Exponential Functions

Exponents are used as mathematical shorthand to describe a product. Rather than write 6 x 6 x 6 x 6 x 6 to indicate the number of possible different outcomes when five dice are rolled, it is easier to express this with an exponent as 6⁵, meaning that 6 is to be multiplied by itself five times. (Sometimes, the word power is used as a synonym for exponent, so 6⁵ might be read as “the 5^thpower of 6″ or “6 raised to the 5^th power.”)

Exponents occur in algebraic notation when a variable is multiplied by itself. For instance, the volume of a cube with side length s is given by the expression s³. Exponents are also used in scientific notation to express either very large or very small numbers in a standard form. For example, the number 53,400 is written in scientific notation as 5.34 x 10⁴. The exponent 4 indicates that 5.34 multiplied by 10 four times yields 53,400.

Exponential functions contain a variable written as an exponent, such as y = 3^x.Investors know the importance of an exponential function, since compound interest can be described by one. The formula A = p(1 + r)^t is an exponential function in which the amount in the account (A) depends on the length of time (t) of an investment (p) deposited at a given rate (r).

Other examples:

• In computer science, information is measured in bits and bytes. One byte is equal to 2³, or 8, bits. A kilobyte is 2¹⁰ bytes, and a megabyte is 2²⁰bytes.
• If the number of insects in a population triples each week, there will be a x 3ⁿ insects in n weeks, where a is the original number of insects. If the original population was 25, in 5 weeks there would be 25 x 3⁵ = 6075insects. The graph of a x 3ⁿ for a = 25 is shown below.

• When $10,000 is invested and earns 3.5% interest annually, there will be:
  • 10,000 x 1.035 = $10,350 in the account after one year.
  • 10,000 x (1.035)² = 10,000 x 1.035 x 1.035 = $10,712.25 after two years.
  • 10,000 x (1.035)ⁿ in the account after n years.

  If the money is left in this account until retirement, it will grow to $39,592.60 in 40 years.

Mathematical Definition

An exponent is a mathematical notation indicating the number of times a quantity is multiplied by itself. For instance, the expression 2³ indicates that 2 is to be multiplied by itself three times; that is, 2³ = 2 x 2 x 2. In the expression 2³, the exponent is 3.

Following is a definition of exponent from a mathematics dictionary:

Exponent: A number placed at the right of and above a symbol. The value assigned to the symbol with this exponent is called a power of the symbol, although power is sometimes used in the same sense as exponent. If the exponent is a positive integer and x denotes the symbol, then xⁿ means x if n = 1, and it means the product of n factors each equal to x if n > 1. E.g., 3¹ = 3; 3² = 3 x 3 = 9 (the second power of 3 is 9); x³ = x * x * x. If x is a nonzero number, the value of x⁰ is defined to be 1 (if x 0, x⁰ can be thought of as the result of subtracting exponents when dividing a quantity by itself: x²/x² = x⁰ = 1). A negative exponent indicates that in addition to the operations indicated by the numerical value of the exponent, the quantity is to be reciprocated. Whether the reciprocating is done before or after the other exponential operations have been carried out is immaterial; e.g., 3^-2 = (3²)^-1 = (9)^-1, or 3^-2 = (3^-1) = (1/3)² = 1/9. The following laws of exponents are valid when m and n are any integers (positive, negative, or zero) and a and b are real or complex numbers (not zero if in a denominator or if the exponent is 0 or negative):

(1) aⁿa^m = a^{n + m};
(2) a^m/aⁿ = a^{m – n};
(3) (a^m)ⁿ = a^mn;
(4) (ab)ⁿ = aⁿbⁿ;
(5) (a/b)ⁿ = aⁿ/bⁿ, if b 0.

If the exponent on a symbol x is a fraction p/q, then x^p/q is defined as (x^1/q)^p, where x^1/q is the positive q^th root of x if x is positive, and the (negative) q^th root if x is negative and q is odd. It follows that x^p/q = (x^p)^1/q and that the above five laws are valid if m and n are either fractions or integers (i.e., rational numbers), provided a and b are positive numbers.

(Source: James, Robert C. and Glenn James. Mathematics Dictionary (5^thedition). New York: Chapman & Hall, 1992.)

Exponential functions are among the many types of functions that algebra students should study. According to the National Council of Teachers of Mathematics (NCTM), “Students should use technological tools to represent and study the behavior of polynomial, exponential, rational, and periodic functions, among others. … As they do so, they will come to understand the concept of a class of functions and learn to recognize the characteristics of various classes.” (Principles and Standards for School Mathematics (PSSM) , NCTM, 2000, p. 296.)

In the video for Workshop 6 Part I, students in Orlando Pajon’s class study exponential functions in the context of population growth. Principles and Standards for School Mathematics suggests that this is an appropriate way to investigate exponential functions from a variety of perspectives. On page 297, PSSM suggests the following situation for exploration:

During 1999 the population of the world hit 6 billion. The expected average growth rate is predicted to be 2 percent a year.

