Click here to go to the Modelling Exponential Decay link to see more examples of solving problems using the exponential regression function to model decay.

Read pages 371-376 Example 1 in your textbook, Principles of Mathematics 12.

Complete the Your Turn questions on page 373 (a, b, c and d) and page 376 (1a, 1b and 1c) for practice in solving problems using exponential regression to model growth and decay.

Click here to verify your answers.

Page 373 (a) and page 376 (b)

L1 = x-values =
L2 = y-values =

Regression Equation:

Often you are not given a table of values for a real-world scenario. Instead, you are given information about the problem, and you must develop a table of values or equation yourself before beginning. Often, students have difficulty forming the correct exponential function in problems involving growth or decay. Example 5 demonstrates how to create an exponential function for two real-world scenarios.

Form the exponential function that models each problem situation.

  1. The population of a town of 10 000 residents in Northern Alberta is predicted to grow at an annual rate of 6%. What is the population, f(t), after t years?
  2. Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half the players are eliminated. Determine how many rounds are in the tournament.
  1. Work from y = abx.

This scenario relates population growth to time. Therefore, the population, f(t), is the dependent variable, y, and time, t, in years is the independent variable, x.

Recall from Lesson 6A that the a-value of the exponential equation equals the y-intercept of the graph. In real world problems, this is often referred to as the initial value or starting value. For this problem, the initial value is the current population of 10 000. Therefore, the a-value in this equation is 10 000.

The annual rate of population increase, 6%, is the growth rate of the exponential function. To put this into the equation, use the following rule:

For exponential growth: b = 1 + growth rate (written as a decimal)

This example is population growth, so b = 1 + 0.06 = 1.06.

The exponential function that models the population growth is f(t) = 10 000(1.06)t.

  1. Work from y = abx.

This scenario relates decreases in population to time. Therefore, the number of players is the dependent variable, y, and time in rounds is the independent variable, x.

The original number of players, 128, corresponds to the initial value, a.

Half the players are eliminated each round; therefore, the decay rate is 50%. To put this into the equation, use the following rule:

For exponential decay: b = 1 − decay rate (written as a decimal)

This example is population decrease, so b = 1 − 0.50 = 0.5.

The exponential function that models the decrease in number of players is y = 128(0.5)x.