2.5 Exponential function context and data modeling
Define an exponential function.
Explain how an exponential function can model growth patterns.
How can you construct an exponential function model from two input-output pairs?
What transformations can be applied to construct exponential function models based on characteristics of a contextual scenario or data set?
How can technology be used to construct exponential function models for a data set?
The population of a city is growing exponentially. In 2010, the population was 5000, and in 2020, the population was 10\,000. Write an exponential function to model the growth of the city's population.
Given a table of values for an exponential function, describe the procedure to find the base and the initial value.
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 3 | 6 | 12 | 24 |
The function f(x) = 3(2)^x models the amount of money in a bank account after x years. Write an equivalent form of the function that reveals how much the amount increases every 2 years.
The number of bacteria in a culture doubles every hour. If there are 50 bacteria at the start, write an exponential function to model the number of bacteria after x hours.
A car depreciates in value by 15\% each year. If the car is worth \$20\,000at the start, write an exponential function to model the car's value after x years.
In a lab experiment, a scientist observes that a colony of bacteria grows by a factor of 5 every 3 hours. If the initial number of bacteria is 200, write an exponential function to model this scenario.
A certain type of bacteria doubles in number every 20 minutes. At 12:00, there were 200 bacteria present. Use an exponential function to predict how many bacteria will be present at 3:00.
The price of a certain stock follows the exponential function f(t) = 1000(1.05)^t, where t is the time in years. What will be the price of the stock after 5 years?
The number of subscribers to a streaming service is modeled by the function \\f(x) = 1000(1.2)^x, where x is the number of months since the service was launched. If the service wishes to have 5000 subscribers, after how many months will this goal be achieved according to the model?
A scientist is studying a population of microorganisms. She observes that the population doubles every 2 hours initially, but after some time, the growth rate decreases. Can this situation be modeled accurately by an exponential function? Explain your reasoning.
You are given the exponential function f(x) = 3(2)^x. Write an equivalent form of this function that reveals how much the function value increases every 3 units of x. What does this new form tell you about the behavior of the function?