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We need to take care of a few nuts and bolts of exponential functions before we put everything together. First of all, what is an exponential function? What do they look like? What are they used for? These are all questions we are going to dive into. Don't worry. We'll keep things interesting with a few real life situations like zombies taking over the world or population explosions. We'll also hit a few key vocabulary words about exponential functions that will make things easier as we get into this chapter. Buckle up and get ready for a ride.
What is an exponential function anyways? There is one big thing to remember, the x is in the exponent. It doesn't necessarily have to be an x, it can be another variable.
These are all examples of exponential functions:
y = 5x
y = 3(0.5)x
f(x) = 2.9(3)x
f(t) = et
Don't get confused about the function notation. If you remember back to Algebra 1 and Algebra 2, f(x) is just the dependent variable which stands for y.
Exponential functions are found in everyday life. Some exponential functions help calculate loans and savings accounts. Some functions calculate the population growth of a city. And some functions calculate the amount of mildew that will eventually take over your kitchen sink. May the bleach be with you.
General exponential functions are in the form:
y = abx
f(x) = abx
where a stands for the initial amount, b is the growth factor (or in other cases decay factor) and cannot also be = 1 since 1x power is always 1. Notice the second equation was put in function notation, get used to seeing it both ways!
Exponential growth functions have b > 1, while exponential decay functions have b < 1. Also, the a value can tell us if the exponential curve is concave up (opening upwards) or concave down (opening downwards). If the a value is a positive number, the function will be concave up. If the a value is a negative number, the function will be concave down.
It's 2095 and cloning technology is now upon us. If our technology was advanced enough, we could clone ourselves. We might have a problem on our hands.
Assume that human cloning is a thing that is happening.
Let's say that each day, each clone would clone themselves. To build this exponential function, the amount of clones would double each day . Our function would look like this:
f(x) = 2x
where f(x) is the number of clones and x is the number of cloning days that took place
To find the number of clones of yourself after 10 cycles, we can simply substitute a 10 for the x.
f(10) = 210 = 1024
That means there would be 1024 copies of yourself after 10 days! Hopefully you stocked up on deodorant.
Some exponential functions can calculate the demise of the human race by the zombie apocalypse. It could start with one infection in one house. Yikes. Suppose a strange contagious virus began from a mutated gene. If one person was infected, they would die and the virus would take over their brain functions and their desire to eat anything living. The virus then would exponentially spread zombie to person, by a scratch or a bite, and new zombies would be created.
We could calculate the number of people that would be zombies after a time period with an exponential function like this:
f(t) = 2.1(4.9)t
where t is time in days and f(t) is the number of zombies that are "alive."
Let's graph this function and look at what happens over a certain time:
Notice the number of zombies increases without bound, even after only a few short days. Long story short, earth, we have a problem. There are zombies everywhere! This type of exponential graph is called exponential growth. Not only can the graph tell us that, but if you look at the original equation, we can tell it is an exponential growth equation. How? Since the general form of an exponential equation is:
f(x) = abx
Our zombie apocalypse equation has a = 2.1 and b = 3.05. Exponential growth functions have a b > 1. Also the a value is positive which shows a concave up graph. Can't zombies learn how to control themselves? There's got to be some type of medicine they can take to calm their need of feeding on people. Get ahold of yourselves, zombies.
Check out this table and let's build an exponential function:
Notice that f(x) changes by a product of 3. Each value is multiplied by 3 to get to the next value. This tells us it is an exponential function (compared to a linear function which would add or subtract a constant value).
So, what does that mean for me, you may ask? Well, it's pretty simple. What we need to know is: what happens at zero? f(0) = 9. Our initial value is 9. What are we multiplying each time by? 3. That means the base is 3. Hold on while we pull the rabbit out of the hat:
f(x) = 9(3)x
Tada! Congratulations, we just built our first exponential function based on a table.
Would you rather go on a ski slope or a water slide? Whichever you choose, that's the shape of an exponential function. We're going to show you their graphing behavior and effects of parameters in the equations. We're going to use vocabulary like growth and decay. Go grab your swimming suit or ski gear and let's jump on this slope, maybe we'll catch a little air.
