today we are going to start chapter 7 chapter 7 is about exponential and logarithmic functions specifically 7.1 is about exponential functions growth and Decay talking about what those terms mean and what those graphs look like and what the equations that look that go along with those graphs also look like an exponential function is a lot different than the other functions that we've dealt with last couple chapters we've talked about polinomial functions like x to the 2 x to the 3r where we see the X as the base and a numerical exponent onent an exponential function means that the variable is actually going to become the exponent so this is what the exponential function looks like y or F ofx equal a * B raised to the X and the key thing there is that that variable is now the exponent instead of being the base over here to write out an exponential function you really just need two things you need an A and you need a b the a is your representative of your initial amount a lot of times in word problems you're going to have some starting amount that is what the value represents it also in terms of a graph is going to represent your Y intercept and we're going to see that in the next slide your B value is your growth factor that's really how much your a is changing every time that you go through some amount of X so it's really what you're multiplying a by over and over again that b value the base is the thing that's going to show the actual growth or the Decay if B Val if the B value is between zero and one something like 0. five3 that is going to show Decay if that b value is greater than one it's going to show growth and again we're going to see when we actually graph these what it means to be decaying what it means to be growing so to actually talk about being able to graph some exponential functions to actually see what these look like we're just going to start by going through and creating some tables when we normally graph them we don't need to worry about being able to create these tables so we start off with plugging in the zero 10 * 3 to the 0o power what this means when we see this here Order of Operations we have to simplify out the 3 to the 0o power exponents comes before the multiplication here 3 to the zero we know is 1 1 time 10 is going to give us 10 we do the same thing for the one 3 to the 1 is three * 10 is 30 we do the same thing for the rest of these we will get 90 and 270 when we get to the negative exponents 10 * 3 to the 1st we still need to simplify this down but when we see that negative exponent what a exponent does is it takes the base and is going to flip that base the word that we're looking for here is reciprocal we take what that Bas is we find the reciprocal of it which the reciprocal of three is 1/3 and we raise it to the positive exponent that we see so it's going 3 to the 1st becomes 1/3 raised to the positive 1 which 1/3 to the 1st is 1/3 * 10 is going to be 10/3 we do the same thing for -2 so it's going to be 3 the2 which means this is 1/3 to the postive 2 so that's going to be 1/3 * 1/3 which is 1 9th * the 10 is going to be 10 over9 same thing for the -3 this will become 10 over 27 to be able to graph these we can just go through and make a a fairly EAS fairly easily fairly nice x- axis because we have nice numbers here the y axis we just going to have to be a little bit more lenient with the first one and the important one that we place is the 010 this is our y intercept so we could say this is approximately 10 place that value there then we go through we can place the rest of these let's start off with the negatives negative 1 is 10/3 which it may help if we change this to a decimal 3.3 repeating which would be about there this one is going to be 1.1 repeating which is pretty close to the xaxis this one here is 370 repeating which is again very close to the x axis the one two three the one is 30 which is going to be way up here the two and the three we don't necessarily need to worry about because they're just going to get so extremely big off the graph we draw this curve that goes through it if we notice how it kind of Curves this way it's getting closer and closer to the x-axis but it's not actually going to go through the xaxis so we're just going to leave it like that and that's what our exponential function looks like let's take a look at this other one 10 * .5 to the X we plug in the zero and again order of operation says we need to raise it to the zero power first before we do the multiplication 0.5 0 is 1 * 10 is going to give us 10 again we raise it to the first so 0.5 to the 1 is 0.5 * 10 is 5 this will give us 2.5 and 1.25 we do the same thing for the negative-1 10 * .5 ra to the 1 so here's where we have to do the reciprocal of 0.5 which if you think 0.5 is really 1 half to begin with the reciprocal of that is going to be two so it's going to be 2 to the positive 1 which is 2 * 10 so this going to be 20 then 40 and then 80 when we plug the rest of those in we make the XY table the exact same way we can put the 10 here to say this is going to be our Y intercept and then we just go through and plug the put place the rest of these 1 five 2 2.5 3 1.25 one negative or neg 120 which will be up here and we see that our curve is going the other direction so there's there's our curve going the other way in comparison to this one what we talked about on the last slide back here was if B is between zero and one it shows Decay B is greater than one it shows exponential growth our B value was a three since that's greater than one this is what exponential growth is exponential growth means as you look at this graph from left to right and it's going to apply to the other one as well is that as we look at this from left to right our graph is going to grow namely the Y value it's what happens to the yvalue to be able to describe that exponential growth and decays how does y change in this case our Y is getting a lot bigger that's why it's a growth over here our B value was 0. five which is between the zero and one that's why we have a Decay if we look at our yv values as we're going from left to right the yv values are getting smaller and smaller so that's why again it's Decay now if you take a look at both of these both of these graphs are going through the Y AIS they both go through at that 010 which is the a value that we have out in front of what's being raised to the power a is 10 here a is 10 here but the other thing that you may notice is that they're both getting closer and closer to the x-axis without actually going through the x-axis when that happens what we do is we call that x-axis in ASM toote again the way that we say that the p is silent it's called an ASM toote what that means is the graph is going to get closer and closer to that line but it's not actually going to go through that line as a little bit of a disclaimer there are graphs that will