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)
Why a cannot be equal to 0?
Notice the use of the independent variable (x) as an exponent. This is important in order to have an exponential function.
If a = 0, then y = 0 × bx
= 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 × 0x
= 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 × 1x
= 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
y = 0.5 × 2x
y = -3 × 0.4x
y = ex
y = 10x
Can you tell what b equals to for the following graphs?
0.5 × 2x
, and 10x
For 0.5 × 2x
, b = 2
, b = e and e = 2.71828
, b = 10
Therefore, if you graph 0.5 × 2x
, and 10x
, the resulting graphs will show exponential growth since b is bigger than 1.
However, if you graph -3 × 0.4x
, the resulting graph will show exponential decay since b is equal 0.4 and 0.4 is between 0 and 1.
Jul 30, 21 06:15 AM
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