What is exponential growth? Whenever something is increasing or growing rapidly as a result of a constant rate of growth applied to it, that thing is experiencing exponential growth.
The figure above is an example of exponential growth. In fact, it is the graph of the exponential function y = 2^{x}
The general form of an exponential function is y = ab^{x}. Therefore, when y = 2^{x}, a = 1 and b = 2.
The following table shows some points that you could have used to graph this exponential growth. Try to locate some of these points on the graph!
x | 2^{x} | y |
1 | 2^{1} | 2 |
2 | 2^{2} | 4 |
3 | 2^{3} | 8 |
0 | 2^{0} | 1 |
-1 | 2^{-1} | 1/2 = 0.5 |
-2 | 2^{-2} | 1/4 = 0.25 |
-3 | 2^{-3} | 1/8 = 0.125 |
There are many real-life examples of exponential growth. For example, suppose that the population of Florida was 16 million in 2000. Then every year after that, the population has grown by 2%. This is an example of exponential growth.
Notice that the rate of growth is 2% or 0.02 and it is constant. This is important since the rate of growth cannot change.
Let us find the exponential function.
Year 2001 or 1 year after:
16,000,000 + 16,000,000 x 0.02 = 16,000,000 (1 + 0.02 )
16,000,000 + 16,000,000 x 0.02 = 16,000,000(1.02)
16,000,000 + 16,000,000 x 0.02 = 16,000,000(1.02)^{1}
Year 2002 or 2 years after:
16,000,000(1.02) + 16,000,000(1.02) x 0.02 = 16,000,000(1.02) [1 + 0.02]
16,000,000(1.02) + 16,000,000(1.02) x 0.02 = 16,000,000(1.02)(1.02)
16,000,000(1.02) + 16,000,000(1.02) x 0.02 = 16,000,000(1.02)^{2}
Following this pattern, suppose that
Then y = 16,000,000(1.02)^{x}
Comparing this exponential function with y = ab^{x}, we see that a = 16,000,000 and b = 1.02.
General rule for modeling exponential growth
Exponential growth can be modeled with the function
y = ab^{x} for a > 0 and b >1
y = ab^{x}
x is the exponent
a is the starting amount when x = 0
b is the base, rate, or growth factor and it is a constant and it is greater than 1.
Jan 20, 20 03:17 PM
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