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 . So, what happens when is less than 1? Let’s analyze .
Example A
Graph and compare it to .
Solution: Let’s make a table of both functions and then graph.
3  
2  
1  
0  
1  
2  
3 
Notice that is a reflection over the axis of . Therefore, instead of exponential growth, the function decreases exponentially, or exponentially decays . Anytime 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 and . However, to be a decay function, . The exponential decay function also has an asymptote at .
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 . c) is an exponential decay function because is between zero and one. d) is neither growth or decay because is negative.
Example C
Graph . Find the intercept, asymptote, domain, and range.
Solution: To graph this function, you can either plug it into your calculator (entered Y= 2(2/3)^(X1)+1) or graph 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 intercept is:
The horizontal asymptote is , the domain is all real numbers and the range is .
Intro Problem Revisit This is an example of exponential decay, so we can once again use the exponential form , but we have to be careful. In this case, a = 10,000, the starting population, xh = 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 intercept, asymptote, domain, and range.
1.
2.
3.
4. Determine if the following functions are exponential growth, exponential decay, or neither.
a)
b)
c)
d)
Answers
1. intercept: , asymptote: , domain: all reals, range:
2. intercept: , asymptote: , domain: all reals, range:
3. intercept: , asymptote: , domain: all reals, range:
4. a) exponential growth
b) exponential decay; recall that a negative exponent flips whatever is in the base. is the same as , which looks like our definition of a decay function.
c) exponential decay
d) neither;
Explore More
Determine which of the following functions are exponential growth, exponential decay or neither.
Graph the following exponential functions. Find the intercept, the equation of the asymptote and the domain and range for each function.
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