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Polynomial Functions: Graphing and Analysis Study Guide

This guide covers the essential concepts for understanding and graphing polynomial functions. It details how to determine end behavior using the leading term test, analyze the impact of root multiplicity on graph crossings and touches, and predict the number of turning points based on the polynomial's degree. Includes worked examples for practical application.

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Function Fundamentals

Domain and Range

Learn to identify the set of valid inputs and resulting outputs, including how to handle restrictions like division by zero and even roots.

Function Composition

Understand how to nest functions within one another and the requirements for the inner function's range to match the outer function's domain.

Inverse Functions

Master the concept of reversing functions, the Horizontal Line Test for one-to-one functions, and the algebraic steps to find an inverse.

Polynomial and Rational Functions

Polynomial Properties and End Behavior

Learn how degree, leading coefficients, and multiplicity determine the roots, turning points, and long-term trajectory of polynomial graphs.

  • Degree of a polynomial
  • Leading coefficient
  • End behavior
  • Leading Term Test
  • Even degree polynomials
  • Odd degree polynomials
  • Positive leading coefficient
  • Negative leading coefficient
  • Roots (zeros) of a polynomial
  • Multiplicity of roots
  • Odd multiplicity
  • Even multiplicity
  • Graph touching x-axis (bouncing)
  • Turning points
  • Maximum number of turning points

Rational Function Discontinuities

Master the identification of removable discontinuities (holes) and non-removable discontinuities (vertical asymptotes) through factorization.

Asymptotic Analysis

Understand how to calculate horizontal, vertical, and slant asymptotes to predict function behavior at boundaries and infinity.

Exponentials and Logarithms

Exponential Laws and Logarithms

Master the fundamental rules of exponents and the properties of logarithms (product, quotient, power, and change of base) to simplify and solve complex equations.

Growth and Decay Modeling

Learn to apply exponential models to real-world scenarios including compound interest, radioactive half-life, and continuous population growth.

Linear Systems and Matrices

Matrix Operations and Properties

Learn the rules for matrix addition, scalar multiplication, and the non-commutative nature of matrix multiplication, including identity and inverse matrices.

Determinants and Invertibility

Understand how to calculate determinants for 2x2 and 3x3 matrices and use them to determine if a system has a unique solution or is singular.

Solving Systems with Matrices

Sequences and Series

Arithmetic and Geometric Progressions

Master the formulas for explicit and recursive terms, as well as the summation of arithmetic and geometric series, including infinite convergence.

Summation Notation

Learn to use Sigma notation to concisely represent and manipulate the sums of complex sequences.

Mathematical Induction

Understand the two-step proof technique involving a base case and an inductive step to verify statements for all natural numbers.

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