5.2 Introduction
If n is a counting number, then we can express "multiplying x, n times" as:
In Biology, one area of study is the population growth of different species. How fast one species grows can have a profound impact on other species within an ecosystem. If the rate of growth of a population is 2%, it means that next year's population will be the current year's population plus an additional 2% of the current year's population. So, if there are 1000 wildebeests in a particular herd this year, with an assumed growth rate of 2%, we'll have:
1000 + .02 × 1000 = 1200
wildebeests next year.
Can we develop a general formula for the size of the population at time n? Let's use a table to develop a function to model this situation using the same type of process that we used to develop our first linear equation; tables are a very powerful tool that can be used to determine a general formula from specific examples.
| Start of Year | Population | Simplified Expression |
|---|---|---|
| 0 | 1000 | 1000 |
| 1 | 1000 + .02 × 1000 | 1000(1.02) |
| 2 | 1000(1.02) + .02 × 1000(1.02) | 1000(1.02)(1.02) |
| 3 | 1000(1.02)(1.02) + .02 × 1000(1.02)(1.02) | 1000(1.02)(1.02)(1.02) |
To get the 3rd column, I factored out the second column and then combined terms: we first encountered factoring when we studied quadratic functions.
As an example, when n=1, we have:
1000 + .02 × 1000 = 1000 × 1 + .02 × 1000 = 1000(1 + .02) = 1000(1.02).
Can you see a pattern in the table? If we ignore time 0 for the moment, then we can see that at the start of year n, we'll have a factor of 1000 times factors of 1.02.
Let's use the symbol 1.02n to represent "multiply 1.02 n times" where n is a counting number; this way we won't have to write out the longer English version.
Since there's nothing special about the number 1.02, we can then use a variable x to abstractly represent 1.02. In mathematical notation, then we have that
\begin{matrix} x^n=\underbrace{x\cdot x\cdot \ldots\cdot x} \\{\; \; \; \; \; \; \; n \; times \end{matrix}
where n is called the exponent and x is called the base. Note that since we're multiplying x a certain number of times, in order for things to make sense, n should be a counting number.
We can now describe the population at the start of year n very succinctly as:
P(n) = 1000(1.02)^n
where we assume n is a counting number.
Explore!
To the nearest whole number, how many wildebeests are there at the start of year 6?
Right now, in the way that we've defined our exponents, we can only use counting numbers. It would be great if we could find out our wildebeest population at ANY point in time. In other words, can we define exponents so that our formula also works at the start when our time is 0, at times in the past when our times are negative, and at fractional times, when our times are "between" years?
To set specific goals, let's see if we can make some sense of :
- P(0) --- this would represent the population at time 0
- P(-1) --- this would be 1 year in the past
- P(1.5) --- this would be at time 1 year and 6 months.
Before we answer these questions, let's get more familiar with xn where n is a counting number. We'll see how we can algebraically manipulate such creatures, and then we'll see if the doors to understanding the more conceptually challenging questions begin to open for us. In the next section, we'll develop some of our basic rules of exponents.
