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Current time:0:00Total duration:5:27

Initial value & common ratio of exponential functions


Video transcript

so let's think about a function I'll just give an example let's say H of n is equal to 1/4 times 2 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 exponential 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 the variable T is equal to is equal to 5 x times 3 to the T once again this is an exponential function now there's a couple of interesting things to think about an exponential function factor we'll explore many many many of them but I'll get a little used to the terminology so it says one thing that you might see is the notion of an initial value initial initial 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 0 and when we evaluate that that's going to be 1/4 times 2 to the 0 well 2 to the 0 power is just 1 so it's equal to 1/4 so the initial value at least in this case it seemed to just be that number that sits out here we have the initial value times some some number to this exponent and we'll come up with the name for this number as well but let's see if this was true over here for F of T so if we look at its initial value F of 0 is going to be 5 times 3 to the 0 power and the same thing again 3 to 0 is just 1 5 times 1 is just 5 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 0 in for the exponent then the the number raised to exponent it's going to be one and you're 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 said well why does it call it 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 1/4 now what is H of one going to be equal to it's going to be 1/4 times 2 to the first power so it's going to be 1/4 times 2 what is H of 2 going to be equal to well it's going to be 1/4 times 2 squared so it's going to be times 2 times 2 or we could just view this as this is going to be 2 times H of 1 and actually I should have done this when I wrote this one out but this we could write as 2 times H of 0 so notice if we were to take the ratio between H of 2 and H of 1 it would be 2 if we were to take the ratio between H of 1 and H of 0 it would be 2 that is the common ratio between successive whole number inputs into our function so H of I could say H of n plus 1 over H of n is going to be equal to is going to be equal to actually I can work it out mathematically 1/4 times 2 to the n plus 1 over 1/4 times 2 to the N that cancels 2 to the n plus 1 divided by 2 to the N is just going to be equal to 2 that is your common ratio so 4 for the function H for the function f our common ratio is 3 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 it you 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 a sound of 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 the parts of an exponential function why they are called what they are called