Exponential function
An exponential function is a function with the general form y = abx and the following conditions:
- a is a constant and a is not equal to zero (a ≠ 0)
- b is bigger than zero (b > 0)
- b is not equal to 1 (b ≠ 1)
Notice the use of the independent variable (x) as an exponent. This is important in order to have an exponential function.
Why a cannot be equal to 0?
If a = 0, then y = 0 × b
x = 0 since zero times anything is zero.
Therefore, a cannot be zero since it will make the function equal to zero.
Why b must be bigger than zero? In other words, b cannot be zero and b cannot be a negative number.
b cannot be zero since y will be equal to a × 0
x = a × 0 = 0
b cannot be negative either. This can create some problems. For example, suppose b = -1, we get y = a (-1)
x
When x = 0.5, y = a(-1)
0.5 = a √(-1) and √(-1) is a complex number.
Why b cannot equal to 1?
If b = 1, then y = a × 1
x = a × 1 = a since 1 to any power is equal to 1 and a times 1 is a.
Notice that x disappears as an exponent when b = 1. Therefore, b cannot be equal to 1.
When b > 1, you can model growth and b is the growth factor.
When 0 < b < 1, you can model decay and b is the decay factor.
What does the graph of an exponential function look like?
As you can see from the figure above, the graph of an exponential function can either show a growth or a decay.
The figure on the left shows exponential growth while the figure on the right shows exponential decay.
Examples of exponential functions
1. y = 0.5 × 2
x
2. y = -3 × 0.4
x
3. y = e
x
4. y = 10
x
Can you tell what b equals to for the following graphs?
0.5 × 2
x, e
x, and 10
x
For 0.5 × 2
x, b = 2
For e
x, b = e and e = 2.71828
For 10
x, b = 10
Therefore, if you graph 0.5 × 2
x, e
x, and 10
x, the resulting graphs will show exponential growth since b is bigger than 1.
However, if you graph -3 × 0.4
x, the resulting graph will show exponential decay since b is equal 0.4 and 0.4 is between 0 and 1.
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May 26, 22 06:50 AM
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