# Types of (Math) Relationships, Part 2

Last month we looked at the details of linear functions. Recall that linear functions can be identified by a constant addition or subtraction relationship in the output of a function (usually the y values). This month we will examine exponential functions which can be identified by a constant multiplication or division relationship. Exponential functions and linear functions show up in similar problem types, and these questions often start with identifying whether the given relationship is linear or exponential.

**Is this exponential or linear?**

A radioactive substance decays at an annual rate of 13 percent. If the initial amount of the substance is 325 grams, which of the following functions f models the remaining amount of the substance, in grams, t years later?

The question to ask yourself is, “Do I multiply/divide or add/subtract to find the next term?”

Since this question mentions a decrease of 13 percent, we can ask ourselves, “Do I normally multiply or add when using percents?”

Multiplication is used when calculating percentages, so we know the answer must be exponential (A or B). The next question is, “What am I multiplying by?” This is where the most common mistake occurs. Often times when calculating a 13% decrease, students tend to do the following:

325 (.13) = 42.25

325 – 42.25 = 282.75

This solves for the answer, but it requires two steps. The equations above model solving the problem in 1 step. So how do we fix this? Well, we are taking 100% of the original amount then subtracting 13% of the original amount. Subtracting the percents, 100 – 13 = 87, we can see that we are calculating 87% of the original amount. We can see that multiplying by .87 will give us the same answer as before:

325 (.7) = 282.75

Therefore, the answer to this question is A.

**Now let’s look at the basic structure of an exponential equation:**

a represents the initial amount,

b represents what you are multiplying by,

and the exponent will always be a variable.

__Example__

In planning maintenance for a city’s infrastructure, a civil engineer estimates that, starting from the present, the population of the city will decrease by 10 percent every 20 years. If the present population of the city is 50,000, which of the following expressions represents the engineer’s estimate of the population of the city t years from now?

We can quickly deduce that the answer must be C or D since a decrease by 10% means we are are going to multiply by 0.9. Next, this decrease only happens every 20 years. Ask yourself, “If I plug in 20 to the equation, what should happen?” When plugging in 20, we should see only 1 decrease. This means that the answer choice must be D. Letter C would result in the town decreasing by 10% 400 times after 20 years!

__Test Yourself__

The population of mosquitoes in a swamp is estimated over the course of twenty weeks, as shown in the table.

Time (weeks) | Population |

0 | 100 |

5 | 1,000 |

10 | 10,000 |

15 | 100,000 |

20 | 1,000,000 |

Which of the following best describes the relationship between time and the estimated population of mosquitoes during the twenty weeks?