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

Initial value & common ratio of exponential functions

In an exponential function of the form f(x) = a*b^x, the initial value is usually taken to be the value of f(0), or "a". The common ratio refers to the rate of change in an exponential function. In the form given above, the common ratio is "b". For example, in the function f(x) = 2*3^x, the initial value is 2 and the common ratio is 3.

Want to join the conversation?

  • old spice man green style avatar for user Aaron
    Is "growth rate" another way of saying common ratio or is it something different?
    (17 votes)
    Default Khan Academy avatar avatar for user
  • primosaur sapling style avatar for user Evan Leonard
    is another name of the common ratio the y-intercept?
    (12 votes)
    Default Khan Academy avatar avatar for user
    • leafers ultimate style avatar for user Alex
      No, the y-intercept is the value when x is equal to 0. For an exponential function the y-intercept is the "initial value" not the common ratio.
      Consider a standard exponential function of the form y(x) = a•rˣ, if you put inx = 0 you get: y(0) = a•rˣ = a•r⁰ = a•1 = a, so the y-intercept is a, which is called the initial value, not r, which is called the common ratio.
      (21 votes)
  • leaf blue style avatar for user 20mdolan
    I don't get how the ratios between h(1) and h(0) , and h(2) with h(1) both equal 2
    (6 votes)
    Default Khan Academy avatar avatar for user
    • purple pi purple style avatar for user doctorfoxphd
      Well, if h(x) = ¼ ∙2ⁿ,
      then as Sal says,
      h(0) = ¼ ∙2⁰ which means that h(0) = ¼ ∙1 because 2⁰ = 1
      and h(1) = ¼ ∙2¹ which means that h(1) = ¼ ∙2¹ because 2¹ = 2
      and h(2) = ¼ ∙2² which means that h(2) = ¼ ∙2 ∙2 → because 2² = 2 ∙2 = 4

      Lining them up without all the explanations,
      h(0) = ¼ ∙2⁰ = 1
      h(1) = ¼ ∙2¹ = 2
      h(2) = ¼ ∙2² = 4

      To find the ratios, we divide each result by the previous result, so h(2) ÷ h(1) = 4 ÷ 2 = 2
      and, h(1) ÷ h(0) = 2 ÷ 1 = 2

      Another way to look at this is
      h(0) = ¼ ∙2⁰ = 1
      h(1) = ¼ ∙2¹ = 2
      h(2) = ¼ ∙2² = 4

      and h(n) = ¼ ∙2
      In each case the ∙2 is visible in the original formula and all of the formulas we set up for evaluating the function
      (16 votes)
  • starky ultimate style avatar for user Abigail
    Is there a way to figure out...
    5/180 = 6^n
    ...without just guessing and guessing?
    (6 votes)
    Default Khan Academy avatar avatar for user
  • male robot donald style avatar for user jonahmolina90
    Im trying some problems but in the hints, they dont show a step and it's completely shutting me down on every problem.

    (1/3)^n = 27 --> (1/3)^2 = 3^3 --> (1/3)^n = (1/3)^3 --> n = -3

    please help me. Im getting stuck on all the problems at this step. Everything prior is OK, its just here I get stuck and dont know the steps/process.
    (6 votes)
    Default Khan Academy avatar avatar for user
    • stelly blue style avatar for user Kim Seidel
      I assume you are trying to find "n".
      When we look at: (1/3)^n = 27, we need to think about 2 things.
      1) Does "n" need to be positive or negative?
      2) What exponent will change 3 into 27?

      1) Does "n" need to be positive or negative? I try to do this first. Notice, you are starting with 1/3. If we apply positive exponents, we always get a fraction with a numerator of 1: (1/3)^2 = 1/9; (1/3)^3 = 1/27; etc. But, we need 27 in the numerator. This tells us the exponent needs to be negative. We need to flip the fraction to get the 27 on top.

      2) What exponent will change 3 into 27? After step one, this is usually the easy part.
      3^3 = 27.

