Intro to exponential functions

in this video I want to introduce you to the idea of an exponential function exponential function and really just show you how fast these things can grow so let's just write an example exponential function here so let's say we have Y is equal to 3 to the X power notice this isn't X to the 3rd power this is 3 to the X power our independent variable X is the actual exponent so let's make a table here to see how quickly this thing grows and maybe we'll graph it as well so let's take some x values here so let's say let's start with X is equal to negative 4 then we'll go to negative 3 negative 2 0 1 2 3 & 4 and let's figure out what our Y values are going to be for each of these X values now here Y is going to be 3 to the negative 4 power which is equal to 1 over 3 to the 4th power 3 to the third is 27 times 3 again is 81 so this is equal to 1 over 81 when X is equal to negative 3 y is 3 let me do this in a different color that color is hard to read y is 3 to the negative 3 power well that's 1 over 3 to the 3rd power which is equal to 1 over 27 so we're going from a super small number to a less super small number and then 3 to the negative 2 power is going to be 1 over 9 right 1 over 3 squared and then we have 3 to the 0 power 3 to the 0 which is just equal to 1 so we're getting a little bit larger a little bit larger but we're going to see we're about to explode now we have 3 to the first power that's equal to 3 then we have 3 to the second power right Y is equal to 3 to the second power that's 9 3 to the third power 27 3 to the fourth power 81 if we were to put the 5th power 243 let's graph this just to get an idea of how quickly we're exploding so let me draw my axes here we draw my axes so this that's my x-axis and that is my y-axis that is my y-axis and let me just do it in increments of five because I really want to get the general shape of the graph here so let me just draw a straight a line as I can let's say this is 5 10 15 actually I won't get to 81 that way I want to get to 81 actually well that's good enough let me draw it a little bit differently than I've drawn it so let me draw it down here because all of these values you might notice our positive values because I have a positive base so let me draw it like this like that good enough and then let's say have 10 20 30 40 50 60 70 80 that is 80 right there that's 10 that's 30 that'll be good for approximation and then let's say that this is negative 5 this is positive 5 right here and actually let me stretch it out a little bit more say this is negative 1 negative 2 negative 3 negative 4 then we have 1 2 3 & 4 so when X is equal to 0 or equal to 1 right when X is equal to 0 Y is equal to 1 which is right maybe right around there when X is equal to 1 Y is equal to 3 which is maybe right around there when X is equal to 2 y is equal to 9 which is right around there when X is equal to 3 y is equal to 27 which is right around there when X is equal to 4 y is equal to 81 Y is equal to 81 so you see very quickly this is just exploding if I did 5 go to 243 which would wouldn't even fit on my screen when you go to negative 1 we get smaller and smaller so negative 1 we're at 1/9 I mean eventually not even going to see this it's going to get closer and closer to 0 as this approaches larger and larger negative numbers or I guess I should say smaller negative numbers so 3 to the negative thousand the negative million we're getting numbers closer and closer to zero without actually ever approaching zero so as we go from negative infinity X is equal to negative infinity we're getting very close to zero we're slowly getting our way ourselves away from zero but then BAM we once we start getting into the positive numbers we just explode we just explode and we keep exploding an ever increasing rate so the idea here is just to show you that exponential functions are really really dramatic that they're they're kind of well they're you can always construct a faster expanding function for example you could say Y is equal to X to the X even faster expanding but out of the ones that we deal with in everyday lives this is one of the fastest so given that let's do some word problems that just give us an appreciation for exponential functions so let's say that someone sends out a chain letter on let's say in week one week one someone sends a chain letter to ten people to ten people and the chain letter says you have to now send this chain letter to ten more new people and if you don't you're going to have bad luck and your hair is going to fall out and and and you'll marry a frog or whatever else so all of these people agree and they go and each send it to ten people the next week so in week week two they go out and each send it to ten more people so each of those original ten people are each sending out ten more of the letters so now a hundred people have the letters right each of those ten sent out ten so that a hundred letters were sent out so let me see sent ten were sent here 100 were sent in week three what's going to happen each of those hundred people who got this one they are each going to send out ten assuming that everyone is really into chain letters so a 1,000 people are going to get in and so the general pattern here is the people who receive it so in week in week n right where n is the week we're talking about how many people receive the letter in weekend we have ten n to the nth people receive receive I before E except after C receive the letter so if I were to ask you how many people are getting the letter on the sixth week on the sixth week how many people are actually going to receive that letter well what's ten to the sixth power ten to the six is equal to one with six zeros which is 1 million people are going to receive that letter in just six weeks just sending out ten letters each and obviously in the real world most people Chuck these in the basket so you don't have this good of a hit rate but if you did if every ten people you sent it to also send it to ten people and so on and so forth by the six week you would have a million people and by the ninth week you would have a billion people and frankly the week after that you would run out of people I'll see you in the next video