hey there class we're going to begin chapter 5 now and this one is talking about exponential functions we're going to look at exponential functions some word problems with them um and later in chapter 5 we're going to be looking at logarithmic functions so that's what chapter five is about we're going to do about four sections in chapter five and so this first one here will be your your page of online notes so have your copy ready and we're gonna write exponential function right here okay so an exponential function we have the definition says that outputs are multiplied by a fixed number for each unit increase in the inputs so the general the basic form of an exponential function looks like this f of x equals b to the x okay b is a constant they call that the base and x is your variable it's in the exponent okay so another way to look at that would be y equals b to the x so we have our x power up in the exponent okay this is called an exponential function the two kinds we're going to look at is growth and decay so these are just the basic kinds okay we're just kind of hitting the highlights of these things these shapes i want you to draw x-axis and y-axis here x y i want you to mark off your y-axis a little bit here and put you a point through the first 0 1 right there and an exponential growth goes through that point and goes up from left to right it increases so the y-intercept this is an example of exponential growth the y-intercept that means it goes through the y-axis at the point 0 1 horizontal asymptote means that it has an invisible boundary down here it's not going to cross that it's not going to go below that and it's never actually going to reach that it never reaches the x-axis the x-axis serves as a boundary here okay on both growth and decay so growth increases it goes through 0 1 so that's just a basic picture of an exponential growth this is a basic picture of exponential decay i want you to label the x-axis y-axis mark off your y-axis a little bit put your point through there and decay decreases it falls down from left to right and again it's going to get close to that x-axis but never reach it so the y-intercept is again 0 1 and the horizontal asymptote is the x-axis okay so this is just some mathematical words in here okay asymptote means it's an invisible boundary okay it's never going to cross that x-axis i'm going to look at just some of these functions view a graph of each of these using your graphing calculator determine if it's growth or decay so part a right here i want you to look that this is exponential it has the x in the exponent okay this one is exponential it has the x and the exponent this one has the x and the exponent all right it's different than linear it's different than quadratic it's different than cubic it's different than all of those so i want you to look in your graphing calculator here i want you to press y equals and let's clear out my previous problem i was working on y equals i'm going to type in 3 to the x power and use your arrow key to get off of that there plus 2. 3 to the x plus 2. i'm going to do zoom 6. okay view a graph of this determine if it's growth or decay so there's that graph of that one and if you notice that one it goes through up a little bit higher there it goes through the point 0 3 why is that because they normally go through zero one but then this plus two moves it up two more spaces to the third spot so this one goes through zero three so we're just viewing the graph of it but i'm just actually going to sketch it here a little bit this one goes through 0 3 and it would be a growth okay view a graph determine if it's growth or decay all right next one this is a example of an exponential 1 3 to the x so i'm going to do parentheses around my 1 3 1 divided by 3 raised to the x power there i'm going to do zoom 6. okay and that one goes through zero one you can look at your table for that let me see what my table is at table settings zero count by ones and now look at my table it goes to the point 0 1 0 1 there you can see that let me go back to my graph it goes through that first point there and i'm just sketching that just viewing it really in your calculator i'm just sketching it so you can have a copy on your notes and that would be an example of a decay exponential decay so this is the shape of these graphs now the letter e i don't know if we've had that on some of our examples before but it shows up here in this page e is on your calculator i think we did it in week one but e is over here on the divide key take a look at where e is at i want you to go to the quit button there and i want you to press in the e okay there's the e e is a number okay he's kind of like pi he's a number e is 2.72 um we're going to use e on this calculation go to y equals and i'm going to clear that one out and do e to the x and do zoom six all right it's a growth it goes through zero one you can double check that on your table take a look there zero one and that was a growth and now i'm going to actually calculate some things for this problem so i'm going to use this one here to calculate these and i'm going to round it three places it says to find find f of 1. now remember find means find the y value that goes with that x value that's f of x that's your x value so i'm using this problem here and so that means plug in 1 for x right there plug in negative 4 for x right there plug in positive 4 for x right there all of these x values can be found in your table so i'm actually going to use my table to find these for this problem i'm going to go to my table find f of one so here is x is one and to the right of that is my y value that goes with that x value that's what find means that's function notation find f of one and so when i highlight it there i look down at the bottom here and it tells me all the digits the table only shows you part of the digits but if you