Growth and Decay
Exponential Functions
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.
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, 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
y = a(1 + r)t
= 150000(1 + 0.08)t
= 150000(1.08)t .
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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Exponential Functions
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
y = a(1 + r)t
= 150000(1 + 0.08)t
= 150000(1.08)t .
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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