#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: $f(x) = ab^x$
  • a is the initial value (y-intercept).
  • b is the base (a positive number not equal to 1).
  • x is the exponent.

#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.

#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.
  • $f(1) = ab^1 = ab$
  • $f(2) = ab^2 = ab * b = ab^2$
  • $f(n) = ab^n = a * b * b * b ... (n times)$

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

#Real-World Applications

• Exponential functions model situations like:
  • Compound interest
  • Population growth
  • Radioactive decay
• Example: Population growth at 5% per year: $P * 1.05^n$ (where P is the initial population and n is the number of years). 🌆

#↕️ Trends: Increasing/Decreasing & Concavity

#Increasing vs. Decreasing

• Exponential functions are always either increasing or decreasing.
• Increasing: When b > 1 (growth).
• Decreasing: When 0 < b < 1 (decay).

#Concavity

• Concave Up: When b > 1 (growth).
• Concave Down: When 0 < b < 1 (decay).

Image: Demonstrating reflections of exponential functions.

#Extrema

• Exponential functions do not have extrema (max or min values) on an open interval because they are always increasing or always decreasing. They only have extrema on a closed interval.

Image: Graph illustrating concavity, inflection points, and extrema.

#Additive Transformations

#Vertical Shifts

• An additive transformation is $g(x) = f(x) + k$.
• It shifts the graph of f(x) vertically by k units. ➕
  • k > 0: Shifts the graph up.
  • k < 0: Shifts the graph down.

Image: Vertical shifts of exponential functions.

#🛑 Limits

#Behavior as x Approaches Infinity

• b > 1 (Growth): As x → ∞, f(x) → ∞. The function increases without bound. 🪐
• 0 < b < 1 (Decay): As x → ∞, f(x) → 0. The function approaches zero. 0️⃣

#Behavior as x Approaches Negative Infinity

• b > 1 (Growth): As x → -∞, f(x) → 0. The function approaches zero.
• 0 < b < 1 (Decay): As x → -∞, f(x) → ∞. The function increases without bound. 🔻

Image: Summary of limits for exponential functions.

#Final Exam Focus

#High-Priority Topics

• Growth vs. Decay: Understanding how the base b affects the function's behavior.
• Transformations: Knowing how vertical shifts affect the graph.
• Limits: Understanding the behavior of the function as x approaches infinity and negative infinity.
• Real-world applications: Applying exponential functions to model growth and decay scenarios.

#Common Question Types

• Multiple Choice: Identifying growth/decay, transformations, and limits.
• Free Response: Modeling real-world situations, analyzing graphs, and applying transformations.

#Practice Questions

Practice Question

#Multiple Choice Questions

1. Which of the following functions represents exponential decay? (A) $f(x) = 3(2)^x$ (B) $f(x) = 0.5(1.2)^x$ (C) $f(x) = 2(0.8)^x$ (D) $f(x) = 1.5(3)^x$

2. The graph of $f(x) = 2^x$ is shifted vertically upwards by 3 units. Which of the following represents the new function? (A) $g(x) = 2^{x+3}$ (B) $g(x) = 2^x + 3$ (C) $g(x) = 3(2^x)$ (D) $g(x) = 2^x - 3$

3. What is the limit of the function $f(x) = 5(0.7)^x$ as x approaches infinity? (A) 0 (B) 5 (C) Infinity (D) -Infinity

#Free Response Question

A population of bacteria starts with 100 cells and doubles every hour.

(a) Write an exponential function P(t) to model the population of bacteria after t hours.

(b) How many bacteria will there be after 4 hours?

(c) How many hours will it take for the population to reach 6400 cells?

(d) Sketch a graph of the function P(t) for the first 5 hours, labeling the key points.

Scoring Breakdown

(a) 2 points: - 1 point for the correct base (2). - 1 point for the correct initial value (100). - Correct function: $P(t) = 100 * 2^t$ (b) 2 points: - 1 point for correct substitution. - 1 point for the correct answer. - $P(4) = 100 * 2^4 = 100 * 16 = 1600$ cells. (c) 3 points: - 1 point for setting up the equation correctly. - 1 point for using logarithms or equivalent method. - 1 point for the correct answer. - $6400 = 100 * 2^t$ gives $2^t = 64$ and $t = 6$ hours. (d) 3 points: - 1 point for correct shape (exponential growth). - 1 point for labeling the y-intercept (0,100). - 1 point for labeling at least one other point correctly (e.g. (1,200),(2,400), etc.).

You've got this! Remember to take deep breaths, stay focused, and trust in your preparation. You're ready to rock this exam! 🥳