highlight and look down at the bottom it shows you all the digits so i'm rounding three places two point seven one eight okay that's f of one find f of negative four means to find the x value of negative four so i'm just looking at my table i'm going over to my x column and i'm going to f of negative four here and i'm going to highlight the right side over here and i'm around that three places point zero one eight okay find means find the y value that goes with that x value that's f of x now find f of four and it is going to be 54.598 round three places there okay so that's just your basic shapes for exponential we're going to just look at a little word problem with it turn the page please all right let me make sure it's lined up all right this one is a word problem for exponential and it's talking about advertising if you'll just read through that it's talking about weekly sales possibly drop it says after advertising ends so this is going to be the model of an exponential decay okay so you have advertising done so once the advertising is done a lot of people are buying that product but when the advertising ends possibly the sales decay they decline so this is an example and it's given to us by this exponential formula there's our x and it has s s represents the sales and x represents the number of weeks number of weeks after the campaign ends okay s is the sales x is number of weeks after campaign ends after it ends okay what is the level of sales when the advertising campaign ends so immediately when when it ends would be at week number zero okay one week after it ends you'd plug in a one for x two weeks after seven weeks after whatever what is the level when it ends so that would be starting at week zero so you're gonna let x equal zero and you're going to plug in that number into our formula gave us the formula this is our decay formula okay i'm going to look in my table for this problem okay so i'm going to type that in y1 press the y equals and here we go i'm going to type that in 1000 parentheses 2 raised to the negative 0.5 x and use my arrow to get down off of that and press the other parentheses there let me double check my formula okay so my table i go to second and then the word table there and i'm going to the zero plug in x is zero so i'm looking in my table where x is zero and the sales there is one thousand so right when the campaign ends the sales is one thousand next question what is the level of sales one week after well x was the number of weeks so one week after i'm gonna let x equal one plug in one into the formula and we have our table right here to look at it x is one i go over to the x column x is one right there and that sales is 707.11. you can see where the sales declined okay this says use a graph to estimate the week find the week find the week number in which sales equals 500. so i'm going back up to my formula for the letter s right there and i'm going to plug in 500 and i have my formula here 2 to the negative 0.5 x okay that's already in y1 i'm going to put this in y2 if i do y equals i'm going to put this in y2 500 i'm going to get a window for this and i'm going to look at this graph i'm going to look at this decay and i'm going to look at this 500 remember it draws 500 it's going to draw a straight line across at 500 horizontal line so let's talk a minute about our window x is the number of weeks and we have our x min x max s is our sales that is the letter y that is our sales and we're going to have our y min and y max okay i need a window for this um our window comes from our table and this answer is actually in our table i don't know if you saw it yet but let's go back to our table this question says estimate the week in which sales is 500. well if you look for at the number of weeks here week zero the sales was a thousand week one the sales was 707 week two the sale was 500. so there's the answer week three was 353 but listen i'm going to look at my x column here and for the number of weeks my minimum i'm going to start at 0 and i'm just going to go let's say 6 and then 7 and then 8 and then 9 i'm just going to go to about 10 right there that's enough and then sales my highest one was a thousand and my lowest one was 44 my y value so sales my lowest one was 44 so i'm going to go to zero and my highest one was a thousand within that range so i'm just going to type this in for the word window and i just want you to see the graph here i'm going to adjust my window 0 and 10 leave your scale leave your scale alone and then 0 and 1 000. i just want to remind you of how to work this problem if that answer is not in the table and press graph so there's my y1 there's my decay here's my y2 my 500 coming right across there and what i do is calculate number number five not number two that's the zero calculate number five the intersect calculate number five the intersect second calculate number five and you press enter enter enter it'll blink on it until that intersection is two x was the number of weeks so that means on week number two why was the sales the sales was 500 so is week number two so x equals two for that intersection okay just remember to see your graph you can make a window adjustment based on what the info in the table is so knowing what your variables stand for helps you adjust your window according to this model will sales ever fall to zero sales is my y variable will the sales right here see them decreasing will they ever fall to zero well if you look at your table you remember you can keep going down for your sales here they're going to keep decreasing but remember what we said earlier that our x-axis our x-axis is a asymptote horizontal asymptote and that means that graph never reaches it it might get close to it but you see that e to the negative 4 there that just means that's a very small number it's never actually going to reach 0 it's going to keep