Exponential decay equations and graphs

Formula and graph for exponential Decay

Exponential Decay functions model many real world scenarios. Probably the most well known example of exponential decay in the real world involves the half-life of radioactive substances


of Equation & Graph of Exponential Decay Function

  • Property #1) rate of decay starts great and decreases ( Read on, to learn more about this property, which is the primary focus of this web page)
  • Property #2) The domain is
    all real numbers . $$ \{x: x \in \mathbb{R}\} $$
  • Property #3) The range is
    all positive real numbers greater than 0 .$$ \{ y: y > 0 \} $$

    Read more here

  • Property #4) It is a one-to-one function
  • Property #5) The graph is asymptotic with
  • Property #6) The value of 'b' in the general equation must be
    less than 1
  • Property #7) The inverse of exponential decay is
    a logarithmic functions.

    Of note: Exponential Growth is not the inverse of exponential decay.

General Formula for Equation

of Exponential Decay

The Graph

exponential decay graph and what it is

Property #1) Rate of decay of exponential decay decreases , becoming less and less as the graph approaches the x-axis. (but never actually touches the x-axis) !

As the graph on the left shows, at first, exponential really decreases greatly, but the rate of decay of becomes less and less until the becomes almost nothing.

Table of Values

exponential decay table of values

The table of values for the exponential decay equation $$y = \big( \frac 1 9 \big) ^x $$ demonstrates the same property as the graph. The rate of decay is great at first

Let's look at some values between $$ x=-8$$ and $$ x = 0$$.

At first, between x = -7 and x = -8, the value of the function changes by more than 38 MILLION! --the rate of decay is HUGE!

But the rate of decay becomes less and less. Between x = -7 and x = -6, the function decays by a little over 4 million.

But the rate of decay becomes less and less.

Role of 'a'

what is the role of A in the equation

For $$y = \color{red}{a} \cdot b^x $$ , a determines the y-intercept.

Typically, in real world scenarios like half life this y-intercept is the 'starting amount' of the substance or thing that is decaying.

exponential growth the role of A

To try to understand how something could become less and less but never actually hit a value. Think about Zeno's a common twist on the famous Paradox, animated below:
zenos paradox
In this animation, the athlete can jump forever, but each time that she jumps she can only go $$\frac 1 2 $$ as far as she went the prior time.

Problem 1

What is the graph of the following exponential decay function?

Exponential decay of y equals 3x
Problem 2

What is the graph of the exponential decay function below?

Problem 3

Can you graph the exponential decay function whose equation is given below?

Problem 4

Can you graph the exponential decay function whose equation is given below?

Graphs of Exponential Decay Functions and Equations

Below you can compare the graphs for three different exponential decay equations

Graphs of exponential decay equations
back to Exponential Growth next to Half Life