About this video
Given an exponential function g with initial value -2 and common ratio 1/7, the video writes g(t) = -2 * (1/7)^t using the general form A * r^t. It explains that g(0) = -2 because (1/7)^0 = 1, and that each increase of t by 1 multiplies by 1/7, which is why it is the common ratio.
You'll be able to
- write an exponential function in the form A * r^t from its initial value and common ratio.
- explain why A is the value at t = 0 and r is the ratio between successive terms.
Key words
- initial value
- The value of the function at t = 0; called A in A * r^t.
- common ratio
- The number r you multiply by each time t increases by 1; the ratio between successive terms.
Transcript
- [Voiceover] g is an exponential function with an initial value of -2. So, an initial value of -2, and a common ratio of 1/7, common ratio of 1/7. Write the formula for g(t). Well, the fact that it's an exponential function, we know that its formula is going to be of the form g(t) is equal to our initial value which we could call A, times our common ratio which we could call r, to the t power.
It's going to have that form. And they tell us what the initial value is. It's -2. So this right over here is -2. And we know that the common ratio is 1/7. So this is 1/7. So let me just write it again a little bit neater. g(t) is going to be equal to our initial value, -2, times times our common ratio, 1/7, to the t power.
And hopefully this makes sense. Initial value is this number. Well, if t is equal to 0, then 1/7 to the zero power is 1. And so g(0), you could do that at time as being equal to zero if your t is time, would be equal to -2. So that would be our initial value. And then if you think about it, every time you increase t by one, you're going to mulitply by 1/7 again.
And so, the ratio between successive terms is going to be 1/7. And so that's why we call that the common ratio. Hopefully, you found that interesting.
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