#Polynomial and Nonlinear Functions: Your Ultimate Guide 🚀

Hey there! Let's get you prepped for the AP SAT (Digital) Math section with a deep dive into polynomial and nonlinear functions. Think of this as your cheat sheet for tonight – quick, clear, and designed to make everything click. We'll cover the key concepts, throw in some memory aids, and make sure you're feeling confident for tomorrow. Let's do this!

#Polynomial Functions and Their Graphs

# Defining Polynomial Functions

• Polynomial functions are in the form: $$f(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ$$ where:
  • a₀, a₁, ..., aₙ are real numbers.
  • n is a non-negative integer.
• Degree: The highest power of x (e.g., in x³ + 2x² - 5x + 1, the degree is 3).
• Leading Coefficient: The coefficient of the term with the highest degree (e.g., in 3x³ + 2x² - 5x + 1, it's 3).
• Classification:
  • Linear: Degree 1
  • Quadratic: Degree 2
  • Cubic: Degree 3
  • And so on...
• End Behavior: Determined by the degree and the sign of the leading coefficient.
  • Even degree, positive leading coefficient: Both ends go up. ⬆️⬆️
  • Odd degree, positive leading coefficient: Left end goes down, right end goes up. ⬇️⬆️
  • Negative leading coefficient: Flips the end behavior.

# Characteristics of Polynomial Graphs

• Turning Points: Maxima and minima. The number of turning points is at most one less than the degree.
• Multiplicity of Zeros (Roots):
  • Odd Multiplicity: Graph crosses the x-axis.
  • Even Multiplicity: Graph touches the x-axis but doesn't cross.
• Fundamental Theorem of Algebra: A polynomial of degree n has exactly n complex roots (counting multiplicity).

# Analyzing Polynomial Functions

• Factoring: Find the zeros (roots) and reveal the function's structure.
• Rational Root Theorem: Helps find potential rational roots.
• Synthetic Division: Divide by a linear factor to find roots and factor polynomials.
• Transformations: Shifts, reflections, stretches, and compressions help analyze and sketch graphs.

#Nonlinear Functions: Exponential and Logarithmic

# Exponential Functions

• Form: $$f(x) = b^x$$, where b is a positive number not equal to 1. - Base b: Determines growth or decay.
  • If b > 1: Exponential growth (like population growth).
  • If 0 < b < 1: Exponential decay (like radioactive decay).
• Vertical Asymptote: The y-axis (x = 0).

# Logarithmic Functions

• Inverse of Exponential Functions: Denoted as $$f(x) = log_b(x)$$.
• Graph: Reflects the corresponding exponential graph across the line y = x.
• Horizontal Asymptote: The x-axis (y = 0).
• Properties of Logarithms:
  • Product Rule: $$log_b(xy) = log_b(x) + log_b(y)$$
  • Quotient Rule: $$log_b(x/y) = log_b(x) - log_b(y)$$
  • Power Rule: $$log_b(x^n) = n * log_b(x)$$

# Other Nonlinear Functions

• Rational Functions: Quotients of polynomials (e.g., $$f(x) = \frac{x² + 3x + 2}{x - 1}$$).
• Absolute Value Functions: Involve the absolute value of an expression (e.g., $$f(x) = |x - 3|$$).
• Square Root Functions: Include the square root of a variable or expression (e.g., $$f(x) = \sqrt{x + 2}$$).

#Applications of Polynomial and Nonlinear Functions

# Solving Equations and Inequalities

• Combine algebraic techniques:
  • Factoring (e.g., $$x² - 4x + 3 = 0$$).
  • Substitution (e.g., replace y with x² + 1 in y = 2x + 3).
  • Logarithms or exponents (e.g., solve $$2^x = 8$$).

# Modeling Real-World Scenarios

• Polynomials: Optimization problems (e.g., maximizing the area of a rectangle).
• Exponentials: Growth and decay (e.g., compound interest, bacterial growth).
• Logarithms: Diminishing returns (e.g., earthquake intensity, pH scale).
• Interpret Function Parameters:
  • Growth rate in population models.
  • Decay constant in radioactive decay.
  • Carrying capacity in logistic growth models.

# Graphical Analysis and Interpretation

• Key Features:
  • Zeros (roots): Break-even points or equilibrium states.
  • Maxima/minima: Optimal values or turning points.
  • End behavior: Long-term trends or asymptotic behavior.
• Tools: Graphing calculators or software for visualization.
  • Plot multiple functions to compare behavior.
  • Zoom in for local behavior.
  • Use trace to find precise values.

#Final Exam Focus

Okay, you're almost there! Here’s what to focus on tonight:

• High-Priority Topics:
  • End behavior of polynomials.
  • Exponential growth and decay.
  • Logarithmic properties and their inverses.
  • Interpreting graphs in context.
• Common Question Types:
  • Matching graphs to equations.
  • Solving for roots and intercepts.
  • Modeling real-world scenarios.
• Last-Minute Tips:
  • Time Management: Don't get stuck on one question. Move on and come back if you have time.
  • Common Pitfalls: Double-check your work, especially when dealing with negative signs and exponents.
  • Strategies: Use your calculator to visualize functions and verify answers.

#Practice Questions

a) What is the initial population of bacteria?
b) What is the population after 5 hours?
c) How long will it take for the population to double?
d) Sketch a graph of this function for the first 10 hours, labeling key points.

a)  **1 point:** Correctly substituting <math-block>t=0</math-block> into the equation and stating the initial population is 100. b)  **1 point:** Correctly substituting <math-block>t=5</math-block> into the equation and calculating the population (approximately 271.8).
c)  **2 points:** Setting <math-block>200 = 100e^{0.2t}</math-block>, solving for <math-block>t</math-block> using logarithms (approximately 3.47 hours).
d)  **2 points:** Correctly labeling axes, showing exponential growth, and indicating key points such as the initial population and the population at <math-block>t = 5</math-block>.

You've got this! Go get 'em! 💪