takes a while to get used to it but you're definitely going to um so I'm going to put some line segments on here again because this is where people start to get a little confused because negative numbers I don't I don't know I guess are weirder um so as we move from left to right say we have a slope of like -10 and then maybe negative 1 and - 1110th well let's look at those on a number line so they would go -101 1110th as you go in that direction as you go to the right on a number line things get bigger so uh these rates of change are increasing so this is a function that is decreasing its rate of change is negative the function you can see visually is concave up and if a function is concave up it means the rate of change is increasing make sure you're memorizing these things uh as you do more in calculus so this is I know it's called pre-calculus which maybe I don't know what the pre is for like prerequisite before I tend to think this is just a preview of calculus you talk about this all the time in calculus all right let's look at one more so we've got this function so here f is definitely decreasing because as you move from left to right the y- values get smaller um when we say a function is decreasing we can say that its rate of change or its slope is negative and then we want to talk about the overall shape of the curve so shape of the curve is concavity f here is concave down and again make some note cards start memorizing them um if a function is concave down you're always going to be able to say that its rate of change is decreasing but again this is the one that's like a little bit weird so let's put some line segments on here this is another thing you could do anytime you're a little bit confused just throw some line segments on estimate some slopes put them on a number line and see if you're increasing or decreasing um so here we go uh -110 and then -2 and then 8 I put them on a number line and if you go from 1/10 to -2 to8 you're going to the left and if you're going to the left you are decreasing so these are the increasing decreasing concave up concave down curves the ones that I didn't do are the functions that are just in increasing with zero concavity no concave up no concave down or decreasing with no concavity no concave up no concave down those are just linear functions so I didn't put them in here um anyway I hope this was helpful and good luck