Lesson 3.1, page 376 Exponential Functions

Objective: To graph exponentials equations and functions, and solve

applied problems involving exponential functions and their

graphs.

Look at the following…

Polynomial Exponential

2( ) 4 3 1 ( ) 4 3xf x x x f x

Real World Connection Exponential functions are used to

model numerous real-world applications such as population growth and decay, compound interest, economics (exponential growth and decay) and more.

REVIEW Remember: x0 = 1 Translation – slides a figure

without changing size or shape

Exponential Function

The function f(x) = bx, where x is a real number, b > 0 and b 1, is called the exponential function, base b.

(The base needs to be positive in order to avoid the complex numbers that would occur by taking even roots of negative numbers.)

Examples ofExponential Functions, pg. 376

1( ) 3 ( )

3

( ) (4.23)

xx

x

f x f x

f x

See Example 1, page 377. Check Point 1: Use the function

f(x) = 13.49 (0.967) x – 1to find the number of О-rings expected to fail

at a temperature of 60° F. Round to the nearest whole number.

Graphing Exponential Functions

1. Compute function values and list the results in a table.

2. Plot the points and connect them with a smooth curve. Be sure to plot enough points to determine how steeply the curve rises.

Check Point 2 -- Graph the exponential function y = f(x) = 3x.

(3,1/27)1/273

(2, 1/9)1/92

(1, 1/3)1/31

(3, 27)273

9

3

1

y = f(x) = 3x

(2, 9)2

(1, 3)1

(0, 1)0

(x, y)x

Check Point 3: Graph the exponential function

(3,1/27)1/273

(2, 1/9)1/92

(1, 1/3)1/31

(3, 27)273

9

3

1

(2, 9)2

(1, 3)1

(0, 1)0

(x, y)x 1( )

3

x

y f x

1( )

3

x

y f x

Characteristics of Exponential Functions, f(x) = bx, pg. 379

Domain = (-∞,∞) Range = (0, ∞) Passes through the point (0,1) If b>1, then graph goes up to the right and

is increasing. If 0<b<1, then graph goes down to the right

and is decreasing. Graph is one-to-one and has an inverse. Graph approaches but does not touch x-axis.

Observing Relationships

Connecting the Concepts

Example -- Graph y = 3x +

2.

The graph is that of y = 3x shifted left 2 units.

2433

812

271

90

3

1

1/3

y= 3 x+2

1

2

3

x

Example: Graph y = 4 3x

3.963

3.882

3.671

30

1

5

23

y

1

2

3

x

The graph is a reflection of the graph of y = 3x across the y-axis, followed by a reflection across the x-axis and then a shift up of 4 units.

The number e (page 381) The number e is an irrational

number. Value of e 2.71828 Note: Base e exponential functions

are useful for graphing continuous growth or decay.

Graphing calculator has a key for ex.

Practice with the Number e

Find each value of ex, to four decimal places, using the ex key on a calculator.a) e4 b) e0.25

c) e2 d) e1

Answers:a) 54.5982 b) 0.7788c) 7.3891 d) 0.3679

Natural Exponential Function

Remember e is a number

e lies between 2 and 3

xf (x) e

Compound Interest Formula

A = amount in account after t years P = principal amount of money

invested R = interest rate (decimal form) N = number of times per year interest

is compounded T = time in years

1 nt

rA P

n

Compound Interest Formula for Continuous Compounding

A = amount in account after t years

P = principal amount of money invested

R = interest rate (decimal form) T = time in years

rtA Pe

See Example 7, page 384.Compound Interest Example

Check Point 7: A sum of $10,000 is invested at an annual rate of 8%. Find the balance in that account after 5 years subject to a) quarterly compounding and b) continuous compounding.