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…
f ( x)  4 x  3 x  1
2

Polynomial
f ( x)  4  3
x
Exponential
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 of
Exponential Functions, pg. 376
f ( x)  3
1
f ( x)   
3
x
f ( x)  (4.23)
x
x
See Example 1, page 377.

Check Point 1: Use the function
f(x) = 13.49 (0.967) x – 1
to 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.
2.
Compute function values and list the
results in a table.
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.
x
y = f(x) = 3x
(x, y)
0
1
(0, 1)
1
3
(1, 3)
2
9
(2, 9)
3
27
(3, 27)
1
1/3
(1, 1/3)
2
1/9
(2, 1/9)
3
1/27
(3,1/27)
x
Check Point 3: Graph the
1
y

f
(
x
)

 
exponential function
3
x
1
y  f ( x)   
3
x
(x, y)
0
1
(0, 1)
1
3
(1, 3)
2
9
(2, 9)
3
27
(3, 27)
1
1/3
(1, 1/3)
2
1/9
(2, 1/9)
3
1/27
(3,1/27)
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=
x
+
2
3
.
The graph is that of y = 3x shifted left 2 units.
x
y= 3 x+2
3
1/3
2
1
1
3
0
9
1
27
2
81
3
243
Example:
Graph y = 4  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.
x
y
3
23
2
5
1
1
0
3
1
3.67
2
3.88
3
3.96
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) e0.25
c) e2
Answers:
a) 54.5982
c) 7.3891
d) e1
b) 0.7788
d) 0.3679
Natural Exponential Function
f(x)  e

x
Remember
 e is a number
e lies between 2 and 3
Compound Interest Formula
 r
A  P 1  
 n





nt
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
Compound Interest Formula for
Continuous Compounding
A  Pe




rt
A = amount in account after t years
P = principal amount of money invested
R = interest rate (decimal form)
T = time in years
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.