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# Exponential decay intro

CCSS.Math:

## Video transcript

what we're going to do in this video is quickly review exponential growth and then use that as our platform to introduce ourselves to exponential decay so let's review exponential growth let's say we have something that I'll do this on a table here let's let me make that straight so let's say this is our X and this is our Y now let's say when X is zero Y is equal to three and every time we increase X by one we double Y so Y is going to go from three to six if x increases by 1 again so we go to 2 then we're going to double Y again and so 6 times 2 is 12 this right over here is exponential growth and you can even go for negative X's when X is negative 1 well if we're going back 1 and X we would divide by 2 so this is going to be three halves 3 halves I notice if you go from negative 1 to 0 you once again you keep multiplying by 2 and this will keep on happening and you can describe this with an equation you could say that Y is equal to and sometimes people might call this your y-intercept or your initial value is equal to 3 essentially what happens when x equals 0 is equal to 3 times our common ratio and a common ratio is well what are we multiplying by every time we increase X by 1 so 3 times our common ratio 2 to the to the X to the X power and you can verify that pick any of these when X is equal to 2 it's going to be 3 times 2 squared which is 3 times 4 which is indeed equal to 12 and we can see that on a graph so let me draw a quick graph right over here so I'm having trouble drawing a straight line all right there we go and let's see we could go and they're going to be on the slightly different scale my x and y axes so this is x axis y axis and we go from negative 1/2 - one two - yes we're going all the way up to 12 so let's say that this is three six nine and let's say this is twelve and we could just plot these points here when X is negative 1 Y is 3 has so looks like that then at y equals 0 X when x is 0 Y is 3 when x equals 1 y has doubled it's now at 6 when X is equal to 2 y is 12 and you will see this telltale curve and so there's a couple of key features that we've what we've already talked about several of them but if you go to increasingly negative x values you will ask asymptotes towards the x axis it'll never quite get to 0 as you get to more and more negative values but it'll definitely approach it as you get to more and more positive values it just kind of skyrockets up we always we've talked about in previous videos how this will pass up any linear function or any linear graph eventually now let's compare that to exponential decay exponential exponential decay an easy way to think about it instead of growing every time you're increasing X you're going to shrink by about load by a certain amount you are going to decay so let's set up another table here with x and y values that was really a very I'm supposed to distance opposed to what I press shift it should create a straight line but my my computer I've been eating next to my computer maybe there's crumbs in the keyboard or something okay so here we go we have X and we have Y and so let's start with let's say we start in the same place so when x is 0 Y is 3 but instead of doubling every time we increase X by 1 let's go by half every time we increase X by 1 so when X is equal to 1 we're going to multiply by 1/2 and so we're going to get to three-halves and then when x is equal to 2 we'll multiply by 1/2 again and so we're going to get 2/3 we're going to get to three forts and so on and so forth and if we were to go to negative values when X is equal to negative one well to go if we're going backwards in X by one we would divide by one half and so we would get to six or going from negative one to zero as we increase X by one once again we're multiplying we're multiplying by 1/2 and so how would we write this as an equation I encourage you to pause the video and see if you can write it in a similar way well it's going to look something like this it's going to be y is equal to you have your you could you can have your y intercept here the value of y when x is equal to zero so it's three times what's our common ratio now well every time we increase X by one we're multiplying by 1/2 so 1/2 and we're going to raise that to the X power and so notice these are both Exponential's we have some you could say y-intercept or initial value and it's being multiplied by some common ratio to the power X some common ratio to the power X but notice when you're growing our common ratio and it actually turns out to be a general idea when you're growing your common ratio the absolute value of your common ratio is going to be greater than one let me write it in so the absolute value of 2 in this case is greater than 1 but when you're shrinking the absolute value of it is less than 1 and that makes sense because it's the if you have something where the absolute value is less than 1 like 1/2 or 3/4 or 0.9 every time you multiply it you're going to get a lower and lower and lower value and you could actually see that in a graph let's graph the same information right over here and let me do it in a different color I'll do it and I'll do it in a blue color so when X is equal to negative 1 y is equal to 6 when X is equal to 0 Y is equal to 3 when X is equal to 1 Y is equal to 3 halves when X is equal to 2 y is equal to 3/4 and so on and so forth and notice because our common ratios are the reciprocal of each other that these two graphs look like they've been flipped over they look like they've been flipped horizontally or flipped over the y-axis they're symmetric around that y-axis and what you'll see an exponential decay is that things will get smaller and smaller and smaller but they'll never quite exactly get to zero it'll approach zero it'll ask them to towards the x axis as X becomes more and more positive just as for exponential growth if X becomes more and more more negative we asked them to towards the x axis so that's the introduction I do is a very specific example but in general if you have an equation of the form y is equal to a times some common ratio to the X power we could write it like that just to make it a little bit clearer well just a bunch of different ways that we could write it this is going to be exponential growth so if the absolute value of R is greater than 1 then we're dealing with growth because every time you multiply every time you increase extra melt your multiplying by more and more ours is one way to think about it and if the absolute value of R is less than 1 you're dealing with decay you are shrinking as x increases and I'll let you think about what happens when what happens when R is equal to 1 what are we dealing with in that situation and it's a bit of a trick question because it's actually quite straight I'll just tell you if R is equal to 1 well then this thing right over here is always going to be equal to 1 and you boil down to just the constant equation y is equal to a so this would just be a horizontal line