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Exponential Decay

Rational functions with x as an exponent in the denominator

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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?


In the last concept, we only addressed functions where \begin{align*}|b|>1\end{align*}. So, what happens when \begin{align*}b\end{align*} is less than 1? Let’s analyze \begin{align*}y=\left(\frac{1}{2}\right)^x\end{align*}.

Example A

Graph \begin{align*}y=\left(\frac{1}{2}\right)^x\end{align*} and compare it to \begin{align*}y=2^x\end{align*}.

Solution: Let’s make a table of both functions and then graph.

\begin{align*}x\end{align*} \begin{align*}\left(\frac{1}{2}\right)^x\end{align*} \begin{align*}2^x\end{align*}
3 \begin{align*}\left(\frac{1}{2}\right)^3 = \frac{1}{8}\end{align*} \begin{align*}2^3=8\end{align*}
2 \begin{align*}\left(\frac{1}{2}\right)^2 = \frac{1}{4}\end{align*} \begin{align*}2^2=4\end{align*}
1 \begin{align*}\left(\frac{1}{2}\right)^1 = \frac{1}{2}\end{align*} \begin{align*}2^1=2\end{align*}
0 \begin{align*}\left(\frac{1}{2}\right)^0 = 1\end{align*} \begin{align*}2^0=1\end{align*}
-1 \begin{align*}\left(\frac{1}{2}\right)^{-1} = 2\end{align*} \begin{align*}2^{-1}=\frac{1}{2}\end{align*}
-2 \begin{align*}\left(\frac{1}{2}\right)^{-2} = 4\end{align*} \begin{align*}2^{-2}=\frac{1}{4}\end{align*}
-3 \begin{align*}\left(\frac{1}{2}\right)^3 = 8\end{align*} \begin{align*}2^{-3}=\frac{1}{8}\end{align*}

Notice that \begin{align*}y=\left(\frac{1}{2}\right)^x\end{align*} is a reflection over the \begin{align*}y\end{align*}-axis of \begin{align*}y=2^x\end{align*}. Therefore, instead of exponential growth, the function \begin{align*}y=\left(\frac{1}{2}\right)^x\end{align*} decreases exponentially, or exponentially decays. Anytime \begin{align*}b\end{align*} is a fraction or decimal between zero and one, the exponential function will decay. And, just like an exponential growth function, and exponential decay function has the form \begin{align*}y=ab^x\end{align*} and \begin{align*}a>0\end{align*}. However, to be a decay function, \begin{align*}0 < b < 1\end{align*}. The exponential decay function also has an asymptote at \begin{align*}y=0\end{align*}.

Example B

Determine which of the following functions are exponential decay functions, exponential growth functions, or neither. Briefly explain your answer.

a) \begin{align*}y=4(1.3)^x\end{align*}

b) \begin{align*}f(x)=3 \left(\frac{6}{5}\right)^x\end{align*}

c) \begin{align*}y = \left(\frac{3}{10}\right)^x\end{align*}

d) \begin{align*}g(x)= -2(0.65)^x\end{align*}

Solution: a) and b) are exponential growth functions because \begin{align*}b>1\end{align*}. c) is an exponential decay function because \begin{align*}b\end{align*} is between zero and one. d) is neither growth or decay because \begin{align*}a\end{align*} is negative.

Example C

Graph \begin{align*}g(x)=-2 \left(\frac{2}{3}\right)^{x-1}+1\end{align*}. Find the \begin{align*}y\end{align*}-intercept, asymptote, domain, and range.

Solution: To graph this function, you can either plug it into your calculator (entered Y= -2(2/3)^(X-1)+1) or graph \begin{align*}y=-2 \left(\frac{2}{3}\right)^x\end{align*} and shift it to the right one unit and up one unit. We will use the second method; final answer is the blue function below.

The \begin{align*}y\end{align*}-intercept is:

\begin{align*}y=-2 \left(\frac{2}{3}\right)^{0-1}+1=-2 \cdot \frac{3}{2}+1=-3+1=-2\end{align*}

The horizontal asymptote is \begin{align*}y=1\end{align*}, the domain is all real numbers and the range is \begin{align*}y < 1\end{align*}.

Intro Problem Revisit This is an example of exponential decay, so we can once again use the exponential form \begin{align*}f(x)=a \cdot b^{x-h}+k\end{align*}, but we have to be careful. In this case, a = 10,000, the starting population, x-h = 5 the number of years, and k = 0, but b is a bit trickier. If the population is decreasing by 5%, each year the population is (1 - 5%) or (1 - 0.05) = 0.95 what it was the previous year. This is our b.

\begin{align*}P = 10,000 \cdot 0.95^5\\ = 10,000 \cdot 0.7738 = 7738\end{align*}

Therefore, the city's population in 2017 is 7,738.

Guided Practice

Graph the following exponential functions. Find the \begin{align*}y\end{align*}-intercept, asymptote, domain, and range.

1. \begin{align*}f(x)=4 \left(\frac{1}{3}\right)^x\end{align*}

2. \begin{align*}y=-2 \left(\frac{2}{3}\right)^{x+3}\end{align*}

3. \begin{align*}g(x)= \left(\frac{3}{5}\right)^x-6\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*}


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*}

Explore More

Determine which of the following functions are exponential growth, exponential decay or neither.

  1. \begin{align*}y= -\left(\frac{2}{3}\right)^x\end{align*}
  2. \begin{align*}y= \left(\frac{4}{3}\right)^x\end{align*}
  3. \begin{align*}y=5^x\end{align*}
  4. \begin{align*}y= \left(\frac{1}{4}\right)^x\end{align*}
  5. \begin{align*}y= 1.6^x\end{align*}
  6. \begin{align*}y= -\left(\frac{6}{5}\right)^x\end{align*}
  7. \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.

  1. \begin{align*}y= \left(\frac{1}{2}\right)^x\end{align*}
  2. \begin{align*}y=(0.8)^{x+2}\end{align*}
  3. \begin{align*}y=4 \left(\frac{2}{3}\right)^{x-1}-5\end{align*}
  4. \begin{align*}y= -\left(\frac{5}{7}\right)^x +3\end{align*}
  5. \begin{align*}y= \left(\frac{8}{9}\right)^{x+5} -2\end{align*}
  6. \begin{align*}y=(0.75)^{x-2}+4\end{align*}
  7. Is the domain of an exponential function always all real numbers? Why or why not?
  8. 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.
    1. 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.
    2. What will the price be after the item has been on display for 5 weeks?
    3. After how many weeks will the item be half its original price?

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 8.2. 


Exponential Decay Function

Exponential Decay Function

An exponential decay function is a specific type of exponential function that has the form y=ab^x, where a>0 and 0<b<1.
Exponential Function

Exponential Function

An exponential function is a function whose variable is in the exponent. The general form is y=a \cdot b^{x-h}+k.


A model is a mathematical expression or function used to describe a physical item or situation.

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