Exponential Functions

Olivia King
7 min read
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Study Guide Overview
This study guide covers exponential functions including their basic form (f(x) = ab^x), identifying a as the initial value and b as the base. It differentiates between exponential growth (b > 1) and decay (0 < b < 1), explores domain, and provides real-world applications. The guide also examines increasing/decreasing trends and concavity, additive transformations (vertical shifts), and limits as x approaches infinity and negative infinity. Finally, it offers practice questions and exam tips focusing on growth/decay, transformations, limits, and real-world applications.
#AP Pre-Calculus: Exponential Functions - Your Ultimate Review 🚀
Hey there! Let's get you prepped for the exam with a super-focused review of exponential functions. We'll break down everything you need to know, highlight key points, and make sure you're feeling confident. Let's do this! 💪
#Exponential Functions: The Basics
#What is an Exponential Function?
- An exponential function has the form:
- a is the initial value (y-intercept).
- b is the base (a positive number not equal to 1).
- x is the exponent.
- The variable is in the exponent, not the base. This is what makes it exponential! 💡
#Exponential Growth vs. Decay
- Growth (b > 1): As x increases, f(x) increases rapidly. The larger the base, the faster the growth. 📈
- Decay (0 < b < 1): As x increases, f(x) decreases rapidly. The smaller the base, the faster the decay. 📉
Image: Exponential function formula and growth/decay conditions.
- Remember: 'a' must be greater than 0 for the function to be defined.
#Domain
- The domain of an exponential function is all real numbers (-∞, ∞). You can plug in any number for x! 🫂
#How Exponential Functions Work
- When x is a natural number (1, 2, 3,...), it indicates how many times to multiply the base by itself.
Image: Graph of y = 2^x, illustrating exponential growth.
#Real-World Applications
- Exponential functions model situations like: ...

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