Interpreting nonlinear expressions — Harder example

- [Instructors] We're told the equation above models the number of subscribers, S, of an online styling service q quarter years after the service launches. Based on the model, which of the following statements is true. And let's see, they have a bunch of statements on how much the subscribers increased by either each quarter or each year. And these first two, they're talking about increasing about fixed amount each quarter a year. And here it's a percentage each quarter or year. So let's look at this expression. And the immediate thing that probably jumps out of you is that we have an exponent here. This is an exponential expression. And so let's actually just make a little table here to think about how does S grow with Q and what I'm going to do. So this is Q and this is S and I'm going to pick some Q values that'll make the math a little bit simple 'cause we noticed that the exponents here is Q over four. So let's say what happens at when Q is equal to zero 'cause zero over four is pretty easy to calculate. It's not going to be a fraction. Let's say when only four quarters have gone by 'cause four over four is pretty easy. And maybe when eight quarters have gone by which is the same thing as two years. So when Q is equal to zero, this is going to be 1.06 to the zero power, which is this one. So we have 1000 subscribers. And so you could view it as that's the initial amount of subscribers. Then when Q is equal to four, what do we have? Well, this exponents is going to be four over four. So it's just going to be one. So it's going to be 1000, 1000 times 1.06. And then when Q is equal to eight, eight divided by four is two. So it's going to be 1.06 to the second power. So that's going to be 1000. We could write it times 1.06, actually let me just write it this way. Times 1.06 to the second power. So let's think about how it's growing at least every four quarters which is the same thing as one year. Well, it looks like we are growing or we are multiplying by 1.06 or another way of thinking about it is we are increasing by 6%. Once again, when we go from quarter four to quarter eight which is another year goes by, we're multiplying by another 1.06. So that's growing by another 6%. So it looks like we're growing by 6% every four quarters or every year. And so the choice that does that, the number of subscribers increases by 6% each year, that one looks good. Now let's think about the other choices, just to feel super. If we're doing this on a test, we would just keep going. But just to feel really confident that it's not these other choices. The number of subscribers increases by 60 each quarter year. Well, this would be a linear equation. It would look something like that, like this. S is equal to your initial number of subscribers plus the number of quarter years times 60, it would look something like this for choice A which clearly we don't have over here. So we can rule that out. The number of subscribers increases by 60 each year. Well, this would be something like subscribers is equal to 1000, our initial number of subscribers plus Q over four times 60 because then every four quarters you would increase by a year, and obviously you could simplify this. It would be 15 times Q but this is clearly not what is happening here. And then the number of subscribers increases by 6% each quarter year. Well, the exponential that would describe that would be our initial number of subscribers times 1.06, 1.06 not to the Q over four, that would just be to the q so that every time we are a new quarter year or every time a quarter goes by, you would increase by 6% but here we clearly have Q over four. So every four quarters we're increasing by 6%.