§ 9.1

§ 9.1

Exponential Functions

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 9.1


Consider the function xxf 2)(

Does this function look different from other functions we have worked with before?

If you said, “yes” – you are right. It is different. The difference is that x is in the exponent – not the base.

Functions like this one are called “exponential functions.”



You will need a calculator for evaluating exponential expressions. Any scientific calculator will work for this purpose.

Many real-life situations, including population growth, growth of epidemics, radioactive decay, and other changes that involve rapid increase or decrease, can be described using exponential functions.



Definition of the Exponential FunctionThe exponential function f with base b is defined by

where b is a positive constant other than 1 (b > 0 and ) and x is any real number.

1b xx bybxf or

Page 640



Definition of the Exponential FunctionThe exponential function f with base b is defined by


1b xx bybxf or

Page 640



Examples of the Exponential Function

The exponential function f with base b is defined by


not

• Variable is the base and not the exponent.

• The base cannot be 1.

• The base of an exponential function must be positive.

• Variable is both the base and the exponent

1b xx bybxf or

40

1

1x

2

1)(3)( 10g(x) 2

xxx xjxhxf

xx xxxxxxF )(J )1()(H 1)G( x2



Characteristics of Exponential Functions of the Form T

1) The domain of consists of all real numbers. The range of consists of all positive real numbers.

2) The graphs of all exponential functions of the form pass through the point (0,1) because . The y-intercept is 1.

3) If b > 1, has a graph that goes up to the right and is an increasing function. The greater the value of b, the steeper the increase.

4) If 0 < b < 1, has a graph that goes down to the right and is a decreasing function. The smaller the value of b, the steeper the decrease.

5) The graph of approaches, but does not touch, the x-axis. The x-axis, or y = 0, is a horizontal asymptote.

xbxf xbxf

xbxf

xbxf

xbxf

xbxf

010 0 bbf

xbxf

Page 644See graph on page 644.



EXAMPLEEXAMPLE

Graph in the same rectangular coor-dinate system. How is the graph of g related to the graph of f ?

xx xgxf 3 and 3

SOLUTIONSOLUTION

We begin by setting up a table showing some of the coordinates for f and g, selecting integers from -2 to 2 for x.

x

-2

-1

0

1

2

xxg 3 xxf 3

9/132 2 f

3/131 1 f

130 0 f

331 1 f

932 2 f

9332 22 g

3331 11 g

1330 00 g

3/13/131 11 g

9/13/132 22 g



CONTINUECONTINUEDD We plot the points for each function and connect them with a

smooth curve. Because of the scale on the y-axis, some points on each function are not shown. The graph of g is a reflection of the graph of f across the y-axis.

-2.00, 0.11 -1.00, 0.330.00, 1.00

1.00, 3.00

2.00, 9.00-2.00, 9.00

-1.00, 3.00

1.00, 0.332.00, 0.110

1

2

3

4

5

6

7

8

9

-2 -1 0 1 2

xxf 3 xxg 3



The Natural Base eAn irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72 More accurately,

The number e is called the natural base. The function

is called the natural exponential function.

71828.2e

xexf

Remember that e is not a variable. It’s just an irrational number.

Pages 645-646

Not on tests.



EXAMPLES from homeworkEXAMPLES from homework

Do 2-10 (even) using calculator.

Do 12




Do 12-16 •Opposite causes graph to appear in quadrants 3 and 4 since f(x) will always be negative as in 14 and 16

•-1 in number 13 causes graph to cross the y axis at (0, 0) rather than (0, 1)




Do 18




20




26


Exponential Functions in Application

EXAMPLEEXAMPLE

In college, we study large volumes of information – information that, unfortunately, we do not often retain for very long. The function

describes the percentage of information, f (x), that a particular person remembers x weeks after learning the information.

Find the percentage of information that is remembered after 4 weeks.

2080 5.0 xexf

Page 646-647



Because we want to know the percentage of information retained after 4 weeks, we replace x with 4.

2080 5.0 xexf

CONTINUECONTINUEDDSOLUTIONSOLUTION

This is the given function.

20804 45.0 ef Replace x with 4.

20804 2 ef Multiply -0.5 and 4.

2014.0804 f Evaluate the exponent.

83.304 f Finish simplifying.

Therefore, four weeks after learning information, a certain person retains about 30.83% of that information.



Formulas for Compound InterestAfter t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas:

1) For n compoundings per year:

2) For continuous compounding:

nt

n

rPA

1

.rtPeA

Pages 647-648



EXAMPLEEXAMPLE

Find the accumulated value of an investment of $5000 for 10 years at an interest rate of 6.5% if the money is (a) compounded semiannually, (b) compounded monthly, (c) compounded continuously.

SOLUTIONSOLUTION

We are trying to determine what the accumulated value of an investment is. Therefore, we are looking for A. We first determine the values for t, P, and r. Since the investment will accumulate for 10 years, t = 10. Since the initial investment is $5000, P = 5000. And since the interest rate is 6.5%, r = 0.065. Now we are ready to use the appropriate formulas to answer the questions.



(a) Since the investment is being compounded semiannually, n = 2. We now solve for A.

nt

n

rPA

1

CONTINUECONTINUEDD

This is the equation to use.

102

2

065.015000

A Replace P with 5000, r with 0.065,

n with 2 and t with 10.

200325.015000 A Divide and multiply.

200325.15000A Add.

90.15000A Evaluate the exponent.



Therefore, the accumulated value of the investment is $9,479.19.

nt

n

rPA

1

CONTINUECONTINUEDD


1012

12

065.015000

A Replace P with 5000, r with 0.065,

n with 12 and t with 10.

1200054.015000 A Divide and multiply.

1200054.15000A Add.

19.9479A Multiply.

(b) Since the investment is being compounded monthly, n = 12. We now solve for A.



Therefore, the accumulated value of the investment is $9,541.92.

CONTINUECONTINUEDD


10065.05000 eAReplace P with 5000, r with 0.065, and t with 10.

65.05000eA Multiply.


92.9541A Multiply.

(c) Since the investment is being compounded continuously, there is no n value. We now solve for A.


rtPeA



Therefore, the accumulated value of the investment is $9,600.

CONTINUECONTINUEDD

9600A Multiply.

You may wish to remember the compound interest formula. Almost everyoneneeds to either borrow or invest – so that’s a formula that is applicable for many!



EXAMPLEEXAMPLE

The 1986 explosion at the Chernobyl nuclear power plant in the former Soviet Union sent about 1000 kilograms of radioactive

cesium-137 into the atmosphere. The function describes the amount, f (x), in kilograms, of cesium-137 remaining in Chernobyl x years after 1986. If even 100 kilograms of cesium-137 remain in Chernobyl’s atmosphere, the area is considered unsafe for human habitation. Find f (80) and determine if Chernobyl will be safe for human habitation by 2066.

305.01000x

xf



SOLUTIONSOLUTION

305.01000x

xf This is the given function.

CONTINUECONTINUEDD

In finding f (80), we are finding how many kilograms of cesium-137 are in Chernobyl 80 years after 1986, or in 2066.

30

80

5.0100080 f Replace x with 80.

3

8

5.0100080 f Divide.

157.0100080 f Evaluate the exponent. 15780 f Multiply.

Chernobyl will not be safe for human habitation by 2066 with approximately 157 kilograms of cesium-137.

§ 9.1

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