#🚀 AP SAT (Digital) Math: Nonlinear Functions - Your Night-Before Guide 🚀

Hey there! Feeling the pre-exam jitters? No sweat! This guide is your express ticket to mastering nonlinear functions. We'll break down everything you need to know, focusing on what's most likely to show up on the test. Let's make sure you're feeling confident and ready to crush it! 💪

#📈 Exponential Functions

# Basic Structure and Properties

• General Form: f(x) = a ⋅ bˣ
  • a = initial value
  • b = base (growth factor)
• Key Feature: The graph never crosses the x-axis. 🚫
• Y-intercept: Always equals the initial value a. 📍
• Rate of Change: Constant percent change over equal intervals. 🔄

# Growth and Decay Behavior

• Growth: b > 1 (function increases rapidly). 🚀
• Decay: 0 < b < 1 (function approaches zero asymptotically). 📉

Exponential growth (left) and decay (right) graphs. Note how growth increases rapidly, while decay approaches zero.

#💰 Exponential Growth and Decay

# Modeling Real-World Scenarios

• Equations:
  • A(t) = A₀ ⋅ (1 + r)ᵗ
  • A(t) = A₀ ⋅ eʳᵗ
  • A₀ = initial amount
  • r = growth/decay rate
  • t = time
• Doubling Time/Half-Life: t = ln(2) / ln(1 + r)
• Applications: Population growth, radioactive decay, compound interest. 🌍

# Financial Applications

• Continuously Compounded Interest: A(t) = P ⋅ eʳᵗ
  • P = principal amount
  • r = annual interest rate
  • t = time in years
• Use: Financial planning, investments, loans. 🏦

#🧮 Key Features of Quadratic Graphs

# Parabola Characteristics

• General Form: f(x) = ax² + bx + c, where a ≠ 0
• Graph: A parabola (U-shaped curve). -...