If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Intro to exponential functions

An exponential function represents the relationship between an input and output, where we use repeated multiplication on an initial value to get the output for any given input. Exponential functions can grow or decay very quickly. Exponential functions are often used to model things in the real world, such as populations, radioactive materials, and compound interest. Created by Sal Khan and CK-12 Foundation.

Want to join the conversation?

  • leaf green style avatar for user Sivaramari31
    What does he mean by "exploding"
    (20 votes)
    Default Khan Academy avatar avatar for user
  • piceratops seed style avatar for user Sachi Goli
    what happened to little x=-1?
    (16 votes)
    Default Khan Academy avatar avatar for user
  • starky seedling style avatar for user hailee.292682
    how do you convert graph to equation in exponential functions?
    (1 vote)
    Default Khan Academy avatar avatar for user
    • mr pink green style avatar for user David Severin
      So the standard form for a quadratic is y=a(b)^x. So one basic parent function is y=2^x (a=1 and b=2). Learning the behavior of the parent functions help determine the how to read the graphs of related functions. You start with no shifts in x or y, so the parent funtion y=2^x has a asymptote at y=0, it goes through the points (0,1) (1,2)(2,4)(3,8),... So we find the common ratio by dividing adjacent terms 8/4=4/2=2/1=2. Thus, we find the base b by dividing the y value of any point by the y value of the point that is 1 less in the x direction which shows an exponential growth. It works the same for decay with points (-3,8). (-2,4)(-1,2)(0,1), So 1/2=2/4=4/8=1/2. It will also have a asymptote at y=0. Next, if we have to deal with a scale factor a, the y intercept will tell us that. With 2(2)^x, you double all the y values to (0,2)(1,4)(2,8)(3,16) - note that 16/8=8/4=4/2=2, so we still get the same base, but the y intercept tells us that the scale factor is 2. Similarly, if we have (0,3) and (1,6) our base is 6/3=2, but the scale factor is 3, so we have y=3(2)^x. This will work the same for decay functions, but the base will be a fraction less than 1.
      So the next easiest is to shift up and down by adding a constant to the end. This can be easily be determined by a change in the asymptote. If you see an asymptote at say y=3, then "act like" this is the y axis and see how far points are away from the this line. Thus y=2^x + 3 would have points (0,4) 1 away from asymptote, (1,5) two away from asymptote, etc. Thus, you would have to do (5- 3)/(4 - 3) to get 2/1=2 as the base. This is a good introduction, which is good for all but shifts in the x direction such as y = 3 (4)^(x+2) - 5. This shifts from the origin to (-2,-5) which makes the asymptote at y=-5, but it is a little harder to determine the x axis shift back 2.
      (20 votes)
  • spunky sam orange style avatar for user Sooraj
    What is the difference between exponential functions and quadratic functions
    (8 votes)
    Default Khan Academy avatar avatar for user
  • piceratops seed style avatar for user bwolfwagner18
    At , why did you not solve for 3^-1, and skip right to 3^0? Is there a reason or is that just a mistake?
    (7 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user George Marianne Garcia
    Whats the exponential form of 10,000
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user alonenotlonely57
    How do you multiply 1.04 times an exponent of 1/12
    (4 votes)
    Default Khan Academy avatar avatar for user
  • starky ultimate style avatar for user Michele Franzoni
    Why is a function with irrational exponents defined only for a base greater or equal than zero?

    Check this out: if you go to Desmos and type in -2^pi you get -8.824
    BUT if you type in the equivalent form e^(ln(-2^pi) you get undefined as a result. Can someone shed some light on the reason why this happens?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • female robot ada style avatar for user Breamomah
    What if it is 3 raised to power 1.02?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • leaf red style avatar for user Mena Aly
    why does the base in an exponential function have to be positive?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • scuttlebug green style avatar for user Tzviofen ✡
      If it is negative, the function is totally different. for example:
      y = (-2)^x
      x = 1: y = -2
      x = 2: y = +4
      x = 3: y = -8
      x = 4: y = +16
      etc.
      Basically (for x integers), when x is even, y is positive, and when x is odd, y is negative, though the absolute value of y is growing exponentially.
      (2 votes)

Video transcript

In this video, I want to introduce you to the idea of an 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 third 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. Let's start with x is equal to negative 4. Then we'll go to negative 3, negative 2, 0, 1, 2, 3, and 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 fourth power. 3 to the third is 27 times 3 again is 81. So this is equal to 1/81. When x is equal to negative 3, y is 3. We'll do this in a different color. This color is hard to read. y is 3 to the negative 3 power. Well, that's 1 over 3 to the third power, which is equal to 1/27. 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 1/9, right? 1 over 3 squared, and then we have 3 to the 0 power, which is just equal to 1. So we're getting a little bit larger, a little bit larger, but you'll see that we are about to explode. Now, we have 3 to the first power. That's equal to 3. So 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 fifth power, 243. Let's graph this, just to get an idea of how quickly we're exploding. Let me draw my axes here. So that's my x-axis and that is my y-axis. And let me just do it in increments of 5, because I really want to get the general shape of the graph here. So let me just draw as 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. 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, are positive values because I have a positive base. So let me draw it like this. Good enough. And then let's say I 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. Let's say this is negative 1, negative 2, negative 3, negative 4. Then we have 1, 2, 3, and 4. So when x is equal to 0, we're equal to 1, right? When x is equal to 0, y is equal to 1, which is right around there. When x is equal to 1, y is equal to 3, which is right around there. When x is equal to 2, y is equal to 9, which is right around there. When x equal to 3, y is equal to 27, which is right around there. When x is equal to 4, y is equal to 81. You see very quickly this is just exploding. If I did 5, we'd go to 243, which wouldn't even fit on my screen. When you go to negative 1, we get smaller and smaller. So at negative 1, we're at 1/9. Eventually, you're not even going to see this. It's going to get closer and closer to zero. As this approaches larger and larger negative numbers, or I guess I should say smaller negative numbers, so 3 to the negative thousand, 3 to 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! Once we start getting into the positive numbers, we just explode. We just explode, and we keep exploding at an ever-increasing rate. So the idea here is just to show you that exponential functions are really, really dramatic. Well, 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 life, 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 in week 1. In week 1, someone sent a chain letter to 10 people. And the chain letter says you have to now send this chain letter to 10 more new people, and if you don't, you're going to have bad luck, and your hair is going to fall out, and you'll marry a frog, or whatever else. So all of these people agree and they go and each send it to 10 people the next week. So in week 2, they go out and each send it to 10 more people. So each of those original 10 people are each sending out 10 more of the letters. So now 100 people have the letters, right? Each of those 10 sent out 10, so that 100 letters were sent out. 10 were sent. Here, 100 were sent. In week 3, what's going to happen? Each of those 100 people who got this one, they're each going to send out 10, assuming that everyone is really into chain letters. So 1,000 people are going to get it. And so the general pattern here is, the people who receive it, so in week n where n is the week we're talking about, how many people received the letter? In week n, we have 10 to the nth people receive-- i before e except after c-- the letter. So if I were to ask you, how many people are getting the letter on the sixth week? How many people are actually going to receive that letter? Well, what's 10 to the sixth power? 10 to the sixth is equal to 1 with six zeroes, which is 1 million people are going to receive that letter in just 6 weeks, just sending out 10 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 10 people you sent it to also sent it to 10 people and so on and so forth, by the sixth 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.