Exponential functions are always curved and continuous, and they sort of look like "half of a parabola." You will notice that all exponential functions rise on the left or the right, and on the opposite side they look like they are converging to one y value. This y value is called a horizontal asymptote.
The Big Bad Horizontal Asymptote: there is an imaginary line in each exponential function that the curve will keep trying to touch but will never quite get there. This is one of those very tough concepts for the human mind to grasp. Picture yourself facing a wall and moving towards it. Every step you take is a little bit smaller. If you decrease your step by half each time, you will never actually make it to the wall. You can keep walking forward forever, but you will never get there. Ever.
Look at the graph of this function:
f(x) = 1.3x
Notice the graph above is increasing "without bound" from left to right. In precalculus terms, that means that as x approaches infinity, the value of y increases exponentially towards infinity. This graph is an exponential growth function. You might be confused whether 1.3 is a or b. Since the base is 1.3 and there is no number in front, a = 1 and b = 1.3. Notice the function has a b value that is greater than 1.
Here are some facts about the figure above:
Graph and describe:
f(x) = -2(3.9)x
The graph above is decreasing from left to right and the y values are going deeply negative as x increases. Notice how the y value on the graph is at around -200000 at x = 10. This also means that it is an exponential growth function. What? Exponential growth that is negative? Notice the equation has a b value that is greater than 1, but the a value is negative. The a value is causing the right end behavior of the function to fall.
Here are some facts about the graph above:
Graph and describe:
f(x) = 2(0.6)x + 50
The lovely graph above is decreasing from left to right but notice the y values are getting closer and closer to 50 as the x values increase. Also notice that the b value is 0.6. If the value of b is between 0 and 1, this tells us it is an exponential decay. Yum, zombie decay. Just kidding. This type of decay just means the y values are getting slightly less for each x value.
If you glance back at the original problem, you can see there is a +50 on the end of the function, and that shows the horizontal asymptote is at y = 50.
Here are some facts about the graph:
Graph and describe:
This graph is increasing from left to right and as you can see, the horizontal asymptote is at y = -10. Although the exponential curve is under the x-axis and has all negative y-values, it is trending towards one y-value of -10 which means it is another exponential decay function.
Here are some facts about this graph:
Graph this exponential function:
When graphing functions of any type, it's best to start out with an table. Choose different values of x and find the values of y, which is f(x). The table below shows how to calculate the y values for various x values we chose.
Now, we can simply graph the coordinates on a grid and we should get something that looks like this:
This is a classic exponential decay graph. More like the decay in our refrigerator. Decay that cannot and must not be stopped, as it may contain the antidote to the zombie virus.
Are you ready to spend some mulah? Or are you ready to save some dough? This section is all about using the equations for loans, investments, and present value which you can use when you are shopping for that car or house.
These formulas can arm you with ways to calculate how much money you are actually spending if you leave a balance on your credit card. We will learn how to plug values into these equations and quickly see how much you will pay over the life of the loan. Just won the lottery? You are in luck. You will be able to calculate how much money you can make over a period of time on your investments. Young mulah, baby.
Exponential functions really come into play when we are talking about loans or savings accounts.
The general equation for investments is:
where A is the amount of the investment or loan after a certain number of years t. P is the principal, or amount you started with. The r is the interest rate and the n is the number of times that the investment or loan is compounded every year. For your reference, here is a compounding table to help you decide what to use for n.
So you won the lottery, $1 million dollars it is! And because you are such a savvy saver, you have decided to invest this into an account that will get 5% annual interest rate which is compounded monthly. Let's calculate how much you will have in this account after 20 years if you didn't spend any of it.
Here is what we know:
P = 1000000
r = 5% = 5/100 = 0.05
n = 12
t = 20
Substituting those values in:
You can almost triple your initial investment thanks to exponential growth and compounding! The way compounding works is that every month, the monthly percentage rate is applied to the cumulative total of your savings. You get more money the more you save because your balance is always on the rise.
