#🚀 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).
• Opens Upward: a > 0 (minimum point). ⬆️
• Opens Downward: a < 0 (maximum point). ⬇️
• Vertex: The minimum/maximum point (h, k)
  • h = -b / 2a
  • k = f(h)

# Important Points and Lines

• Axis of Symmetry: x = -b / 2a (vertical line through the vertex). ↔️
• Y-intercept: (0, c) 📍
• X-intercepts (Roots/Zeros): Where the parabola crosses the x-axis (0, 1, or 2 intercepts). ➗

A parabola showing the vertex, axis of symmetry, and x-intercepts.

#🧩 Solving Quadratic Equations

# Solution Methods

• Methods: Factoring, completing the square, quadratic formula, graphing.
• Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a
• Factoring: Best for integer solutions.
• Completing the Square: Useful for finding vertex form. 🔄

# Discriminant Analysis

• Discriminant (Δ): Δ = b² - 4ac
  • Δ > 0: Two distinct real solutions. 👯
  • Δ = 0: One repeated real solution. ☝️
  • Δ < 0: Two complex solutions (no x-intercepts). 👻

#🎭 Properties of Polynomial Functions

# General Characteristics

• Form: f(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ
  • a₀, a₁, ..., aₙ are constants
  • n is a non-negative integer
• Degree: Highest power of the variable (n).
• Leading Coefficient: Coefficient of the term with the highest degree (aₙ).
• Continuity: Continuous with no asymptotes. 〰️
• Graphs: Can have multiple turning points. 🎢

# Behavior and Applications

• End Behavior: Determined by degree and leading coefficient. ⬆️⬇️
• Modeling: Projectile motion, economic trends, etc. 🎯
• Approximation: Higher-degree polynomials can approximate complex curves. 📈

#🔚 End Behavior and Zeros of Polynomials

# End Behavior Patterns

• Even Degree:
  • Positive leading coefficient: rises on both ends. ⬆️⬆️
  • Negative leading coefficient: falls on both ends. ⬇️⬇️
• Odd Degree:
  • Positive leading coefficient: falls left, rises right. ⬇️⬆️
  • Negative leading coefficient: rises left, falls right. ⬆️⬇️

# Zeros and Their Properties

• Zeros (Roots): Where f(x) = 0.
• Multiplicity:
  • Odd multiplicity: graph crosses x-axis. ✖️
  • Even multiplicity: graph touches x-axis. 〰️
• Number of Zeros: ≤ degree of polynomial.
• Complex Zeros: Always occur in conjugate pairs. 👯

#➗ Rational Functions and Graphs

# Function Characteristics

• Form: f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0
• Domain: Excludes x-values where the denominator is zero. 🚫
• Graph Features: Vertical asymptotes, horizontal asymptotes, and holes. 🕳️

# Asymptote Analysis

• Vertical Asymptotes: Where the denominator equals zero. ↕️
• Horizontal Asymptotes:
  • Degree of numerator < degree of denominator: y = 0
  • Degree of numerator = degree of denominator: y = ratio of leading coefficients
• Oblique (Slant) Asymptotes: Degree of numerator is one more than degree of denominator. ↗️

A rational function graph showing vertical and horizontal asymptotes.

#🧮 Solving Rational Equations

# Solution Process

• Identify: x-values making denominators zero (not solutions). 🚫
• Multiply: By the least common denominator (LCD) to clear fractions.
• Solve: Resulting polynomial equation.
• Check: Potential solutions in the original equation.
• Discard: Solutions making denominators zero. 🗑️

# Solution Interpretation

• Context: Consider the problem's context. 🧐
• Domain: Ensure solutions are within the original function's domain.
• Reasonableness: Analyze if solutions make sense in the real world. 🤔
• Graph: Visualize solutions on the graph. 📈

#🎯 Final Exam Focus

• High Priority: Exponential functions, quadratic equations, and rational functions. 💯
• Common Question Types: Graph interpretation, solving equations, and modeling real-world scenarios.
• Time Management: Don't spend too long on one question. Move on and come back if time allows. ⏱️
• Common Pitfalls: Forgetting to check for extraneous solutions, misinterpreting graphs, and incorrect application of formulas. ❌
• Strategies: Practice, practice, practice! Review your notes, and do practice problems. 📝

#🧪 Practice Questions

Practice Question

#Multiple Choice Questions

1. The function f(x) = 2(3)^x represents: a) Exponential decay b) Linear growth c) Exponential growth d) Quadratic growth

2. What is the vertex of the parabola represented by f(x) = x² - 4x + 7? a) (2, 3) b) (-2, 19) c) (2, 11) d) (-2, -1)

3. Which of the following is a vertical asymptote of the function f(x) = (x+1)/(x-2)? a) y = 1 b) x = -1 c) x = 2 d) y = 0

#Free Response Question

A population of bacteria grows exponentially. Initially, there are 500 bacteria. After 2 hours, the population has grown to 1500 bacteria.

(a) Write an equation to model the population of bacteria, P(t), after t hours. (b) How many bacteria will there be after 5 hours? (c) How long will it take for the population to reach 10,000 bacteria?

Scoring Breakdown:

(a) Equation (3 points): - 1 point for using the exponential growth form P(t) = P₀ * b^t - 1 point for correctly identifying P₀ = 500 - 1 point for calculating the growth factor b = 3^(1/2) - Correct equation: P(t) = 500 * (3^(t/2)) or P(t) = 500 * (√3)^t

(b) Population after 5 hours (2 points): - 1 point for correct substitution of t = 5 - 1 point for correct calculation: P(5) ≈ 8660 bacteria

(c) Time to reach 10,000 (3 points): - 1 point for setting up the equation 10000 = 500 * (3^(t/2)) - 1 point for using logarithms correctly to solve for t - 1 point for correct calculation: t ≈ 6.3 hours

You've got this! Stay calm, trust your preparation, and go ace that exam! 🎉