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
Example
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 . $$ \{x: x \in \mathbb{R}\} $$
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Property #3) The range is
all positive real numbers greater than 0 .$$ \{ y: y > 0 \} $$
- Property #4) It is a one-to-one function
- Property #5) The graph is asymptotic with
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Property #6) The value of 'b' in the general equation must be
less than 1
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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
![Picture of Exponential Decay function example](images/exponential-decay-formula-graph.png)
In this example, 'a' stands for the initial amount, and 'b' is any real number that is less than 1.
The graph's asymptote at x -axis .
When B is greater than 1 , you are not dealing with exponential decay but rather exponential growth (Exponential Growth Lesson ).
What about when b is exactly equal to 1 ?
In this case, you are not dealing with an exponential equation, but rather a linear equation.(Link)
$ y = ab^x \\ y = a(1)^x \\ y = a $As you can see from the work above, and the graph, when b is 1, you end up with the equation of a horizontal line. ( Link )
The Graph
![exponential decay graph and what it is](images/how-exponential-decays-graph-changes.png)
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](images/table-of-values-exponential-decay.png)
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.
Role of 'a'
![what is the role of A in the equation](images/exponential-decay-the-role-of-A_RED.png)
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](images/exponential-decay-role-of-A-example-graphs.png)
![Exponential decay of y equals 3x](images/diagram_exponential_decay.gif)
![](images/graph_exponential_decay.gif)
![Click For Answer](images/graph_problem_3A.gif)
![Click for answer](images/graph_problem_4B.gif)
Graphs of Exponential Decay Functions and Equations
Below you can compare the graphs for three different exponential decay equations
![Graphs of exponential decay equations](images/graph_equation_exponential_decay.gif)