Interpretting exponential expression
The expression ( 5 * 2^T ) gives the number of leaves in a plant as a function of the number of weeks since it was planted. What does two represent in this expression? So pause this video and see if you can figure it out on your own.
All right, so let's look at the expression right over here. We could write it as defining a function, so we could say leaves as a function of time is equal to ( 5 * 2^T ) power.
And so we could try this out a little bit. If we say, well, what is ( L(0) )? That would be ( T = 0 ); that's when we're 0 weeks after it was planted, so this is right when it was planted. Well, that's ( 5 * 2^0 ), which is just ( 2^0 ) is just one, so it's equal to five.
And so when you see an exponential expression or an exponential function like this, that is why this number out here is often referred to as your initial value. Initial value.
And so let's explore this a little bit more. What is ( L(1) )? What happens after one week? Well, that's going to be ( 5 * 2^1 ), or ( 5 * 2 ). So, going from when it was planted to the first week, we are multiplying by two. The number of leaves doubles.
Well, what happens after two weeks? The number of leaves after two weeks? Well, that's going to be ( 5 * 2^2 ). Well, that's the number that you had in the first week times two. So it looks like every week we are doubling; we are multiplying by two.
And that's why this number right over here, which is what the question is about, the two, this is often referred to as the common ratio. Common ratio. Because between any two successive weeks, the ratio between say week two and week one is two. Week two is double week one, and week one is double week zero.
So let's see which of these choices actually match up to that. There were initially two leaves in the plant? Well, we know that there weren't two leaves in the plant; our initial value was five, so let me cross that one out.
The number of leaves is multiplied by two each week? Well, that's exactly what we just described, so I like that choice.
Let's look at the last one just for good measure. The plant was planted two weeks ago? Well, no, they don't tell us anything about that. This is a general expression for ( T ) weeks after it was planted, so they're not saying when it was actually planted, so we could rule that out. And we feel good about that second choice.