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Exponential Decay

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Exponential Decay Function
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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 |b|>1 . So, what happens when b is less than 1? Let’s analyze y=\left(\frac{1}{2}\right)^x .

Example A

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

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

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

Notice that y=\left(\frac{1}{2}\right)^x is a reflection over the y -axis of y=2^x . Therefore, instead of exponential growth, the function y=\left(\frac{1}{2}\right)^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=ab^x 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 \left(\frac{6}{5}\right)^x

c) y = \left(\frac{3}{10}\right)^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 \left(\frac{2}{3}\right)^{x-1}+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 \left(\frac{2}{3}\right)^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:

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

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)=a \cdot b^{x-h}+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 .

P = 10,000 \cdot 0.95^5\\= 10,000 \cdot 0.7738 = 7738

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 \left(\frac{1}{3}\right)^x

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

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

4. Determine if the following functions are exponential growth, exponential decay, or neither.

a) y=2.3^x

b) y=2 \left(\frac{4}{3}\right)^{-x}

c) y=3\cdot 0.9^x

d) y=\frac{1}{2} \left(\frac{4}{5}\right)^{x}

Answers

1. y -intercept: (4, 0) , asymptote: y=0 , domain: all reals, range: y < 0

2. y -intercept: \left(0, -\frac{16}{27}\right) , asymptote: y=0 , domain: all reals, range: y<0

3. y -intercept: (-5, 0) , asymptote: y=-6 , domain: all reals, range: y>-6

4. a) exponential growth

b) exponential decay; recall that a negative exponent flips whatever is in the base. y=2 \left(\frac{4}{3}\right)^{-x} is the same as y=2 \left(\frac{3}{4} \right)^{x} , which looks like our definition of a decay function.

c) exponential decay

d) neither; a < 0

Vocabulary

Exponential Decay Function
An exponential function that has the form y=ab^x where a>0 and 0<b<1 .

Practice

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

  1. y= -\left(\frac{2}{3}\right)^x
  2. y= \left(\frac{4}{3}\right)^x
  3. y=5^x
  4. y= \left(\frac{1}{4}\right)^x
  5. y= 1.6^x
  6. y= -\left(\frac{6}{5}\right)^x
  7. y= 0.99^x

Graph the following exponential functions. Find the y -intercept, the equation of the asymptote and the domain and range for each function.

  1. y= \left(\frac{1}{2}\right)^x
  2. y=(0.8)^{x+2}
  3. y=4 \left(\frac{2}{3}\right)^{x-1}-5
  4. y= -\left(\frac{5}{7}\right)^x +3
  5. y= \left(\frac{8}{9}\right)^{x+5} -2
  6. y=(0.75)^{x-2}+4
  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 x 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?

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