Exponential Functions #
Introduction #
An exponential function is any function in the form \( a(b^x) \). The variables are defined as follows: \( a \) is the starting amount, \( b \) is the common ratio, and \( x \) can be the amount of intervals passed, such as years or steps. Since any number (with a few undefined exceptions) raised to the power of 0 is 1, \( a \) is also the y-intercept of the function.
Exponential Growth #
Suppose a new car goes up in value by 15% every year. The car was bought for $10,000. For a growth rate, we add the percentage of growth to 1. Since the function is increasing by 15% every year, we add 0.15 to 1. We can construct the function:
\( f(x)=10,000(1.15^x) \),
where \( x \) is the time passed in years.
Plugging in 1 for \( x \), we can conclude that in 1 year, the car will be worth $11,500. Note that we could have also gotten our answer by repeatedly multiplying by 1.15, revealing the reason for exponentiation in this problem.
Exponential Decay #
Now suppose we buy another car for $15,000. It will go under heavy usage. The car will now depreciate. In this example, the value of the car will go down by 20% every year. When the value or amount of something depreciates/decays, we subtract the rate from 1. We will have to subtract 0.20 from 1. This information leads to the function:
\( f(x)=15,000(0.80^x) \)
Substituting 1 for \( x \), we find that, in 1 year, the car will be worth $12,000.
Exercises #
- Exercise 1: What is the function rule in the following sequence: 1000, 100, 10, 1, 0.1, 0.01, 0.001…?
- Exercise 2: The amount of bacteria in a certain place will triple every minute. The starting amount of bacteria was 600. How many bacteria will there be in 10 minutes?
- Exercise 3: You have $100 in your savings account. Your bank gives you two options on interest: they can either give you $10,000 every day for a month, or alternatively, they can double your savings every day for a week. Which option would yield more, and by how much?
- Exercise 4: A function in the form \( f(x)=a(b^x) \) satisfies \( f(1)=2 \), \( f(3)=8 \), and \( f(5)=32 \). What is the value of \( a+b \)?
- Exercise 5: Which of these will grow the fastest: \( f(x)=3x \), \( f(x)=x^3 \), or \( f(x)=3^x \)?
- Exercise 6: At a growth rate of 1% every year, how long will it take for a city’s population to double?
- Exercise 7: What is the product of the solutions to \( 2^x=2x \)?
- Exercise 8: A new computer is worth $1,000 but drops in value by 10% every year after its initial release. Another computer sold 1 year later is worth $950. Which computer is the less expensive option: the first or second?
- Exercise 9: What is the y-intercept of the function \( f(x)=6(36^x) \)?
- Exercise 10: To the nearest whole number, what is the value of \( g(9,999,999) \) when \( g(x)=9(0.999^x) \)?