getting smaller and smaller and smaller and smaller remember the x axis is a horizontal asymptote a horizontal asymptote okay so no it never reaches zero it keeps decreasing and decreasing and decreasing and decreasing but never reaches zero the sales all right make sure you can see all of that okay turn the page for the next problem and we're going to do a little bit more with exponential here and let me just make sure it's lined up and this is this exponential because your variable in this case the t is in the exponent okay so this is this example of exponential but it has some extra variables in there because these word problems deal with investing money okay so we're going to do a word problem here and i think the last example is dealing with well it goes back to the e so here's a couple of word problems dealing with these investing money and it's talking about let's just look at both of them okay there's two formulas we're going to look at similar to the exponential function and they are used to solve investment word problems so they're talking about the future value of your investment you're going to put money in an account and it's talking about the future value so the future value is your s variable with your investment of p dollars that's how much money you're putting in invested for t years t is time in years at an interest rate of r okay so that's your four variables s p r and t there's your s p r and t if you have your rate interest rate for example eight percent you have to convert that to 0.08 in your formula and then another form of two formulas circle that one circle that one this other one is talking about investing your future value is s dollars for investing p so p is what you put in s is what you get out t years at an interest rate of r and this one is compounded continuously this one up here was compounded annually that's the difference this one is compounded annually this was compounded continuously so that's the difference between those two okay compounding continuously compounded annually so we have these two investment formulas and i just go along with our with our exponential uh equation here for this section so here's an example with them find the future value for sixty thousand dollars invested for twenty years at five percent compounded annually so this is the first formula compounded annually so we have s equals p parentheses one plus r to the t power okay let's look at these s find the future value that's what we don't know for sixty thousand dollars invested remember p is your dollars invested so on this problem p is sixty thousand for 20 years that's your t time in years t equals 20 20 years 5 that's your r interest rate and you have to convert it to a decimal and we just plug that in the formula s equals 60 000 1 plus r that's the 0.05 raised to the t power which is 20. i plug in the p i plug in the r and i plug in the t this is 1 plus r is 1 plus 0.05 and i just type that in my calculator so i'm not ask it's not asking you to graph it's just asking you to calculate something so i have 60 000 1 plus 0.05 parentheses raised to the 20 and hit enter so there's my answer s equals let's see what i got 159 197.88 159 000 197.86 that's my future inve my future value for 60 000 invested for 20 years compounded annually that's the difference between those two this one this next one is compounded continuously so this is the second formula so if ten thousand dollars is invested for fifteen years at ten percent what is the future value so what is the future value s that's what we don't know this is the second formula p and ten thousand dollars invested that's the p dollars invested so on this one ten thousand dollars fifteen years that's your t time fifteen years r is your interest rate ten percent so that's point ten so this is the second formula p e to the r t p e raised to the r t second formula right there so what is the future value s i don't know so let's plug in these numbers 10 000. e is a number as a button on your calculator you type in the e and that's raised to the rt 15 which is multiplied or times by whatever you want to put there times by point 10. e is raised to that number r multiplied to 10 r times r times t are uh 15 times by point 10 okay so i'm typing that in 10 000. e i'm typing my e after that and it's raised and i'm just typing it in with a parenthesis 15 times by 0.10 and i'm putting parentheses around it and pressing equals here so there's that one forty four thousand eight one six point eight nine all right next question after how many years will the value of the original investment double my original investment was ten thousand dollars when will it double after how many years so they want the future value s to be twenty thousand dollars on this one so s is twenty thousand dollars p is your original investment ten thousand dollars time we don't know when will it double and the r is 0.10 so look at my formula here s equals p e r to the t it's compounded continuously s is going to be i'm going to plug in the s here 20 000. equals p which is ten thousand e to the rt r was point ten and t is time i don't know that's my x variable so after how many years i'm solving for t how do i solve that i'm going to put this in y1 i'm going to put this in y2 and i'm going to calculate number 5 that intersection and it's going to tell me t okay i might need to play around around with a window obviously because this is larger numbers so let's look at my x min x max y min y max okay let's discuss that for a second can you see those over there okay just make sure you can see them all right so what does x stand for in this problem well that's my x variable it stands for t time what does y stand for that's my y my s my t and my s are my two that's my future value so my future value needs to go up to at least twenty thousand dollars so i'm gonna go zero up to let's just say 30 000. okay