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Writing exponential functions from tables

Both linear equations and exponential equations represent relationships between two variables. However, the way that the variables are related to each other in each type of equation is different. A linear equation can be thought of as representing repeated addition on an initial value, while an exponential equation can be thought of as representing repeated multiplication on an initial value. Created by Sal Khan.

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  • piceratops tree style avatar for user calvertb24
    i didnt understand any of this
    (25 votes)
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    • male robot hal style avatar for user Coogle Chess
      If you don't get it, check the hints in the exercise. If you don't get it here are some examples.
      x | f(x)
      ---------
      0 | 4
      1 | 8

      So, we see that when x is 0, f(x) = 4. f(x) is like y. Then, when x=1, f(x)[or y] = 8. We see that we have to multiply 4 by 2 to get 8. So, f(x)=4*(2)^x. OR, y=4*(2)^x. And when we plug in x=1, We get f(x)=8, which is right based on the table. I hope this helps. If it doesn't, just comment.
      (3 votes)
  • duskpin ultimate style avatar for user Radhika Kattula
    For g(x)= a*r^x, Sal immediately used g(0) to find out what a was, but what are you supposed to do if you are not told what g(o) is equal to? Would you still be able to solve the problem and if so how?
    (11 votes)
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    • marcimus pink style avatar for user Tyler
      If you didn't know g(0), you could instead start with another point like (1, g(1)) and then work backwards from there to find g(0).

      For instance, to get from g(0) to g(1), we know that we needed to multiply by 2/3. So, to go backwards, we would need to divide by 2/3, which is the same thing as multiplying by 3/2. So we could just take g(1), which is 2, and multiply that by 3/2 to get back to g(0). Since 2 * 3/2 = 3, we know that g(0) must be 3, and now we're done.

      So, using other points, you can always work backwards to find g(0). I hope this helps.
      (8 votes)
  • aqualine ultimate style avatar for user For Kobe, aka sad 2020 :(
    I dont really understand this pls help :(
    (4 votes)
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    • mr pink green style avatar for user David Severin
      So Sal talks about two functions, linear (with a common difference) and exponential (with a common ratio). It is much easier when x is incremented by 1 such as shown (1, 2, 3, ...) because it is easier to find what we need. For linear equations, we have y = m (slope) x + b (y intercept) and for exponential equations we have y = a (initial value)*r(ratio or base)^x. So in each case, we need to find two things.
      In both cases, the y intercept and initial value are found where x = 0 (y intercept) and the table gives us these, so linear b = 5 and exponential a = 3. We are already 1/2 way there.
      Linear slope is found by the common difference (since slope is change in y/change in x and change in x is 1, divide by 1 does not change anything). We subtract any two consecutive terms 2nd - 1st which gives 7 -5 = 9 - 7 = 11 - 9 = ... = 2. So with a slope of 2 and an intercept of 5, we get y = 2x + 5.
      For exponential, we divide instead of subtract, so 2/3 = (4/3)/2 = (8/9)/(4/3) = ... = 2/3. We have to use our dividing fraction skills to see they are the same, flip denominator and multiply, so 8/9 * 3/4 does reduce to 2/3 which gives us the base. So with an initial value of 3 and a base (common ratio) of 2/3 we get y = 3 (2/3)^x. I hope running them together makes more sense than separate since they can be related to each other is a variety of ways. Does this help?
      (3 votes)
  • aqualine ultimate style avatar for user P Nicholas
    why does he do triangles when he says change for example at
    (0 votes)
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  • winston default style avatar for user MasonG
    I don't get something: Why did Sal do 'Δf/Δx' instead of just solving 'mx + b' for 'x' like 'm1 + 5 = 7', '(2)1 + 5 = 7', so 'm = 2'?
    (2 votes)
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  • blobby green style avatar for user David Leventhal
    How do you make an exponential equation given 2 points in the form y=ab^x? Is there a video for it?
    (2 votes)
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  • starky ultimate style avatar for user kaitlyn
    What do you do if you only have 2 points?
    Here was the problem:
    Complete the equation for the table:
    h(x)=____
    x___|0_|1
    h(x)|10|4
    I have h(x)=10*b^x. How do I find the exponential degree (b) if there are only two points? Thanks!
    (2 votes)
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  • marcimus orange style avatar for user CaSondra Bierman
    isn't anything to the 0 power 1?
    (2 votes)
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  • winston baby style avatar for user Yomna K
    So we always need to find the initial term first by plugging 0 for x, yes?
    (1 vote)
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  • blobby green style avatar for user ArvinCastillo449
    What does this * symbol mean(in math)? and why does it mean it?
    (0 votes)
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Video transcript

