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A small business bought a van for $40,000. The van depreciates by 15% every year after its purchase.
Graph A
Graph B
Graph C
Graph D
There are lots of ways to represent an exponential function. Suppose the population of a city was 20,000 in 1990 and that it increased by 10% each year.
We can represent this situation with a table of values and show, for instance, that the population increased by a factor of 1.1 each year.
| year | population |
|---|---|
| 1990 | 20,000 |
| 1991 | 22,000 |
| 1992 | 24,200 |
| 1993 | 26,620 |
We can also use a graph to show how the population was changing. While the graph looks almost linear, it has a slight upward curve since the population is increasing by a factor of 1.1 and not a constant value each year.
An equation is another useful representation. In this case, if is the number of years since 1990, then the population is a function of , where . Here we can see that the 20,000 in the expression represents the population in 1990, while 1.1 represents the growth factor due to the 10% annual increase each year. We can even use the equation to calculate the population predicted by the model in years before 1990, such as 1985. Because 1985 is 5 years before 1990, we use an input of -5 to get , which is about 12,418 people.
It is useful to describe information about a situation in many different representations such as a description of the situation in words, a table of values, a graph, and an equation. Understanding the connections among the different representations and moving freely between them is important to more fully understanding what is happening and using the information given to predict or estimate values.