it has clear mathematical concepts that enhance the reader in engaging in concepts that will require the reader to fully understand the basic principles of growth and decay that usually involve appreciation and depreciation as well as mathematical induction in proving the truth of a mathematical ex...
An exponential function is a nonlinear function of the form y = abx , where a 6= 0, b 6= 1,
and b > 0. The y-intercept is a in this case and is obtained by putting x = 0. As the
independent variable x changes by a constant amount, the dependent variable y is multiplied
by a constant factor (the y-values change by a factor of b as x increases by 1), which means
consecutive y-values form a constant ratio.
Example:
The function represented by the table
x 0 1 2 3
y 4 8 16 32
is an exponential function since as x increases by 1, y is multiplied by 2. The function is
y = 4 (2x ) in this case. On the other hand, the function represented by the table
x 0 1 2 3
y 2 4 12 48
is not exponential since as x increases by 1, y is not multiplied by a constant factor.
Exponential Growth and Decay
Definition: Exponential growth occurs when a quantity increases by the same factor over
equal intervals of time. A function of the form y = a(1 + r)t , where a > 0 and r > 0, is an
exponential growth function. In this case y = final amount, a = initial amount, r = rate of
growth (in decimal form), 1 + r = growth factor (where 1 + r > 1), t = time. Note that the
function is of the form y = abx , where b is replaced by 1 + r and x is replaced by t.
Definition: Exponential decay occurs when a quantity decreases by the same factor over
equal intervals of time. A function of the form y = a(1 − r)t , where a > 0 and 0 < r < 1, is
an exponential decay function. In this case y = final amount, a = initial amount, r = rate
of decay (in decimal form), 1 − r = decay factor (where 1 − r < 1), t = time. Similarly, the
function is of the form y = abx , where b is replaced by 1 − r and x is replaced by t.
Example:
1. Consider the table below.
x 0 1 2 3
y 5 10 20 40
As x increases by 1, y is multiplied by 2. So, the table represents an exponential
growth function.
1
, 2. Consider the table below.
x 0 1 2 3
y 270 90 30 10
As x increases by 1, y is multiplied by 1/3. So, the table represents an exponential
decay function.
3. Determine whether each function represents exponential growth or exponential decay,
and hence find the percent rate of change.
(a) y = 5(1.07)t
(b) f (t) = 0.2(0.98)t
Solution:
(a) The function is of the form y = a(1 + r)t , where 1 + r > 1. So it represents
exponential growth. In this case the growth factor is 1 + r = 1.07 so that r = 0.07.
Therefore, the rate of growth is 7%.
(b) The function is of the form y = a(1 − r)t , where 1 − r < 1. So it represents
exponential decay. In this case the decay factor is 1 − r = 0.98 so that r = 0.02.
Therefore, the rate of decay is 2%.
4. The inaugural attendance of an annual music festival is 150,000. The attendance y
increases by 8% each year.
(a) Write an exponential growth function that represents the attendance after t years.
(b) How many people will attend the festival in the fifth year? Round your answer to
the nearest thousand.
Solution:
(a) The initial amount is 150000, and the rate of growth is 8%, or 0.08. The exponential
growth function is
Therefore, the festival attendance is represented by y = 150000(1.08)t .
(b) During the first year t = 0 and during the fifth year t = 4. So, in the fifth year, we
have y = 150000(1.08)4 ≈ 204073. Therefore, about 204,000 people will attend the
festival in the fifth year.
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