Consider the following table of values for a linear function f of x is equal to mx plus b and an exponential function g of x is equal to a times r to the x. Write the equation for each function. And so they give us, for each x-value, what f of x is and what g of x is. And we need to figure out the equation for each function and type them in over here. So I copy and pasted this problem on my little scratchpad. So let's first think about the linear function. And to figure out the equation of a line or a linear function right over here, you really just need two points. And I always like to use the situation when x equals 0 because that makes it very clear what the y-intercept is going to be. So, for example, we can say that f of 0 is going to be equal to m times 0 plus b. Or this is just going to be equal to b. And they tell us that f of 0 is equal to 5. b is equal to 5. So we immediately know that this b right over here is equal to 5. Now, we just have to figure out the m. We have to figure out the slope of this line. So just as a little bit of a refresher on slope, the slope of this line is going to be our change in y-- or our change in our function I guess we could say, if we say that this y is equal to f of x-- over our change in x. And actually, let me write it that way. We could write this as our change in our function over our change in x if you want to look at it that way. So let's look at this first change in x when x goes from 0 to 1. So we finish at 1. We started at 0. And f of x finishes at 7 and started at 5. So when x is 1, f of x is 7. When x is 0, f of x is 5. And we get a change in our function of 2 when x changes by 1. So our m is equal to 2. And you see that. When x increases by 1, our function increases by 2. So now we know the equation for f of x. f of x is going to be equal to 2 times 2x plus b, or 5. So we figured out what f of x is. Now we need to figure out what g of x is. So g of x is an exponential function. And there's really two things that we need to figure out. We need to figure out what a is, and we need to figure out what r is. And let me just rewrite that. So we know that g of x-- maybe I'll do it down here. g of x is equal to a times r to the x power. And if we know what g of 0 is, that's a pretty useful thing. Because r to the 0th power, regardless of what r is-- or I guess we could assume that r is not equal to 0. People can debate what 0 to the 0 power is. But if r is any non-zero number, we know that if you raise it to the 0 power, you get 1. And so that essentially gives us a. So let's just write that down. g of 0 is a times r to the 0 power, which is just going to be equal to a times 1 or a. And they tell us what g of 0 is. g of 0 is equal to 3. So we know that a is equal to 3. So so far, we know that our g of x can be written as 3 times r to the x power. So now we can just use any one of the other values they gave us to solve for r. For example, they tell us that g of 1 is equal to 2. So let's write that down. g of 1, which would be 3 times r to the first power, or just 3-- let me just write it. It could be 3 times r to the first power, or we could just write that as 3 times r. They tell us that g of 1 is equal to 2. So we get 3 times r is equal to 2. Or we get that r is equal to 2/3. Divide both sides of this equation by 3. So r is 2/3. And we're done. g of x is equal to 3 times 2/3. Actually, let me just write it this way. 3 times 2/3 to the x power. You could write it that way if you want, any which way. So 3 times 2/3 to the x power, and f of x is 2x plus 5. So let's actually just type that in. So f of x is 2x plus 5. And we can verify that that's the expression that we want. And g of x is 3 times 2 over 3 to the x power. And let me just verify that that's what I did there. I have a short memory. All right. Yeah, that looks right. All right. Let's check our answer. And we got it right.