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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?

Exponential Delay Function

Previously, we have 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*}.

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

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*}.

Let's determine which of the following functions are exponential decay functions, exponential growth functions, or neither and briefly explain our answers.

  1. \begin{align*}y=4(1.3)^x\end{align*}
  2. \begin{align*}f(x)=3 \left(\frac{6}{5}\right)^x\end{align*}
  3. \begin{align*}y = \left(\frac{3}{10}\right)^x\end{align*}
  4. \begin{align*}g(x)= -2(0.65)^x\end{align*}

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.

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

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*}.

Examples

Example 1

Earlier, you were asked to find the city's population in 2017 if the population was 10,000 in 2012 and is declining at a rate of 5% each year.

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.

For Examples 2-4, graph the exponential functions. Find the \begin{align*}y\end{align*}-intercept, asymptote, domain, and range.

Example 2

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

\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*}

Example 3

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

\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*}

Example 4

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

\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*}

For Examples 5-8, determine if the functions are exponential growth, exponential decay, or neither.

Example 5

\begin{align*}y=2.3^x\end{align*}

 exponential growth

Example 6

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

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.

Example 7

\begin{align*}y=3\cdot 0.9^x\end{align*}

exponential decay

Example 8

\begin{align*}y=\frac{1}{2} \left(\frac{4}{5}\right)^{x}\end{align*}

neither; \begin{align*}a < 0\end{align*}

Review

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 Review Problems

To see the Review answers, open this PDF file and look for section 8.2. 

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Vocabulary

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

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

Model

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

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