Analyzing tables of exponential functions | High School Math | Khan Academy

Let's say that we have an exponential function h of n, and since it's an exponential function, it's going to be the form a times r to the n, where a is our initial value and r is our common ratio. We're going to assume that r is greater than 0.

They've given us some information on h of n. We know that when n is equal to 2, h of 2 is 144; that h of 4 is 324; that h of 6 is 729. So based on the information here, let's see if we can actually figure out what a and r are going to be. Like always, pause the video and try to give it a go.

All right, now let's do this together. So I'm going to focus on r first, the common ratio. If we had successive ends, if we had h of 3, then we could just find the ratio between h of 3 and h of 2 and r would just come out of that. Or if the ratio between h of 4 and h of 3, we could solve explicitly for r. But we can get pretty close to that. We can just find the ratio between h of 4 and h of 2.

So, the ratio between h of 4 and h of 2 is going to be equal to, well, we know h of 4 is 324 and h of 2 is 144. We could simplify this a little bit. Let's see, if we simplify this, we would get they're both divisible by 2. If we divide them both by 2, this one in the numerator 324 divided by 2 is 162. 144 divided by 2 is 72.

Let's see if we can divide by 2 again. 81 over 36. Divide by 2. Actually, no, we can't divide by 2 anymore, but we can divide by 9 now, so 81 divided by 9 is 9, and 36 divided by 9 is 4. So, this thing can be rewritten as 9 over 4.

But we could also rewrite this ratio by using this form of an exponential function. We could also say that h of 4 is going to be a times r, now n is 4, so r to the 4th power, and h of 2 is going to be a times r to the second power. This simplifies nicely: a divided by a cancels, which is just one, and then r to the fourth divided by r squared, well, that's going to be r to the (4 minus 2) power or r squared.

So we have a nice little equation set up: r squared needs to be equal to 9 over 4. So let me write that down: r squared is equal to 9 over 4. And r needs to be greater than 0, so we could just say r is going to be the principal root of 9 over 4, which is equal to 3 over 2.

So we were able to figure out r. Now, how do we figure out a? Well, there's a couple of ways to do it. You can think about a as being what h of 0 is equal to. So we could do what I could, I guess, we could call a tabular method where let me set up a little table here.

So a little table: if this is n, and this is h of n, so n is 0; we don't know what h of 0 is just yet. It's going to be a. We don't know what h of 1 is yet. We do know that h of 2 is 144, and we do know, since the common ratio is 3/2, if we take h of 1 and we multiply it by 3/2, we're going to get h of 2. And if we take h of 0 and multiply it by 3/2, we're gonna get h of 1.

So h of 1 is going to be 144 divided by 3/2. So let's write that down: h of 1 is equal to 144 divided by 3/2, which is going to be 144 times 2 over 3.

Let's see, 144 divided by 3 is going to be equal to, is it let's see, 1, 3 goes into… I'll just do this by hand; my brain doesn't work that well while I'm making videos. 3 goes into 14 4 times; 4 times 3 is 12, subtract, so it's going to be 2; I see it's going to be 48. 3 goes into 24 8 times; 8 times 3 is 24, you have no remainder. So this is going to be 48 times 2, which is going to be equal to 96.

So this is going to be 96. And so if we want to figure out h of 0, we just divide by 3/2 again. So h of 0 is 96 divided by 3/2, which is equal to 96 times 2 over 3. 96 divided by 3, let's see, it's going to be 32, so this is going to be 32 times… did I do that right? Yep, 32 times 2, which is equal to 64.

64. And so just like that, we figured out that a is equal to 64, and r is equal to 3/2. So we can write h of n; we can say that h of n is equal to 64 is equal to 64 times (3/2) times (3/2) to the n power.

Now there's another way that we could have tackled this. Instead of this tabular method, we could have just solved for a now since we know r. We know, for example, that h of 2, which is going to be equal to a times, we know what our common ratio is; it's (3/2) to the second power is going to be equal to 144.

And so we could say a times (9/4) is equal to 144. And so we can multiply both sides times the reciprocal of (9/4), so we multiply both sides times (4/9). That cancels with that, that cancels with that, and we are left with a is equal to, let's see, 144 divided by 9 is going to be… is going to be, I want to say it's equal to 16. Is that right? I think that is right. Yep, and then 16 times 4, so this is going to be 16, and 1—16 times 4 is 64, which is exactly what we got before.

And so this actually, this method, now that I look at it, is actually a good bit easier. But they're equivalent, and I like this one because you get to see the common ratio in action, and you get to see that the initial value is the initial value; it is h of zero. But either way, once you figure out r and you know one of the h of... you know what the function is at a value, you can solve for a.

And frankly, if you knew a and you knew what the function was for a given n, you could likewise solve for r. So hopefully, you enjoyed that.