      Putting the 2 steps together, we need n = -3
      (7 votes)
  • spunky sam blue style avatar for user Madelyn
    Please help!
    what is the common ratio?
    2, -6, 18, -54, ...?
    (5 votes)
    Default Khan Academy avatar avatar for user
    • starky ultimate style avatar for user Patrick
      To find the common ratio, we need to find a relationship that gets us from one number to the next. After looking at the first four numbers, we notice that they alternate from positive to negative. That means the common ratio is negative. And if we divide two consecutive numbers, say, 2 and -6, we find the number -3. It's negative, so to check, we multiply all the numbers by -3, and see if the answer is the next term:
      2 * -3 = -6
      -6 * -3 = 18
      18 * -3 = -54
      It's all correct, so the common ratio is -3.
      (6 votes)
  • leafers seedling style avatar for user ياسين الصالح
    at
    isn't this the formula of Explicit geometric sequence ?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • female robot grace style avatar for user Megan Brown
    Are exponential functions and geometric sequences basically the same thing?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • primosaur ultimate style avatar for user Greta
      In a way yes, geometric sequences are a specific type of exponential function. Whether they're in recursive or explicit form they have a formula that you fill in the blanks for that's slightly different then other functions.
      (5 votes)
  • blobby green style avatar for user doyledeanna
    can you just do a video on just graphing exponential functions please
    (4 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user Farida Khaled
    At , how did 2^n+1/ 2^n= 2
    (3 votes)
    Default Khan Academy avatar avatar for user

Video transcript

- [Voiceover] So let's think about a function. I'll just give an example. Let's say, h of n is equal to one-fourth times two to the n. So, first of all, you might notice something interesting here. We have the variable, the input into our function. It's in the exponent. And a function like this is called an exponential function. So this is an exponential. Ex-po-nen-tial. Exponential function, and that's because the variable, the input into our function, is sitting in its definition of what is the output of that function going to be. The input is in the exponent. I could write another exponential function. I could write, f of, let's say the input is a variable, t, is equal to is equal to five times times three to the t. Once again, this is an exponential function. Now there's a couple of interesting things to think about in exponential function. In fact, we'll explore many of them, but I'll get a little used to the terminology, so one thing that you might see is a notion of an initial value. In-i-tial Intitial value. And this is essentially the value of the function when the input is zero. So, for in these cases, the initial value for the function, h, is going to be, h of zero. And when we evaluate that, that's going to be one-fourth times two to the zero. Well, two to the zero power, is just one. So it's equal to one-fourth. So the initial value, at least in this case, it seems to just be that number that sits out here. We have the initial value times some number to this exponent. And we'll come up with the name for this number. Well let's see if this was true over here for, f of t. So, if we look at its intial value, f of zero is going to be five times three to the zero power and, the same thing again. Three to the zero is just one. Five times one is just five. So the initial value is once again, that. So if you have exponential functions of this form, it makes sense. Your initial value, well if you put a zero in for the exponent, then the number raised to the exponent is just going to be one, and you're just going to be left with that thing that you're multiplying by that. Hopefully that makes sense, but since you're looking at it, hopefully it does make a little bit. Now, you might be saying, well what do we call this number? What do we call that number there? Or that number there? And that's called the common ratio. The common common ratio. And in my brain, we say well why is it called a common ratio? Well, if you thought about integer inputs into this, especially sequential integer inputs into it, you would see a pattern. For example, h of, let me do this in that green color, h of zero is equal to, we already established one-fourth. Now, what is h of one going to be equal to? It's going to be one-fourth times two to the first power. So it's going to be one-fourth times two. What is h of two going to be equal to? Well, it's going to be one-fourth times two squared, so it's going to be times two times two. Or, we could just view this as this is going to be two times h of one. And actually I should have done this when I wrote this one out, but this we can write as two times h of zero. So notice, if we were to take the ratio between h of two and h of one, it would be two. If we were to take the ratio between h of one and h of zero, it would be two. That is the common ratio between successive whole number inputs into our function. So, h of I could say h of n plus one over h of n is going to be equal to is going to be equal to actually I can work it out mathematically. One-fourth times two to the n plus one over one-fourth times two to the n. That cancels. Two to the n plus one, divided by two to the n is just going to be equal to two. That is your common ratio. So for the function h. For the function f, our common ratio is three. If we were to go the other way around, if someone said, hey, I have some function whose initial value, so let's say, I have some function, I'll do this in a new color, I have some function, g, and we know that its initial initial value is five. And someone were to say its common ratio its common ratio is six, what would this exponential function look like? And they're telling you this is an exponential function. Well, g of let's say x is the input, is going to be equal to our initial value, which is five. That's not a negative sign there, Our initial value is five. I'll write equals to make that clear. And then times our common ratio to the x power. So once again, initial value, right over there, that's the five. And then our common ratio is the six, right over there. So hopefully that gets you a little bit familiar with some of the parts of an exponential function, why they are called what they are called.