actually go through the ASM toote for but for our purposes what we're going to be working with this year they won't actually go through that ASM toote so this line that we have here the x-axis is our ASM toote the way that we write out the x-axis is the x-axis is a horizontal line and any horizontal line is always written out as y equals some number and it's what point does that go through on the Y AIS this is the x or what point does that go through on the y- axis it's the x- axis that we're trying to write out which is the horizontal line so it's going to be y equals something and the point that the x-axis goes through the Y AIS is right there at zero so the equation of the x-axis is y equals 0 so our ASM toote is yal z an easy way to think about this for you Lord of the Ring fans out there is at one point in time Gandalf goes up against the bow Rog and Gandalf is basically the ASM toote to Bow Rog if you don't understand that ask about it in class tomorrow so let's talk about going through and actually going through and graphing one of these if we want to go through and graph this let's take a look at this second one we're going to leave this first one blank for a while is if we look at here first thing we see is we have a number raised to the X power as soon as you see that you have a number raised to the X power that is going to represent your B value so since we know B is8 we can go through and already Define it as a Decay but if we want to talk about the a value because we need to know the a value the a value should be right here since we don't see an actual a value the number that we're going to put in is a one and that's going to help us in terms of being able to sketch the graph when you sketch the graph you have to show two things the first one that you have to show is that Y intercept so we're going to put this to say this is the value of one and our graph is going going to go through that then we have to show a Decay which from what we said before that means as you're going from left to right the graph is going to be decreasing so from left to right the graph is decreasing but as it gets closer and closer to that x-axis it's not actually going to go through the xaxis and when we sketch the graph here we do not want to show that this is going through the x-axis that's is a key part of an exponential function the Y intercept and then showing that that we have an Asm toote on that x axis way that we show that is that it's not going through that x-axis part three labeling the Y intercept well we've already labeled the Y intercept in our sketch we don't need to worry about that the ASM toote which we said was the xaxis again the x-axis is a horizontal line so we just have to write out the equation of that horizontal line any horizontal equation line equation is going to be y equals well what part of the y- axis does that go through it goes through zero on the Y so our horizontal line is y equals 0 that's our ASM toote our domain and range domain and range are talking about first off domain is talking about what x values can this graph reach which when we take a look at this it's what are the possible X values that we can use for X when we put into this equation where well if we take a look at this graph we could see that it's going to go to the left because it's going up but at the same time that it's going up it's also going out so it's going to reach negative Infinity at some point as it goes to the right it's going going to keep on going to the right which means that it's going to reach positive Infinity so there's no spot on the x- axis that this graph cannot reach no XV value that we cannot use the way that we're going to write that out then since we can use every possible x value is our domain is just going to be all real numbers our range on the other hand our range what we're looking at is what possible yv values can we get out of this equation is in what possible yv values can this graph reach well we know that the graph is going up so as it keep keeps going up it's going to reach a positive Infinity but as the graph is going down it's not going to go all the way down it's only going to reach towards the x-axis it's not going to hit the x-axis but the value that it's going to go towards is zero so it's really going to cover every value from zero up to positive Infinity but we can't include that zero so we can say that the graph is going to get close to zero and it's going to go up to positive Infinity continually going up so our yv values are going to be greater than zero now a lot of times what people like to do is they like to write out our domain and range in set builder notation and the way that we use set builder notation is we take an inequality that we have here which is inequality that we already wrote and we just add some brackets around it we're going to put a bracket at the end to say that we're ending a bracket at the beginning and then we're going to say why with this horizontal line here to represent this inequality that we have what this means is these brackets are talking about a set as in a group of numbers this Y is just saying that we're dealing with values of Y we don't know what those values of Y are but this line is going to tell us that we are going to Define these values of Y as Y is greater than zero the way that we read this is those brackets are talking about the set of the Y is just going to be all y's this horizontal line here means such that and then the inequality just means the inequality such that Y is greater than zero and this is set builder notation you guys are going to be using this a lot in FST and and pre-cal this other one I want you guys to take a couple minutes now or either at the end and I want you to do this one for class tomorrow so this is going to be a homework problem for tomorrow yes to be able to do the same thing for this one as what we did for this one before before we actually get into the word problems we're going to talk about how word problems are written they don't actually tell us what that growth factor is we have to figure out what that growth factor is a lot of word prompts are going to use words such as increase or decrease by some percent as in each time this value changes it's going to change by some percentage of the value that is currently at so what we want to do is instead of saying it increases or decreases by some amount we want to figure out what that actual growth factor is think about if you go to a store and you buy buy something that has sales tax all right what you're going to do is to figure out your total price for the sales tax let's say we have a a 7% sales tax we use this one here that sales tax is going to increase that amount well to figure out what that sales tax is you have to multiply that by 07 now let's say we don't actually know what that number is so we're going to call it X we take that