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5.2 Exponential Functions & Graphs


Graph exponential equations and
exponential functions.
Solve applied problems involving
exponential functions and their
graphs.
Exponential Function
The function f(x) = ax, where x is a real number, a > 0
and a  1, is called the exponential function, base a.
The base needs to be positive in order to avoid the
complex numbers that would occur by taking even
roots of negative numbers.
The following are examples of exponential functions:
x
 1
x
f (x)  2
f (x)   
f (x)  (3.57)x
 2
Graphing Exponential Functions
To graph an exponential function, follow the steps
listed:
1. Compute some 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.
Graph
y2
x
x
-3
-2
-1
0
1
2
3
y
Graph
1
y 
2
x
x
-3
-2
-1
0
1
2
3
y
y  2x
1
y 
2
x
Graphs of Exponential Functions
Observe the following graphs of exponential
functions and look for patterns in them.
Example
Graph y = 2x – 2.
The graph is the graph of y = 2x shifted to right 2 units.
Example
x
 1
x

5

0.5

5


5

2
.
Graph y = 5 –
 
2
The graph is a reflection of the graph of y = 2x across
the y-axis, followed by a reflection across the x-axis
and then a shift up 5 units.
0.5x . y
x
Example
Find the exponential function whose graph is
shown.
Application
The amount of money A to which a principal
P will grow to after t years at interest rate r
(in decimal form), compounded n times per
year, is given by the formula
nt
r

A  P 1   .

n
nt
Example
r

A  P 1   .

n
Suppose that $100,000 is invested at 6.5%
interest, compounded semiannually.
a) Find a function for the amount to which the
investment grows after t years.
b) Find the amount of money in the account at t = 0,
4, 8, and 10 yr.
c) Graph the function.
a) Since P = $100,000, r = 6.5%=0.065, and n = 2,
we can substitute these values and write the
following function
0.065 

A t   100,000  1 


2
2t
 $100,000 1.0325 
2t
b) We can calculate the values directly on a calculator.
c) Graph.
The Number e
e is a very special number in mathematics.
Leonard Euler named this number e. The
decimal representation of the number e does not
terminate or repeat; it is an irrational number
that is a constant;
e  2.7182818284…
nt
r

A  P 1   .

n
Example
Find each value of ex, to four decimal places,
using the ex key on a calculator.
a) e3
b) e0.23
c) e0
d) e1
Solution:
a) e3 ≈ 20.0855
b) e0.23 ≈ 0.7945
c) e0 = 1
d) e1 ≈ 2.7183
Graphs of Exponential Functions, Base e
Example
Graph f(x) = ex and g(x) = e–x.
Use the calculator and enter y1 = ex and y2 = e–x. Enter
numbers for x.
Graphs of Exponential Functions,
Base e - Example (continued)
The graph of g is a reflection of the graph of f across the y-axis.
Example
Graph f(x) = ex + 3.
Solution: The graph f(x) = ex + 3 is a translation of
the graph of y = ex left 3 units.
Example
Graph f(x) = e–0.5x.
Solution: The graph f(x) = e–0.5x is a horizontal
stretching of the graph of y = ex followed by a
reflection across the y-axis.
Example
Graph f(x) = 1  e2x.
Solution: The graph f(x) = 1  e2x is a horizontal
shrinking of the graph of y = ex
followed by a reflection across
the y-axis and then across the
x-axis,
followed
by a
translation
up 1 unit.
Example
Find the exponential function given
f(0) = 4 and f(-2) = 16.
Example
a. Determine an exponential function for the
population after t years given the table below.
The population is given in thousands.
t
P(t)
0
10
5
20
10
40
b. What is the population when t = 40?