Final answer:
Exponential functions are used to represent situations where a quantity grows at a rate proportional to its current value. Examples include sales that double each year or a home that appreciates by a consistent percentage annually. Not all consistent increases are exponential; for instance, a constant weekly increase in running distance represents linear growth.
Step-by-step explanation:
Exponential functions represent situations where a quantity grows or decays at a rate proportional to its current value. One of the key characteristics of exponential growth is that the growth rate, as a percentage or fraction, remains constant over time. This means that after identical time intervals, the quantity multiplies by the same factor. Two examples of these situations include: the sales at an auction website doubling each year, and the value of a home increasing by 7.5% per year.
In the case of the auction website, each year the sales are multiplied by a factor of 2, which can be represented as y = 2n, where n is the number of years. The home value increase can also be represented exponentially as y = P(1 + 0.075)n, where P is the initial value of the home and n is the number of years.
However, not all steady increases follow an exponential pattern. If we look at Karen who increases her running distance by a constant 0.5 km each week, this is a linear growth, not exponential, because the amount of increase is constant rather than the rate of increase being consistent as a proportion of the current value. Similarly, in the stadium where each row has only 3 more seats than the previous row, this represents a linear pattern as well.