# Exponential function

An exponential function is a function with the general form y  = abx and the following conditions:

• x is a real number
• 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 × 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

1. y = 0.5 × 2x

2. y = -3 × 0.4x

3. y = ex

4. y = 10x

Can you tell what b equals to for the following graphs?

0.5 × 2x, ex, and 10x

For 0.5 × 2x, b = 2
For ex, b = e and e = 2.71828
For 10x, b = 10

Therefore, if you graph 0.5 × 2x, ex, 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.

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