Graphing experimental data might reveal exponential growth or decay.
Exponential Growth
Exponential functions play key roles in modeling many natural phenomena. Such models are based upon empirical data. For example, most people know that their chances of getting cancer increase as they age. In fact, by looking at data complied by the National Cancer Institute you can readily see that incidence of cancer increases dramatically between the ages of 35 and 80. Look at this data for incidence of all cancers in females from 2001:
Age |
Cancer cases per 100,000 people |
<1 |
23 |
1-4 |
19 |
5-9 |
10 |
10-14 |
12 |
15-19 |
19 |
20-24 |
33 |
25-29 |
59 |
30-34 |
101 |
35-39 |
160 |
40-44 |
265 |
45-49 |
398 |
50-54 |
576 |
55-59 |
803 |
60-64 |
1059 |
65-69 |
1353 |
70-74 |
1603 |
75-79 |
1817 |
80-84 |
1897 |
>80 |
1790 |
Data from U.S. Cancer Statistics Working Group. United States Cancer Statistics: 1999-2001 Incidence and Mortality Web-based Report Version . Atlanta (GA): Department of Health and Human Services, Centers for Disease Control and Prevention, and National Cancer Institute; 2004. Available at: www.cdc.gov/cancer/npcr/uscs . Accessed 11/14/2005. |
Notice that the incidence of cancer nearly doubles every ten years between the ages of 20 and 75. If we graph this data with age on the x-axis and cancer incidence per 100,000 people on the y-axis, you can see that cancer incidence appears to increase exponentially until age 75.
Exponential Decay
Human immunodeficiency virus (HIV) can now be controlled using a combination of powerful antiviral drugs. Upon treatment, the amount of virus in the bloodstream of infected patients falls rapidly then plateaus to levels that are difficult to detect with standard laboratory techniques. The following hypothetical data models the typical course of infection for a patient taking a regimen of antiviral drugs for one year.
Duration of treatment (weeks) |
Viral load (molecules/mL blood) |
0 |
123,550 |
2 |
12,170 |
4 |
975 |
6 |
150 |
8 |
80 |
10 |
55 |
12 |
25 |
14 |
10 |
16 |
10 |
18 |
10 |
20 |
10 |
22 |
10 |
24 |
10 |
26 |
10 |
28 |
10 |
30 |
10 |
32 |
10 |
34 |
10 |
36 |
10 |
38 |
10 |
40 |
10 |
42 |
10 |
44 |
10 |
46 |
10 |
48 |
10 |
50 |
10 |
If you plot duration of treatment on the x-axis and viral load on the y-axis, you can study how the amount of virus in the bloodstream diminishes over time.
By inspecting this graph, it appears that the overall amount of virus in the blood stream exhibits approximate exponential decay during drug treatment. Virologists and mathematicians have collaborated to describe viral dynamics and calculate the half life of virus in the blood stream in the presence and absence of various drugs.
What does the graphical representation of y = a ^{x}, look like?
Given an exponential function, you can determine if it represents exponential growth or exponential decay by looking at the value
of the constant a.
Let's consider exponential decay and exponential growth by inspecting their respective
general shapes of their graphical representations.
Case 1: 0 < a < 1, Exponential Decay
Case 2: a > 1, Exponential Growth
To be accurate in sketching y = a ^{x},
we need to know the exact value of a. In
particular,
if
a is much greater than one (or much less than one) the
exponential function will grow (or decay)
very quickly.
For example, the functions f_{1} (x)
= 10^{ x} and f_{2 }(x)
= (1.1)^{ x} will have
the same general shape as in Figure 2 (i.e. both exhibit
exponential growth), but will differ in their
rates of growth. Specifically, as x gets
large (i.e. x → ∞),
f_{1 }(x) = 10 ^{x} gets
larger faster than f_{2 }(x) = (1.1) ^{x}.
Both functions, however, increase without bound.
*****
In the next section we will summarize
these results and focus on graphing specific exponential
functions by looking at two representative examples. |