The Research Experience for Teachers Program /

Activity Title: “Solve Word Problems in Exponential Functions using Microsoft Excel”

Activity Goals:

The module is designed to help students appreciate the concept of modeling and graphing exponential functions. They will learn exponential functions that shows the relationship between two quantities and interpret key features of graphs and tables in terms of the quantities.They will also use formulas or equations of the functions to calculate outcomes of exponential growth or decay and then sketch graphs showing the order given in the verbal description and demonstrates the key features of the graph using technology.

Introduction/Motivation:

It is estimated that a person should have at least $700 000.00 of retirement funds in order to live comfortably for the next 25 years after he retires from work. The different investmentinstruments and retirement accountssuch as 401(k)are big help to attain this objective.Have you ever wondered how these investment instruments grow the money? In this activity, we will learn that concept by studying the principles of Exponential Functions.

Materials:

Computer and Microsoft Excel

Background:

An Exponential Functions is a function defined by f(x) = ax where a >0 and a ≠ 1. The word exponential is an adjective which means extremely rapid. Exponential functions can be used to express exponential growth and exponential decay of microorganism in biology, amount of radioactive substance given its half-life in chemistry, compound interests and annuities in the business and economics, or even population growth demography.

In general, the formula used to illustrate population growth and population decay after n years are expressed as follows:

Population growth - Pn = P(1 + r)n.

Population decay-Pn = P(1 - r)n.

Where:

Pn = Population after n years.

P = The initial population

n = Number of years

r = Constant rate

Preparation:

Students are expected to have knowledge in constructing table of ordered pairs for thefunction, basic properties of exponents, and can be able to recognize function notation.Likewise, it is assumed that the students have basic knowledge in encoding formulas and graphing tables using Microsoft Excel.

Illustrative Example:

The graph of exponential functions is the primary tool used in describing its behavior and characteristic. Below is the table and the graph of f(x) = 2x. The exponential functions is increasing.

X / -4 / -3 / -2 / -1 / 0 / 1 / 2 / 3 / 4
Y / 0.0625 / 0.125 / 0.25 / 0.25 / 1 / 2 / 4 / 8 / 16

Procedure:

The following are the steps to be followedto solve the word problems.

  1. Open a new Microsoft Excel Workbook.Before solving the problems, savethe workbook using the first letter of your name followed by your last nameas the file name of your document. (e.g. DMorales.xlsx)
  2. Solve all the problems stated on this activity. Make a table to represent the value of the data and sketch the graph to illustrate the progress of the function. Create a table and sketch a graph for each problem.
  3. We will have a uniform configuration for the answers. All answers shall be rounded off to the nearest hundredths (two decimal places).
  1. To round off the answer into the nearest hundredths, select the desired cell in the Excel spreadsheet and right click the mouse.
  2. Choose Format Cells.
  3. Click Number in the category box.
  4. Set the decimal places box into 2and then click okay.

Word Problems

Solve the following problems using the Microsoft Excel Spread Sheets.

  1. Construct a table and sketch a graph of the following exponential functions. Set the values of the domain (x) from -4 to 4. Tell whether the exponential functions is increasing or decreasing.
  1. f(x) = (1/2)-x
  2. f(x) = 3-x
  1. A certain radioactive substance decays one-fourth of itself every day and there are 100 grams initially. How much substance will be left after 12 days? Construct a table illustrating the decay of the substance from day 1 to day 15 andshow the progress by sketching the graph.
  1. Michael deposits $10 000 in an investment instrument that pays 2.5 % interest compounded semi-annually. Construct a table showing the growth of the money for the next 10 years and sketch the graph to represent the progression.

The formula to be used is

where A is the total amount after t years, P is the principal amount, r is the interest rate, and nis the number of times the amount is compounded a year.

  1. The depreciation of a car can be determined by the formula V=C (1-r)t, where V is the value of the car after t years, C is the original cost, and r the rate of depreciation. A brand new car loses 11% of its value as soon as the customer leaves the car dealership. During the first five years, the car depreciates an average of 15% of its value. Jeremiah decided to buy a brand new car at $28,000.Use the above information to answer the following questions.
  1. What is the value ofJeremiah’s car after 5 years?
  2. The averagedepreciationrate of a new car after the first 5 years decreases from 15% to 12% annually. If Jeremiah decided to keep his car, what will be the value of thecar at the end of 10 years?
  3. Construct a table showing how the value of the car depreciated within 10 years and sketch the graph to represent the decline of the value.

Evaluation:

Answer the following questions. Support your answer using the principles of exponential function.

  1. Remember that in linear functions, there is equal first differences in the function valuesof the dependent variable. Using this information, describe the function values of the dependent variable (range) of the exponential functions.
  2. Describe the domain and the range of the exponential functions.

Closing Activity:

All Students should turn in (via email) the completed Excel spreadsheet to assess their performance.

Reference: Advance Algebra, Trigonometry and Statistics

by: Orones, Esparrago and Reyes