4.3
GRAPHS OF EXPONENTIAL
FUNCTIONS
1
Graphs of the Exponential Family: The
Effect of the Parameter a
In the formula Q = abt, the value of a tells us where the graph
crosses the Q-axis, since a is the value of Q when t = 0.
Q
Q
Q=150 (1.2)t
Q=50 (1.4)t
Q=50 (1.2)t
Q=100 (1.2)t
50
150
100
50
t
0
Q=50 (0.8)t
Q=50 (1.2)t
5
10
Q=50 (0.6)t
0
t
5
2
Graphs of the Exponential Family: The
Effect of the Parameter b
The growth factor, b, is called the base of an exponential
function. Provided a is positive, if b > 1, the graph climbs
when read from left to right, and if 0 < b < 1, the graph
falls when read from left to right.
Q
Q=50 (1.4)t
50
Q=50 (1.2)t
Q=50 (0.8)t
Q=50 (0.6)t
0
t
5
3
Horizontal Asymptotes
The horizontal line y = k is a horizontal asymptote of a
function, f, if the function values get arbitrarily close to k
as x gets large (either positively or negatively or both).
We describe this behavior using the notation
f(x) → k as x → ∞
or
f(x) → k as x → −∞.
Alternatively, using limit notation, we write
lim
x 
f ( x)  k
or
lim
f ( x)  k
x 
4
Interpretation of a Horizontal Asymptote
Example 1
A capacitor is the part of an electrical circuit that stores electric
charge. The quantity of charge stored decreases exponentially with
time. If t is the number of seconds after the circuit is switched off,
suppose that the quantity of stored charge (in micro-coulombs) is
given by Q = 200(0.9)t, t ≥0.
The charge stored by a capacitor over one minute.
200 Q, charge (micro-coulombs)
100
0
15
30
t (seconds)
45
60
5
Solving Exponential Equations Graphically
Exercise 42
The population of a colony of rabbits grows exponentially. The
colony begins with 10 rabbits; five years later there are 340
rabbits.
(a) Give a formula for the population of the colony of rabbits as a
function of the time.
(b) Use a graph to estimate how long it takes for the population
of the colony to reach 1000 rabbits.
Solution
Solving 340 = 10b5 we get
b5 = 34, b = 2.0244, R ≈ 10 (2.0244)t
Based on the graph, one would
estimate that the population of
rabbits would reach 1000 in a
little more than 6 ½ years.
6
Finding an Exponential Function for Data
Example: Population data for the Houston Metro Area Since 1900
Table showing population (in
thousands) since 1900
t
0
10
20
30
40
50
N
184
236
332
528
737
1070
t
60
70
80
90
100
110
N
1583
2183
3122
3733
4672
5937
Graph showing population data
with an exponential model
P (thousands)
6000
P = 190 (1.034)t
5000
4000
3000
2000
1000
t (years since 1900)
0
0
20
40
60
80
100
Using an exponential regression feature on a calculator or computer
the exponential function was found to be P = 190 (1.034)t
7