8.2: Exponential Decay Function
The population of a city was 10,000 in 2012 and is declining at a rate of 5% each year. If this decay rate continues, what will the city's population be in 2017?
Guidance
In the last concept, we only addressed functions where
Example A
Graph
Solution: Let’s make a table of both functions and then graph.




3 


2 


1 


0 


1 


2 


3 


Notice that
Example B
Determine which of the following functions are exponential decay functions, exponential growth functions, or neither. Briefly explain your answer.
a)
b)
c)
d)
Solution: a) and b) are exponential growth functions because
Example C
Graph
Solution: To graph this function, you can either plug it into your calculator (entered Y= 2(2/3)^(X1)+1) or graph
The
The horizontal asymptote is
Intro Problem Revisit This is an example of exponential decay, so we can once again use the exponential form
Therefore, the city's population in 2017 is 7,738.
Guided Practice
Graph the following exponential functions. Find the
1.
2.
3. \begin{align*}g(x)= \left(\frac{3}{5}\right)^x6\end{align*}
4. Determine if the following functions are exponential growth, exponential decay, or neither.
a) \begin{align*}y=2.3^x\end{align*}
b) \begin{align*}y=2 \left(\frac{4}{3}\right)^{x}\end{align*}
c) \begin{align*}y=3\cdot 0.9^x\end{align*}
d) \begin{align*}y=\frac{1}{2} \left(\frac{4}{5}\right)^{x}\end{align*}
Answers
1. \begin{align*}y\end{align*}intercept: \begin{align*}(4, 0)\end{align*}, asymptote: \begin{align*}y=0\end{align*}, domain: all reals, range: \begin{align*}y < 0\end{align*}
2. \begin{align*}y\end{align*}intercept: \begin{align*}\left(0, \frac{16}{27}\right)\end{align*}, asymptote: \begin{align*}y=0\end{align*}, domain: all reals, range: \begin{align*}y<0\end{align*}
3. \begin{align*}y\end{align*}intercept: \begin{align*}(5, 0)\end{align*}, asymptote: \begin{align*}y=6\end{align*}, domain: all reals, range: \begin{align*}y>6\end{align*}
4. a) exponential growth
b) exponential decay; recall that a negative exponent flips whatever is in the base. \begin{align*}y=2 \left(\frac{4}{3}\right)^{x}\end{align*} is the same as \begin{align*}y=2 \left(\frac{3}{4} \right)^{x}\end{align*}, which looks like our definition of a decay function.
c) exponential decay
d) neither; \begin{align*}a < 0\end{align*}
Vocabulary
 Exponential Decay Function
 An exponential function that has the form \begin{align*}y=ab^x\end{align*} where \begin{align*}a>0\end{align*} and \begin{align*}0<b<1\end{align*}.
Practice
Determine which of the following functions are exponential growth, exponential decay or neither.
 \begin{align*}y= \left(\frac{2}{3}\right)^x\end{align*}
 \begin{align*}y= \left(\frac{4}{3}\right)^x\end{align*}
 \begin{align*}y=5^x\end{align*}
 \begin{align*}y= \left(\frac{1}{4}\right)^x\end{align*}
 \begin{align*}y= 1.6^x\end{align*}
 \begin{align*}y= \left(\frac{6}{5}\right)^x\end{align*}
 \begin{align*}y= 0.99^x\end{align*}
Graph the following exponential functions. Find the \begin{align*}y\end{align*}intercept, the equation of the asymptote and the domain and range for each function.
 \begin{align*}y= \left(\frac{1}{2}\right)^x\end{align*}
 \begin{align*}y=(0.8)^{x+2}\end{align*}
 \begin{align*}y=4 \left(\frac{2}{3}\right)^{x1}5\end{align*}
 \begin{align*}y= \left(\frac{5}{7}\right)^x +3\end{align*}
 \begin{align*}y= \left(\frac{8}{9}\right)^{x+5} 2\end{align*}
 \begin{align*}y=(0.75)^{x2}+4\end{align*}
 Is the domain of an exponential function always all real numbers? Why or why not?
 A discount retailer advertises that items will be marked down at a rate of 10% per week until sold. The initial price of one item is $50.
 Write an exponential decay function to model the price of the item \begin{align*}x\end{align*} weeks after it is first put on the rack.
 What will the price be after the item has been on display for 5 weeks?
 After how many weeks will the item be half its original price?
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Exponential Decay Function
An exponential decay function is a specific type of exponential function that has the form , where and .Exponential Function
An exponential function is a function whose variable is in the exponent. The general form is .Model
A model is a mathematical expression or function used to describe a physical item or situation.Image Attributions
Here you'll learn how to graph and analyze an exponential decay function.