price that we have we multiply it by 07 but once you get that sales tax you're going to actually have to add that to the original price to get the total well 007x plus X which this is like 1X would give you a total of 1.07 X and that's going to be the total price that you're going to get in one step instead of saying let's multiply by 07 and then add it to the original we can just multiply by 1.07 another way to think about that is when it uses the word increases by 7% think increase means plus I have to add to that 7% well am I adding 7% to I need some other percentage to add 7% to so it's going to be 100% And 100% plus 7% is 107% which is 1.07 and that's the number that we're going to use as the growth factor for that problem if it's a decrease problem as in decreases by 12% you can do that same thing decrease means minus 12% with 12% what are we subtracting 12% from that would be 100% which would give us 88% or 88 as a growth factor so let's actually take a look at a word problem Tony purchased a rare 1959 Gibson L Paul guitar in 2000 for $122,000 expert estimate that its value will increase by 14% per year use a graph to find when the value of the guitar will be $60,000 so since it gives us this information here that it's increasing by 14% each year implying that it's not going to be the same amount that it increases by because 14% of 12,000 is different than 14% 60,000 at some point in time this is going to be an exponential function as in y = a * b x remember the a value is the initial amount so in this case the initial amount is what he purchased it for which was 12,000 so it's going to be y = 12,000 multiplied by B which is our growth factor so we need to figure out our growth factor the percentage is going to give us our growth factor it's 14% it's increasing so we're going to say we're adding 14% each time but what are we adding 14% to that's 100% so it's 114% is what we're actually going to use and that number that we put in here is the decimal equivalent 1.14 raised to the X so there is our exponential equation that we are going to use now all we have to do is answer the question which it says when will it be $60,000 what we want to know is what variable is actually going to be equal to $60,000 what variable is going to be the 60,000 normally X deals with time since we're talking about years here he bought it in 2000 so X is time so what variable is it it's going that variable needs to be y now what you may think is we're going to take this Y and we're going to plug this Y in to this equation here and say that it's going to be 60,000 = 12,000 * 1.14 raised to the X we get rid of that 12,000 by dividing by it and we get 5 = 1.14 raed to the X the problem is we can't just go through here and divide by 1.14 to get the X by itself because this X is not being multiplied by that 1.14 is being raised to it so right now we do not have a way to actually solve that 1.14 raised X we don't know how to get the X by itself we will in a little bit of time but we don't know right now so we have to go back to the drawing board and talk about how we can solve this well we have this equation that says yal 60,000 and another equation that's y equals this this being just y equals something is a horizontal line and the other one is an exponential function so what we can do is we can actually go through and graph these two lines now when we talk about being able to graph them or graph these two equations this being an exponential function is going to have something that's going to go through and it's going to look like this this one being a horizontal line is going to go through and look like this and really to find out what that X values B where these two are equal is going to be that spot right there all we have to do is talk about how to actually graph these as in where are we going to find that intersection at if you take a look remember this is your a value so this is really your Y intercept you're going to have a y intercept of 12,000 and this one being a horizontal line at 60,000 means that it's going across here at 60,000 so what we want to do is we need to make our y value of our window our y max value higher than 60,000 000 since we know that's going across at 60,000 maybe we want to make it 70,000 so that we have enough of a of a big enough window to actually see where it's going to cross and then we can go through and try some different X's but think about this the X's you know if it's 12,000 and it's increasing by 14% each year it shouldn't take too many years to actually reach that 60,000 Mark to be five times as expensive so we don't need to make our X's too big we don't need to sit there and make our x max like 300 or 400 because it's not going to take that many years for it to be 60,000 so what we can do is we can make our X's a little bit bigger 20 just to try to get a better representation of where this would be if that's not enough we just need to make our x max a little bit bigger and when we go through we find where those two cross we just have to find the intersection point which the calculator is going to help us with this to find that intersection Point remember this is where we use the calculator second trace and under second Trace we are going to select intersect and when we do the intersect it's going to ask us first curve second curve guess we run through all that and it should give in intersection Point says X = 12.28 Y = 60,000 which that's the amount that we were looking for which means this is going to be the number the amount of time that it's going to take which we talking about it it just said 2000 we're talking about years so it's going to be 12.28 years to be able to get a value of 60,000 12.28 years means sometime in 2012 but since at the beginning of 2012 since we're just talking about specifically years at the beginning of 2012 it's not quite going to be 60,000 so if we had to estimate then we would really say it's 2013 this though is a better answer because we're talking about an exact answer if it's asking specifically what year we'd want to say 2013 but if it's not saying specifically what year we can just say how much time would it take to be 60,000 which is 12.28 years we have one more question that we're going to take a look at but what I want you guys to do is I want you to do this one for homework as well I want you to go through and write out the equations and see if you can graph it if you have a graphing calculator go through and graph it if you don't I want you to write out the equations and I want you to sketch a graph of what this would look like this time though it says decrease so you have to be careful that's going to be the opposite of what we did in the last one again if if you don't have a graphing calculator write out the two equations that you would use and sketch a graph and then Circle where those two line those two curves would intersect