LESSON 20 (3.1) EXPONENTIAL FUNCTIONS AND THEIR GRAPHS
You should learn to:
1. Recognize, evaluate, and graph exponential functions with base any base.
2. Use exponential functions to model and solve real-life problems.
Terms to know: exponential function with base a, natural base e, compound interest: non-continuous vs.
continuous compounding, exponential growth (or decay), half-life.
The exponential function with base a is written as f  x   a x , where a > 0, a  1, and x is any real number.
Exponential Function: y  a x
Graph:
For exponential functions, the base is constant
(some number) and the exponent is variable.
x
1
You need to be careful not to confuse y  2 , y  3 , y    , etc.
2
x
x
1
2
with y  x , y  x , y  x , etc. (power functions)
2
3
Example 1: Graph the following exponential functions. Find the domain and asymptotes of each.
y  2x
y  5x
y  3x
X
-2
Y
¼
X
-1
Y
1/3
X
-1
Y
1/5
-1
½
0
1
0
1
0
1
1
3
1
5
1
2
2
9
2
4
Domain:
Domain:
Domain:
Range: y  0
Range: y  0
Range: y  0
Asymptote : y  0
Asymptote : y  0
Asymptote : y  0
List the common features for the graphs. These are the characteristics for all exponential graphs of the form
y  a x , a  1.
R: y  0
D:
Asy: y  0
Key Point: (1, a),  1, 1a 
y-int: (0,1)
continuous and always increasing.
Example 2:
Graph the following based on the graphs from #1.
( x 3)
a. y  3 x
b. y  2
y  3x
y  2x
reflected through
shifted right
the y axis.
3 units
c. y  2
( x 1)
2
y  2x
shifted left 1 unit
and down 2 units
In Calculus, by far the most commonly used base for exponential functions is base e. It is known as the natural base.
e  2.718
Example 3: Graph y  e x . Use it to graph the other functions listed. List 2 points and the asymptote for each.
y  ex  3
y  ex
Key points: (0,1), (1, e)
Asy: y  0
y  e x  2
Key points: (0, 2), (1, e  3)
Asy: y  3
Asy: y  2
Key points: (0,1), (1, e  2)
A common business application of the number e occurs in computing interest for investments.
1.
Non-continuous compounding: Interest is calculated a specific number of times each year.
(Daily (365) , weekly (52) , monthly (12) , quarterly (4) , semi-annual (2) , and annual (1) compoundings are
the
most common types)
2.
Continuous compounding: Interest is always being calculated - an infinite number of times.
 r
A  P 1  
 n
Compound Interest:
nt
A = Amount
P = Principal
r = rate
t = time (yrs)
n = number of compounds per year
Continuous Interest:
A  Per t
Example 4: Suppose you have $10,000 to invest in a college fund, and you can leave the money in the fund
for 8 years before you start college. Which of the following would provide you with the most
money for college?
a. A fund providing an interest rate of 5.35%
compounded semi-annually.
 r
A  P 1  
 n
A?
b. A fund providing an interest rate of 5.3%
compounded continuously.
A  Pe r t
nt
A?
P  10, 000
r  .053
P  10, 000
t 8
r  .0535
A  10, 000e.0538
t 8
A  $15, 280.62
n2
 .0535 
A  10, 000  1 

2 

28
A  $15, 255.81
The continuous compoundings would
provide the most money for college.
*The two most common student errors in solving interest problems are:
1.
Not changing the interest percent to a pure decimal rate of r.
2. Using parentheses incorrectly for non-continuous compoundings. Make sure you enclose the exponent  n  t  in
parentheses.
Many common applications of exponential growth (or decay) occur in the sciences.
Example 5: Radioactive Decay
Let Q represent the mass, in grams, of radioactive radium

226
Ra  , whose half-life is 1620 years. The mass of
t
 1 1620
radium present after t years is given by the formula: Q (t)  16  
.
2
a. What is the initial mass?
t 0
b. What amount of radioactive
radium 226 Ra remains after 1000 years?


t  1000
t
 1 1620
Q (t)  16  
2
0
t
 1 1620
Q (t)  16  
2
1000
 1 1620
Q (0)  16  
2
 1 1620
Q (1000)  16  
2
Q (0)  16 grams
Q (1000)  10.430 grams
ASSIGNMENT 20
Pages 193-195 (Vocabulary Check 4-5, 1-5, 7, 9, 13-17, 21, 23-24, 29, 37,
45 (find asymptotes by letting x  100 and x  100. The graph of this function is
called a logistics curve.), 50, 52, 54, 60, 68, 72, 75, 78, 85, 87)