Students might attempt to represent this situation with an expression for the function and compare it to other functions they have explored – linear, step, quadratic, and so forth. In addition, “students might generate an iterative or recursive definition for the function, using the population of a given year (NOW) to determine the population of the next year (NEXT):

NEXT = (1.02) NOW, start at 6 billion

Moreover, students should recognize that they can represent this situation explicitly with the exponential function f(n) = 6(1.02)ⁿ, where f(n) is the population in billions and n is the number of years since 1999.” (PSSM, p. 297).

While numerical representations (e.g., tables) help to show that exponential functions grow very quickly, students may develop a better conceptual understanding of exponential functions by exploring graphical or symbolic representations. Because the graph of an exponential function rises or falls very rapidly, it shows pictorially that the value of the function increases or decreases at a swift rate.

To understand the attributes of exponential functions, students ought to consider various functions from this class. For instance, PSSM suggests that students explore three distinct exponential functions:

• g(x) = 3 2^x + 4
• h(x) = 2 3^x – 1
• k(x) = 2 1.1^x

Investigating these functions with a graphing calculator will reveal common characteristics. PSSM suggests:

To help students notice and describe characteristics of these three functions, teachers might ask, “What happens to each of these functions for large positive values of x? For large negative values of x? Where do they cross the y-axis?” One student might note that the values of each function increase rapidly for large positive values of x. Another student could point out that the y-intercept of each graph appears to be a + c [when the function is written in the form f(x) = a b^x + c]. Teachers should then encourage students to explore what happens in cases where a < 0 or 0 < b < 1. Students should find that changing the sign of a will reflect the graph over a horizontal line, whereas changing b to 1/b will reflect the graph over the y-axis. The graphs will retain the same shape. This type of exploration should help students see that all functions of the form f(x) = a b^x + c share certain properties. (PSSM p. 298)

In addition to understanding exponentials as a class of functions, students should understand exponential functions in comparison to other classes. Exponential functions, when described, often sound like linear functions: “The population of Springfield increases by 1.5 percent every year.” This sounds linear because an increase of 1.5 percent per year seems to indicate a constant rate of change. However, because the growth is stated as a percentage, the increase each year is actually slightly higher than the previous year. Students should learn to distinguish this situation from a linear situation: “The population of Springfield increases by 150 people each year.”

Read what Mike Melville says about the important aspects of exponentials in the video for Workshop 6 Part II:

Transcript from Mike Melville

The first “a-ha moment” was [when] they realized that when you start taking a number, 2, to an exponential power, that it very quickly gets very big. And so that’s a really basic understanding of exponentials. Then when we moved into the class work, the cake and the beverage [problem from the Alice in Wonderland activity], and they realized that all you had to do was see the difference between the two exponents, and you knew what the exponent for the answer was, and that was a big “a-ha” for them. And the next big “a-ha” was, well, what happens when there’s no cake involved and we just have beverage? They got the sense that this is a weird number, it’s a decimal. And their second sense was that she’s shrinking, because it’s less than 1. And then the third one was that it’s never negative … I think they got this really good overview of what it means to have an exponential; that at one end, it’s very small, at the other end, it’s very big, and it never gets to a negative.

The links below are to pages within stable sites and are current as of the date of publication of this workshop. Due to the ever-changing nature of the Web, it is possible that some links may change. Should you reach a non-working link, we recommend entering a couple words from its description into the site’s search function, or into a Web-based search engine.

Affective Domain:

Goleman, Daniel. Working With Emotional Intelligence. New York: Bantam Books, 1998.

Krathwohl, D., B. Bloom, and B. Masia. Taxonomy of Educational Objectives. Book 2: Affective Domain. New York: David McKay, 1956.

Mason, J., L. Burton, and K. Stacey. Thinking Mathematically. London: Addison-Wesley, 1982.

Ornstein, A. C. and T. J. Lasley, II. Strategies for Effective Teaching (3rd edition). Boston: McGraw-Hill, 2000.

Ringness, Thomas A. The Affective Domain in Education. Boston: Little, Brown & Company, 1975.

Wilson, P. S. (ed). Research Ideas for the Classroom: High School Mathematics. New York: MacMillan Publishing Company, 1993.

Instructional Decision Making:

Silver, Harvey F., J. Robert Hanson, et al. Teaching Styles and Strategies: Interventions to Enrich Instructional Decision Making (3rd edition). Woodbridge, NJ: Thoughtful Education Press, 1996.

Sowder, Judith and Bonnie Schapelle. Lessons Learned from Research. Reston, VA: National Council of Teachers of Mathematics, 2002.

Professional Standards for Teaching Mathematics. Reston, VA: National Council of Teachers of Mathematics, 1991.

Exponents & Exponential Functions:

ESCOT Exponential Growth Applet – Rumors
http://www.escot.org/resources/problems/rumors.html

Powers of 10 Web site
http://powersof10.com/

Definitions of Exponential Function
Wikipedia: http://www.wikipedia.org/wiki/Exponential_function
MathWorld: http://mathworld.wolfram.com/Exponent.html

Barbeau, Edward J. Power Play. Washington, DC: Mathematical Association of America, 1997.

Globe Fearon (ed). Access to Math: Exponents & Scientific Notation. Peason Prentice Hall, 1996.