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CCSS.Math: , ,

the expression five times two to the T gives the number of leaves in a plant as a function of the number of weeks since it was planted what does to 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 five times two to the T power and so we could try this out a little bit if we say well what is L of zero that would be T equals zero that's when we're zero weeks after it was planted so this is right when it was planted well that's five times two to the zero which is just 2 to the 0 is just 1 so it's equal to 5 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 initial value and so let's explore this a little bit more what is L of 1 what happens after one week well that's going to be 5 times 2 to the first power or 5 times 2 so going from the going from when it was planted to the first week we are multiplying by 2 the number of leaves doubles well what happens after two weeks the number of leaves after 2 weeks well that's going to be 5 times 2 to the second power well that's the number that you had in the first week times 2 so it looks like every week we are doubling we are multiplying by 2 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 2 and week 1 is 2 week two is double week one 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 on 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 they don't tell us anything about that this is a general 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