x is time well in 15 years what was that one 15 years it went to 44 000. and so i can look at my table for this let me type this in my y1 y equals clear clear let's go ahead and get it in y1 10 000 that was my original investment e raised to the rt which is point 10 x and then use the arrow to get off of that okay and in y2 i'm going to type in the 20 000. go to your table here at table settings let's go back zero count by ones and look at the table here one year two three four five all that stuff so i want to know when this thing doubles so it's going to double there's sixteen thousand dollars there's eighteen there's twenty thousand are already right there at the seventh year eighth year ninth year so listen at least by ten years and then right here fifteen years it was at the forty four thousand so it doubles somewhere around the sixth seventh or eighth year so i'm looking at my table remember your table helps you find a window so it says after how many years is this thing going to double so i'm going to go 0 to 10 there press the word window i know it happens within the 10 or before 10 years so 0 and 10 and go down 0 and 20 or what i do 30 000 sorry 30 000. press graph i'm going to see my investment growing i'm going to see that 20 000 line come right through there and i'm going to second calculate that intersection and it's going to tell me exactly when enter enter and enter after how many years will my ten thousand dollars double to equal twenty thousand dollars so i did the intersection and x represents my t time and year six point nine three years okay after how many years 6.93 years so i'm just walking you through that window there for that word problem all right let's turn the page and look at the last example back to a word problem with the k here and this is just a scientific word problem okay so this one's talking about the amount of an isotope is given by this this formula a is the amount t is the time time in years that this thing decays okay so this word problem just fits in with our section here on exponential growth and exponential decay so the initial amount present is 800 grams of this radioactive isotope so somebody's in a scientific lab they got this product and it's decaying how many grams remain after 20 years t is the time in years so i plug in t equals 20. so i have a of 20 that's what i'm calculating 800 e to the negative 0.02852 times by 20. okay i can find that in my table or i can type it in they gave me my formula i'm just using my formula and plugging in something um i'll just type it in right here instead of going to my table let's see what i want to type in 800 e raised to the negative 0.02852 times about 20 after 20 years and press enter i get 452.24 grams of that radioactive substance after 20 years okay sketch a graph of this function for 0 to 100 years so we're going to have a window to look at x min x max y min y max what does x stand for in this problem if you look at x the x is always on this side of the equation this is my y this is x this is y so x over here is my t time y over here is my a amount so let's type this in y one e to the raise it up to the negative point o two x there it is let's go to my table settings zero count by ones because x represents your years here so they're at zero years is at the 800 one year two year three year four year and all that stuff so they wanted me to graph from zero to a hundred years zero to a hundred they told me that so i'm just kind of looking at my y value here my maximum y value started at eight hundred and i'm just looking at a decay here over the years 15 years i'm just looking that through here wanting me to go to 100 years okay so let's go zero count by table settings zero count by tens i'm just kind of skip around here go back to table zero ten twenty thirty forty fifty years there's a hundred years it gets down to forty six so never reaches zero so i'm just gonna do my minimum here my amount over here it never reaches 0 but i can use 0 as my boundary there okay so i have to set my window window 0 to 100 years and the amount 0 to 800 was my max and press graph so here's that amount decaying over the course of 100 years it got down to like 40 something out there at the end but never reached zero and if you remember this decay has that horizontal asymptote on the x-axis it's never going to reach the zero so this is our t time and this is our a amount it started at the 800 and this is an example of a decay okay if the half-life is the time it takes for half to decay use graphical methods to estimate the half-life so listen the half-life is the a so half of this thing it started out at 800 half of that would be 400. so i'm going to plug in my a as 400 and my problem was 800 e to the negative 0.02852 t okay so this is in y1 i'm going to put this in y2 press your y equals put that in y2 the question says estimate the half-life find t for when half of it decays so a is your amount t is the time and number of years so when does the amount reach half of what it started out as started out at 800 you can see that in the formula or you can it also told you in the problem so i'm putting in the 400 what am i going to see you need to know what you're going to see there you're going to see the horizontal line go through there at the 400. how am i going to solve that calculate number five intersect second calculate 5 enter enter enter 24.30 24.3 years is when that thing reaches a half-life so it takes a scientific radioactive example there 24 years for half of it to decay okay so listen just for your exponential growth and decay you're going to do some word problems some scientific decay and all that you're going to do some investment word problems with that and you got your basic formula of 1 b to the x so x is in the exponent we have our growth and decay so you'll just do some mymathlab for this assignment and thank you very much