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# Analyzing tables of exponential functions

CCSS.Math:

## Video transcript

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 and we're going to assume that R is greater than zero and 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 and like always pause the video and try to give it a go all right now let's let's do this together so I'm going to focus on our first the common ratio and 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 H of 4 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 and we could simplify this a little bit let's see if we if we simplify this we would get they're both divisible by 2 if we divide them both by 2 this 1 in the numerator 324 divided by 2 is 162 144 divided by 2 is 72 let's see we can divide by 2 again 81 over 36 divided by 2 well 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 fourths but we could also rewrite this ratio by using using this this form of an exponential function so we could also say that this is going to be equal to H of 4 is going to be a times R now n is 4 so R to the fourth power and H of 2 is going to be a times R to the second power a are to the second power and this simplifies nicely a divided by a cancel 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 it's going to be R to the 2nd power it's going to be R squared and so we have a nice little equation set up R squared needs to be equal to 9/4 so let me write that down r squared is equal to 9/4 and our needs to be greater than 0 so we could just say R is going to be the principal root of 9 fourths which is equal to three-halves so we were able to figure out R so now how do we figure out how do we figure out a well there's a couple of ways to do it you can think about a is going to be 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 so 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 three-halves if we take H of 1 and we multiply it by 3 halves we're going to get H of 2 and if we take H of 0 and multiply it by three halves we're going to get H of 1 so H of 1 is going to be 144 divided by 3 halves so let's write that down H of 1 is equal to 144 divided by 3 halves Oh which is going to be 144 144 times 2 over 3 and 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 I see it's going to be 48 3 goes into 24 eight times 8 times 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 halves again so H of 0 is 96 divided by 3 halves 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 30 32 times as 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 and R is equal to 3 halves so we can write H of n we can say that H of n is equal to 64 is equal to 64 times 3 halves times 3 halves our common ratio to the nth 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 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 halves to the second power is going to be equal to 144 and so we could say a times 9 over 4 is equal to 1 144 and so we can multiply both sides times reciprocal of 9 over 4 so we multiply both sides times 4 over 9 times 4 over 9 that cancels with that that cancels with that and we are left with a is equal to well 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 time four so this is going to be 16 and one 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 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 0 but either way once you figure out are you and you know one of the H of you know what the function is at a value you can solve for a and in fact that 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