Transcript Exponential Functions Definition of the Exponential Function The exponential function f with base b is defined by f (x) = bx or y.

Exponential Functions
Definition of the Exponential
Function
The exponential function f with base b is defined by
f (x) = bx or y = bx
/
Where b is a positive constant other than and x is any real number.
Here are some examples of exponential functions.
f (x) = 2x
g(x) = 10x
h(x) = 3x+1
Base is 2.
Base is 10.
Base is 3.
Text Example
The exponential function f (x) = 13.49(0.967)x – 1 describes the
number of O-rings expected to fail, when the temperature is x°F.
On the morning the Challenger was launched, the temperature was
31°F, colder than any previous experience. Find the number of Orings expected to fail at this temperature.
Solution Because the temperature was 31°F, substitute 31 for x and
evaluate the function at 31.
f (x) = 13.49(0.967)x – 1
f (31) = 13.49(0.967)31 – 1

f (31) = 13.49(0.967)31 – 1=3.77
This is the given function.
Substitute 31 for x.
1.
2.
3.
4.
5.
6.
Characteristics of Exponential
Functions
The domain of f (x) = b consists of all real numbers. The range of f (x)
x
= bx consists of all positive real numbers.
The graphs of all exponential functions pass through the point (0, 1)
because f (0) = b0 = 1.
If b > 1, f (x) = bx has a graph that goes up to the right and is an
increasing function.
If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is a
decreasing function.
f (x) = bx is a one-to-one function and has an inverse that is a function.
The graph of f (x) = bx approaches but does not cross the x-axis. The xaxis is a horizontal asymptote.
f (x) = bx
0<b<1
f (x) = bx
b>1
Transformations Involving Exponential
Functions
Transformation
Equation
Description
Horizontal translation
g(x) = bx+c
• Shifts the graph of f (x) = bx to the left c units if c > 0.
• Shifts the graph of f (x) = bx to the right c units if c < 0.
Vertical stretching or
shrinking
g(x) = c bx
Multiplying y-coordintates of f (x) = bx by c,
• Stretches the graph of f (x) = bx if c > 1.
• Shrinks the graph of f (x) = bx if 0 < c < 1.
Reflecting
g(x) = -bx
g(x) = b-x
• Reflects the graph of f (x) = bx about the x-axis.
• Reflects the graph of f (x) = bx about the y-axis.
Vertical translation
g(x) = -bx + c
• Shifts the graph of f (x) = bx upward c units if c > 0.
• Shifts the graph of f (x) = bx downward c units if c < 0.
Text Example
Use the graph of f (x) = 3x to obtain the graph of g(x) = 3 x+1.
Solution Examine the table below. Note that the function g(x) = 3x+1 has
the general form g(x) = bx+c, where c = 1. Because c > 0, we graph g(x) = 3
x+1 by shifting the graph of f (x) = 3x one unit to the left. We construct a table
showing some of the coordinates for f and g to build their graphs.
g(x) = 3x+1
(-1, 1)
-5 -4 -3 -2 -1
f (x) = 3x
(0, 1)
1 2 3 4 5 6
Problems
Sketch a graph using transformation of the following:
1.
f ( x)  2x  3
2.
f ( x)  2 x  1
3.
f ( x)  4x1  1
Recall the order of shifting: horizontal, reflection (horz., vert.),
vertical.
The Natural Base e
An 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,
e
2.71828...
The number e is called the natural base. The function f (x) = ex is called the
natural exponential function.
f (x) = 3x f (x) = ex
4
f (x) = 2x
(1, 3)
3
(1, e)
2
(1, 2)
(0, 1)
-1
1
Formulas for Compound
Interest
After t years, the balance, A, in an account with
principal P and annual interest rate r (in
decimal form) is given by the following
rt
 r
formulas:
A  P 1  
 n
1. For n compoundings per year:
2. For continuous compounding: A = Pert.
Example: Choosing Between Investments
You want to invest $8000 for 6 years, and you have a choice between two
accounts. The first pays 7% per year, compounded monthly. The second pays
6.85% per year, compounded continuously. Which is the better investment?
Solution The better investment is the one with the greater balance in the
account after 6 years. Let’s begin with the account with monthly
compounding. We use the compound interest model with P = 8000,
r = 7% = 0.07, n = 12 (monthly compounding, means 12 compoundings per
year), and t = 6.
 
r
A  P 1
n
nt

0.07
 8000 1 
12

12*6
 12,160.84
The balance in this account after 6 years is $12,160.84.
more
Example: Choosing Between Investments
You want to invest $8000 for 6 years, and you have a choice between two
accounts. The first pays 7% per year, compounded monthly. The second pays
6.85% per year, compounded continuously. Which is the better investment?
Solution For the second investment option, we use the model for
continuous compounding with P = 8000, r = 6.85% = 0.0685, and t = 6.
A  Pert  8000e0.0685(6)  12,066.60
The balance in this account after 6 years is $12,066.60, slightly less than the
previous amount. Thus, the better investment is the 7% monthly
compounding option.
Example
Use A= Pert to solve the following problem: Find
the accumulated value of an investment of
$2000 for 8 years at an interest rate of 7% if
the money is compounded continuously
Solution:
A= Pert
A = 2000e(.07)(8)
A = 2000 e(.56)
A = 2000 * 1.75
A = $3500