Introduction to Exponential Functions


Objectives

At the end of this lesson, you should be able to:

  • Identify exponential functions.
  • Obtain the associated exponential formula from a given situation.


Introduction

Let's analyze the following situations:

Situation 1:

A rabbit population initially contains 100 rabbits and doubles every year. If x represents the number of elapsed years and P(x) the rabbit population after x years, the following table shows the values of P(x) for some values of x:

x 0 1 2 3
P(x) 100 200 400 800

Considering that when x increases by 1, the rabbit population doubles, the previous table can be written as follows:

x 0 1 2 3
P(x) 100 = 100×20 200 = 100×21 400 = 100×22 800 = 100×23

From the previous table, we can conclude that the rabbit population follows the model,

P ( x ) = 100 × 2 x

Situation 2:

A laboratory experiments with a drug to wipe out a bacteria population. Initially the population has 1000 bacteria. A scientist observes that the size of the bacteria population halves each day. If x represents the number of elapsed days and P(x) the size of the bacteria population after x days, the following table shows the values of P(x) for some values of x:

x 0 1 2 3
P(x) 1000 500 250 125

Considering that when x increases by 1, the size of the bacteria population halves, the previous table can be written as follows:

x 0 1 2 3
P(x) 1000 = 1000×(½)0 500 = 1000×(½)1 250 = 1000×(½)2 125 = 1000×(½)3

From the previous table, we can conclude that the bacteria population size follows the model,

P x = 1000 × ( 1 2 ) x

In the previous examples, when x increases by 1, the function P(x) multiplies by 2 and 12 respectively. Both of them are examples of exponential functions.

An exponential function f(x) describes a situation where, increasing x by a constant, multiplies f(x) by a constant.



Situations and the Associated Exponential Function

Example 1:

A kangaroo population consist initially of 200 kangaroos and doubles every 3 years. If x represents the elapsed years and P(x) the size of the kangaroo population after x years, what is the associated function, P(x), that models the kangaroo population size by year?

Solution:

The following table shows values of P(x) for some values of x:

x 0 3 6 9
P(x) 200 400 800 1600

In this example, when x increases by 3, the kangaroo population size doubles. The previous table can be written as:

x 0 3 6 9
P(x) 200 = 200×20 400 = 200×21 800 = 200×22 1600 = 200×23

From the previous table, we can conclude that the kangaroo population size follows the model,

P x = 200 × 2 x 3



Example 2:

A bird population consist initially of 10 birds. The size of the population increases by a factor of 3 every 5 years. If x represents the elapsed years and P(x) the bird population size after x years, what is the associated function, P(x), that models the bird population size by year?

Solution:

The following table shows values of P(x) for some values of x:

x 0 5 10 15
P(x) 10 30 90 270

In this example, when x increases by 5, the bird population increases by a factor of 3. The previous table can be written as:

x 0 5 10 15
P(x) 10 = 10×30 30 = 10×31 90 = 10×32 270 = 10×33

From the previous table, we can conclude that the bird population size follows the model,

P x = 10 × 3 x 5

Example 3:

An industrial machine whose purchase price was $10,000 depreciates by a factor of 1/10 every 6 years. If x represents the elapsed years and P(x) the value of the machine after x years, what is the associated function, P(x), that models the value of the machine by year?

Solution:

The following table shows values of P(x) for some values of x:

x 0 6 12 18
P(x) 10000 9000 8100 7290

In this example, when x increases by 6, the machine's value decreases by a factor of 1/10. The previous table can be written as:

x 0 6 12 18
P(x) 10000 = 10000 × ( 9 10 ) 0 9000 = 10000 × ( 9 10 ) 1 8100 = 10000 × ( 9 10 ) 2 7290 = 10000 × ( 9 10 ) 3

From the previous table, we can conclude that the machine's value follows the model,

P x = 10000 × ( 9 10 ) x 6



Example 4:

The half life of the carbon-14 is 5,668 years. That is to say that a specified amount of carbon-14 will be halved in 5,668 years. Suppose we have a sample of 25 grams. If x represents the elapsed years and P(x) the mass of carbon-14 after x years, what is the associated function, P(x), that models the mass of carbon-14 by year?

Solution:

The following table shows values of P(x) for some values of x:

x 0 5668 11336 17004
P(x) 25 12.5 6.25 3.125

In this example, when x increases by 5668, the carbon-14 mass is halved. The previous table can be written as:

x 0 5668 11336 17004
P(x) 25 = 25 × ( 1 2 ) 0 12.5 = 25 × ( 1 2 ) 1 6.25 = 25 × ( 1 2 ) 2 3.125 = 25 × ( 1 2 ) 3

From the previous table we can conclude that the carbon-14 mass follows the next model,

P x = 25 × ( 1 2 ) x 5668
As you could see in the first two examples, the function is increasing, namely the value of the function increases when x increases. In examples 3 and 4 the function is decreasing, so the value of the function decreases when x grows.


Now you should know how to build an exponential function from a given situation. But, if you still don't quite have it, you can use the following applications to see more examples and check your understanding. The applications let you choose the initial data, the increasing or decreasing factor and the period where the change happens.



Definition

If a number is multiplied periodically by a constant, as in the previous examples, the problem models an exponential function.

An exponential function f(x) is a function of the form:

f x = a × b x

or

f x = a × b c x

where a, b and c are constants.

Also if b is not equal to 1 and

  • if b > 1, the exponential function is increasing.
  • if 0 < b < 1, the exponential function is decreasing.


Summary

Now that you have completed this lesson, you should be able to:

  • Identify exponential functions.
  • Determine the associated exponential formula from a given situation.
Kentucky Center for Mathematics
Northern Kentucky University, MEP 475
Highland Heights, KY 41076
ph: 859-572-7690, fax: 859-572-7677
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