What We Review
Introduction
Exponential models are like the secret agents of mathematics, quietly operating to explain a wide array of phenomena. From the rapid spread of a virus to the growth of an investment fund, these models help us predict the future based on current trends. For students preparing for the AP® Precalculus exam, understanding what is an exponential model is crucial. The goal here is to make constructing and understanding these models as straightforward as possible.
What is an Exponential Model?
An exponential model typically refers to a function that grows (or decays) at a constant rate. The classic form of an exponential function is f(x) = ab^x .
- Initial value ( a ): This is the function’s starting point. It’s where everything begins.
- Growth factor ( b ): This denotes how much the function increases (or decreases) over each interval.
For example, suppose there is an initial population of bacteria, and it doubles every hour. Here, the initial value is the number of bacteria you have at the start, and the growth factor is 2. This example will give successive outputs that are proportional over equal-length input-value intervals.
Constructing Exponential Models
Building an exponential model involves key steps.
- Identify the Initial Value ( a ): Understand what the starting point is.
- Determine the Base ( b ): This is your growth or decay factor.
- Create the Model: Use input-output pairs to set up the model.
Example: Constructing a Model from Data
Imagine observing bacteria in a petri dish. Initially, there are 100 bacteria, and their number doubles every hour.
- Identify the Initial Value: At time t = 0 , the initial value is a = 100 .
- Determine the Growth Factor: Since the population doubles hourly, the growth factor is b = 2 .
- Create the Model: The model becomes f(t) = 100 \times 2^t .
This means, at 3 hours, the bacteria population is f(3) = 100 \times 2^3 = 800 .
Transformations in Exponential Functions
Transformations help tweak exponential functions to fit various scenarios. The most common transformations are:
- Vertical Shifts: Moves the graph up or down.
- Horizontal Shifts: Moves the graph left or right.
- Stretching and Compressing: Changes the graph’s width.
Example: Applying Transformations
Consider the function f(x) = 2^x . Suppose it’s modified to f(x) = 2^x + 3 , which shifts the graph 3 units up. Therefore, each output value is now 3 more than before.
Applying Exponential Models
Modeling exponential functions includes predicting future values or estimating growth trends. Understanding the growth factor’s power is key here.
Example: Predicting Future Values
Consider a population of a town that grows by 3% each year. If the current population is 10,000, what will it be in 5 years?
- Initial Value: a = 10,000 .
- Growth Factor: b = 1.03 (3% growth means it’s multiplied by 1.03 each year).
- Model: f(t) = 10,000 \times 1.03^t .
- Future Population in 5 Years: f(5) = 10,000 \times 1.03^5 \approx 11,593 .
The Natural Base ( e )
The natural base e (approximately 2.718) is crucial in continuous growth models. It simplifies complex growth calculations over time.
- Compared to other bases (like 2 or 10), e leads to simpler results in calculus.
Example: Using the Natural Base e
If an investment of $1,000 grows at an annual rate of 5% compounded continuously, the model is f(t) = 1000 \times e^{0.05t} . After 3 years, the value is f(3) = 1000 \times e^{0.15} \approx 1161.83 .
Quick Reference: Key Vocabulary and Definitions
| Term | Definition |
| Exponential Function | A function of the form f(x) = ab^x where a is the initial value and b is the growth factor. |
| Initial Value | The starting point of the exponential function, represented by a . |
| Growth Factor | The constant b that describes how much the output increases for each unit increase in input. |
| Natural Base e | Approximately 2.718, used commonly in continuous growth models. |
Conclusion
Understanding exponential models involves recognizing their starting values, growth factors, and transformations. These models are essential in various fields, from biology to finance. Therefore, practicing with different scenarios will build confidence in creating and applying exponential models effectively. Students are encouraged to seek additional resources or practice problems to master this concept, as it is vital for success in AP® Precalculus.
Sharpen Your Skills for AP® Precalculus
Are you preparing for the AP® Precalculus exam? We’ve got you covered! Try our review articles designed to help you confidently tackle real-world math problems. You’ll find everything you need to succeed, from quick tips to detailed strategies. Start exploring now!
Need help preparing for your AP® Precalculus exam?
Albert has hundreds of AP® Precalculus practice questions, free response, and an AP® Precalculus practice test to try out.
