Exponential Functions have a variable in the exponent

Exponential Functions

f(x) = a (b)^x

^{a = initial value or starting amount}

^{b= (common ratio) growth or decay factor}

Exponential Growth vs. Decay

If b > 1, then Growth

If 0 < b < 1, then Decay

Both Graphs have a

Horizontal Asymptote

(graph never touches

the X-axis)

b^x cannot equal 0

Graphing Exponential Equations

Asymptote

In mathematics, an asymptote is a horizontal, vertical, or slanted line that a graph approaches but never touches.

The asymptote of an exponential function is a horizontal line.

The equation of a horizontal asymptote can be written as

y = constant.

Aysmptote of the Exponential parent function

The asymptote of the exponential parent function is the horizontal line y = 0.

Two types of exponential situations

- Decay

-Growth

Growth Formula

Decay Formula

A 97 mg sample of a radioactive substance is decreasing at a rate of 5% per hour. How much will be left after 6 hours?

• a = Plug in 97 milligrams for the initial amount

• r = 0.05 for the rate of decrease

• t = 6 hours for the time elapsed.

A colony of 300 bacteria doubles in size every 43 minutes. What will the population be 86 minutes from now?

• First, find out how many times the population will double. Divide the number of minutes by how long it takes for the population to double.  86 ÷ 43 = 2

• Now figure out what the population will be after it doubles 2 times. Multiply the population by 2 a total of 2 times.

• 300 × 2 × 2 = 1,200

• After 86 minutes, the population will be 1,200 bacteria