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  2. AP Pre Calculus
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Exponential Functions

Olivia King

Olivia King

7 min read

Next Topic - Exponential Function Manipulation

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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: f(x)=abxf(x) = ab^xf(x)=abx
    • a is the initial value (y-intercept).
    • b is the base (a positive number not equal to 1).
    • x is the exponent.
Key Concept
  • 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. 📉

Exponential function formula displayed as y = ab^x and what defines the formula to be exponential growth or decay.

Image: Exponential function formula and growth/decay conditions.

Quick Fact
  • 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.
    • f(1)=ab1=abf(1) = ab^1 = abf(1)=ab1=ab
    • f(2)=ab2=ab∗b=ab2f(2) = ab^2 = ab * b = ab^2f(2)=ab2=ab∗b=ab2
    • f(n)=abn=a∗b∗b∗b...(ntimes)f(n) = ab^n = a * b * b * b ... (n times)f(n)=abn=a∗b∗b∗b...(ntimes)

Graph displaying the function y=2^x.

Image: Graph of y = 2^x, illustrating exponential growth.

#Real-World Applications

  • Exponential functions model situations like: ...
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Previous Topic - Change in Linear and Exponential FunctionsNext Topic - Exponential Function Manipulation

Question 1 of 12

In the exponential function f(x)=5(2)xf(x) = 5(2)^xf(x)=5(2)x, what is the initial value?

2

5

x

10