Time is money, and money is time. We want to know how much a loan or annuity is worth to you right now. That is called present value. In simple terms, present value means: "What is the future value worth today?"
For our purposes, we will look at a loan for a car since you already might be doing this or will be in the near future. The general form for a loan or annuity is the present value equation:
where PV stands for present value of the loan, R is the monthly payment, r is the annual percentage rate (APR), t is the total years for the loan, and n is the number of times per year the loan will be compounded. (Pssss BIG HINT: If you are asked finance questions involving a monthly payment, this is the equation to use.)
Calculate the present value of a loan if your monthly payments are $350 for 3 years, monthly compounding and the APR is 3.8%.
Here is what we know:
R = 350
r = 3.8% = 0.038
n = 12
t = 3
Substituting those values in:
The amount you will have paid on this car will be almost $12000 (in present value) after 3 years of payments.
Do you know what continuous growth means? Picture a balloon being blown up at a constant rate, rather than in little increments. Pop! Some exponential functions that model continuous growth use the e function. In this section we are going to take a look at a few different types of e functions.
We're going to talk about population growth, bacteria over-growth (eww), carbon dating, virus spread, and even more financial exponential growth. You know that age old question, "What does this have to do with the real world?" The real world is all over the place in this section.
Looking back at the equation for compounding money into a savings account, what happens if we compounded not just daily, but every hour, every second, and even faster than that? The universe would shatter to pieces, right? No, actually this is a real thing. It is called is continuous growth or decay.
Looking at the equation again, what we really want to know is when n approaches infinity. Remember that n is the number of times per year that the investment is compounded.
In the following example, look what happens to the , when we increase the n values (let's say we invested $1 at a rate of 5% for 1 year):
What happened there is we took the following limit:
One of math's forefathers, Leonhard Euler, in the 1700's, discovered this and named this the exp function (or the e function).
f(x) = ex
You have been introduced to the e value before (2.718281828...) which some people call Euler's number. It's an irrational number like π.
What is interesting with the above example is that if the rate was 100% (meaning 100% growth, not 5% growth rate, look what happens!).
Here is 5% rate of continuous compounding for one year if we started with $1:
f(x) = 1e0.05 = 1.051271096
If we had 100% growth rate:
f(x) = e1 = 2.718281828
This is Euler's number! The general equation for continuous growth, whether it be population, or zombie virus spread, or money is:
A = Pert
where A is ending amount, P is the initial amount, r is growth rate, t is time.
How many people will live in New York City in 2024 if there is continuous growth, the growth rate is 2% and the population was 8,336,697 in 2012?
A = 8,336,697e0.02(10)
A = 10,182,640
10 million people. eeeeeeek! This is just an estimate based on previous year growth rate but you can do this with any type of calculation that calls for continuous growth.
If a bacteria growing on a water fountain grows continuously and the number of bacteria is defined by the equation f(t) = 200e0.415t where t is the time in hours, how many bacteria did we start with and how many will there be after 6 hours?
The first thing to remember is that 200 is our initial count of drinking fountain bacteria. Remember back to our equation A = Pert. Don't let the different variables freak you out. P is the same as our 200. We didn't have to do any math to figure out the first part of the problem! Now let's solve for how many bacteria we will have after 6 hours of continuous growth. (Warning: you may not ever drink from a water fountain again.)
f(t) = 200e0.415t
f(6) = 200e0.415(6)
f(6) ≈ 2413
Gross! In just 6 hours, our nasty bacteria colony started a city.
In 2415, an archeologist finds a cell phone from 2012. Carbon continuously decays and can help determine the age of materials. Using the carbon dating technique, how much carbon was present using the following equation: C = 18e-0.0000132t?
Since carbon is the main element in plastic, there is a lot of carbon in cell phones. Our t value is the number of years the cell phone has been laying in the alley, which is 2415 – 2012 = 403 years. Did you notice that the exponent is negative? This is also another type of exponential decay.
C = 18e-0.0000132t
C = 18e-0.0000132(403)
C ≈ 17.9
As you can see, we started with 18 and after 403 years, we didn't lose much carbon. That's because it takes 50,000 years to really start breaking down.