My last two posts discussed the mathematics of linear growth and decay. If you have not read those posts, you might find it helpful to read them before continuing. This post focuses on finding an exponential equation that expresses a relationship between two variables by first constructing a table of data-pairs to better understand the relationship and see the pattern in the relationship.
Most exponential growth/decay relationships involve a time variable t and the amount A of some quantity at time t. Amount could be the current value of an investment account, population of a city, remaining kilograms of radioactive material, assessed value of a truck, etc. The text box and observations below explain how and why the basic fundamental exponential growth/decay formula A = A0*bt/k works, and the role that the parameters A0, b, and k play in the equation. Periodic growth factor is another way to think of the base multiplier b.
Some observations about A = A0*bt/k where b > 0:
• The point (0, A0) is the intercept on the vertical axis of the graph.
• Base multiplier b is a periodic growth or decay factor.
• If 0 < b < 1, the equation models exponential decay.
• If b > 1, the equation models exponential growth.
• Exponential growth/decay is about repeated multiplication by growth/decay factor b.
• A0 and any other point on the graph determines a unique exponential equation.
• If A0 is positive, the graph is above and asymptotic to the horizontal axis.
• If A0 is negative, the graph is below and asymptotic to the horizontal axis.
Most discussions about finding the equation of an exponential relationship don’t start by looking at data-pairs in a table. After only a couple of demonstrations of how to apply the data-pairs approach, students quickly develop the ability to find the three key parameters of an exponential growth/decay relationship. Exponential equations of the form A = A0*bt/k where base b is a rational number are much easier to comprehend than equations of the form A = A0*ekt where e is the irrational math constant = 2.718281828459045 . . . . I will use four familiar math problems that involve an exponential relationship to illustrate the table data-pairs approach. In the comments section of this post, you will find an example that further clarifies my reason for expressing most exponential growth/decay equations as A = A0*bt/k where b is a rational number, and the reason that the solutions of population and radioactive growth/decay problems tend to be expressed in terms of base e only.
Problem 1: Consider a population of bacteria that is growing exponentially 50% every 4 hours and the current population is 60 bacteria. Let t = number of hours in the future and N = the number of bacteria after t hours.
(a) Find an equation that expresses N as a function of t.
(b) Find the population after 10 hours and 45 minutes ago.
(c) Express N as a function of t if the population is increasing 5% every 15 minutes.
The solution is given in the text box below. Problem solvers should carefully read the problem, create a table of data-pairs, determine the equation parameters, and then write the equation that models the problem situation. A companion exponential growth graph with a series of slope/rate triangles is provided to show the role that the equation parameters play in the relationship. Of course, the problem solver should always check the solution by using a computer graphing program to graph the equation.
Problem 2: Suppose a person invests $10,000 in a CD that will earn interest at 6%/year and interest is compounded monthly. Let t = the number of years in the future and V = the value of the investment after t years.
(a) Express V as a function of t.
(b) Find the value of the investment after 10 years and 20 years.
(c) Express V as a function of t if interest is compounded 360 times per year.
Problem 3: The half-life of a radioactive substance equals the time it takes (20 days, 149 years, 5,700 years, etc.) for the substance to lose half its mass. Consider a radioactive substance with a half-life of 60 days that currently has a 100 kg mass. Let time t = the number of days in the future and A = the mass of the remaining substance in kg at time t. Refer to table and companion graph below.
(a) Find a formula that expresses A as a function of time t in days.
(b) Find the mass of the substance after 135 days.
(c) Find a formula for A(t) if the half-life = 6 hours instead of 60 days.
Problem 4: The two exponential growth/decay graphs along with key points on the graphs are shown below.
(a) For graph A: Write an equation that expresses y as a function of x.
(b) For graph B: Write an equation that expresses y as a function of x.
Here are four exercises that you can give to your students. The graphs are a mixture of linear and exponential growth/decay graphs. Using the points on the graph, find the equation of the graph. If you wish, remind them that they should first create a table of data-pairs. Let them do the exercises with a partner and then check their answers by using a computer to graph the equations. We want to create a save environment in which kids feel free to experiment and check their answers for understanding. It’s OK to make a mistake, just fix it. If the first attempt to fix a mistake fails, so what? Try again. This is how real people learn to do anything that is worthwhile. The solutions are given at the end of this post.
• Consider the two mathematically equivalent equations below that model the population growth of a small town where t equals the number of years after 2010.
P = 5,200(1.08)t/4 and P = 5,200e0.019240260t
The first equation immediately tells us the population of the town was 5,200 in 2010, and the population is increasing 8% every 4 years. The second equation tells us the population of the town was 5,200 in 2010, but by just inspecting the second equation, only God can figure out that the population is increasing 8% every four years. (Increasing 8% every 4 years is slightly less than increasing 2% every year.)
• It’s a snap to find the derivative of functions of the form y = Aekt. To find the derivative of functions of the form y = Abx/k where base b is a rational number requires a little more work. I suspect this is the reason that the solutions of population and radioactive grow/decay problems tend to be expressed in terms of base e only. From my point of view, this is not a sufficient reason to do so because converting an exponential function from one base to another base is a simple procedure. My free handout Logarithmic Base Conversion shows how to do this.
• All modern physicists know that the equations they discovered can only give us an approximation of how nature’s laws work. The brilliant physicist Richard Feynman, over and over again, stated this fundamental fact in his lectures and talks. In reference to problem (3) above, if we conducted an experiment with a radioactive material by measuring the remaining mass of the material at various points in time, we would find a discrepancy between the experimental results and the predicted results. No matter how accurately we measure mass and time, the errors can’t be taken out of the experimental observations. We can only say that the remaining mass of radioactive material at time t lies in an area of uncertainly which is the area under a probability distribution curve. This is why least-squares regression equations are used to describe the relationship between two variables.
• The formula for calculating the future value of an account after t years when interest is compounded continuously is FV = Pert where P = the principal and r = the annual interest rate expressed as a decimal. It’s impossible to express this relationship with a base that is a rational number. In a future post, I will give a derivation of this formula in a manner that does not require an understanding of concepts in calculus.
• I have used the handout, Introduction to Exponential Growth and Decay, with college algebra students, pre-calculus students, and as a review for more advanced students. To download the free student and teacher versions of the handout, go to mathteachersresource.com/instructional-content.html. There are other free handouts on properties of exponents, properties of logarithms, solving exponential/logarithmic equations, and logarithmic base conversion.
• All graphs in this post were created with my software, Basic Trig Functions. I designed the software to help teachers quickly make custom content for their classrooms. This software allows you to easily copy any graphic and then import it directly into a document (e.g. lesson plan, class handout, test) or further manipulate it in various graphic processing programs.
My next post will show how to solve Newton’s Law of Cooling problems without understanding differential calculus.
Solutions to exercises:
Graph A: y = -2x + 40
Graph B: y = 40*0.5x/2.5
Graph C: y = 20*1.5x/5
Graph D: y = x + 10