Unit 52: Exponential Growth and Decay

You will likely have 1-2 problems that test your knowledge of exponential growth or decay.  These problems will take either the form of identifying the type of relationship (linear or exponential) or solving exponential equations. We’ll work on the first type of problem below.

Some questions will ask you to identify the type of function that best models the relationship between two variables. The four answer choices will be:

  • decreasing exponential
  • decreasing linear
  • increasing exponential
  • increasing linear

These questions will be asked in the form of graphs, tabular data, or as word problems.

Graphs

The graph problems are typically pretty easy – you will have to pick out the picture that reflects exponential growth or decay.  Remember exponential will always be a curved line.  If the line is straight, it is a linear function and does NOT represent an exponential function.

They might also give you a scatterplot, with or without the line of best fit.  If there is no line of best fit, draw one in. There will be a clear shape.

Moving clockwise from the top left, you should see exponential growth, linear decay, exponential decay, and no relationship.

Word Problems

This question type will give you a scenario in the form of a word problem, and then ask you about what function will best model the relationship that they describe.

For example:

An internet service provider offers a monthly plan where the download speed decreases by 5% from its initial speed at the end of each year due to increased network traffic. If the initial download speed is 100 Mbps, what type of function best models the relationship between the download speed and time in years?

A) Decreasing exponential
B) Decreasing linear
C) Increasing exponential
D) Increasing linear

Strategy:

  1. Understand the scenario. Determine what the independent (input) and dependent (output) variables are. For instance, time might often be the independent variable, and something changing over time, like altitude or amount, would be the dependent variable.
  2. Determine the relationship. Assess how the dependent variable changes in response to the independent variable. Is it increasing or decreasing? Does the rate of change remain constant, or does it increase or decrease over time?

Recognizing Function Types:

  1. Linear Functions: If the rate of change is constant, the relationship is linear. A linear function has the form (f(x)=mx+b) where (m) is the rate of change, and (b) is the starting value.
    • Increasing Linear: If m(m)is positive, the function is increasing.
    • Decreasing Linear: If (m) is negative, the function is decreasing.
  2. Exponential Functions: If the rate of change is proportional to the value of the function itself (meaning it changes more rapidly over time), the relationship is exponential. An exponential function generally has the form (f(x)=ab^x) where (b) is the growth factor.
    • Increasing Exponential: If (b>1), the function is increasing.
    • Decreasing Exponential: If (0<b<1), the function is decreasing.

Going back to our example:

In this scenario, the download speed is decreasing by a percentage each year, which indicates an exponential decay rather than a linear decrease. The correct answer would be A) Decreasing exponential, as this type of function models a quantity that decreases by a certain percentage over equal intervals of time.

Tabular Data

These questions may also take the form of tabular data.  They give you some values, and see if you can determine if it’s linear or exponential.  

For example:

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

Time (weeksPopulation
0100
51,000
1010,000
15100,000
201,000,000

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

A) Decreasing exponential
B) Decreasing linear
C) Increasing exponential
D) Increasing linear

Strategy:

  • An easy way to solve these is to draw yourself a quick graph.  Put time on the x-axis, and whatever else on the y.  This will show you clearly if it’s linear or not. If you can’t get a quick answer by hand, you can always use Desmos to help you plot the points, but this may take a little longer.