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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 |b|>1. So, what happens when b is less than 1? Let’s analyze y=(12)x.

Example A

Graph y=(12)x and compare it to y=2x.

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

x (12)x 2x
3 (12)3=18 23=8
2 (12)2=14 22=4
1 (12)1=12 21=2
0 (12)0=1 20=1
-1 (12)1=2 21=12
-2 (12)2=4 22=14
-3 (12)3=8 23=18

Notice that y=(12)x is a reflection over the y-axis of y=2x. Therefore, instead of exponential growth, the function y=(12)x decreases exponentially, or exponentially decays. Anytime b 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 y=abx and a>0. However, to be a decay function, 0<b<1. The exponential decay function also has an asymptote at y=0.

Example B

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

a) y=4(1.3)x

b) f(x)=3(65)x

c) y=(310)x

d) g(x)=2(0.65)x

Solution: a) and b) are exponential growth functions because b>1. c) is an exponential decay function because b is between zero and one. d) is neither growth or decay because a is negative.

Example C

Graph g(x)=2(23)x1+1. Find the y-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 y=2(23)x 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 y-intercept is:


The horizontal asymptote is y=1, the domain is all real numbers and the range is y<1.

Intro Problem Revisit This is an example of exponential decay, so we can once again use the exponential form f(x)=abxh+k, 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.


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

Guided Practice

Graph the following exponential functions. Find the y-intercept, asymptote, domain, and range.

1. f(x)=4(13)x

2. y=2(23)x+3

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?


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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