Understanding Polynomial Functions and Their Graphs

This comprehensive page introduces the fundamental concepts of polynomial functions and their graphical representations. The content systematically breaks down different types of polynomials and their behaviors.

Definition: A polynomial function is a mathematical function where coefficients are real numbers and exponents are whole numbers.

Vocabulary: A polynomial can be either a monomial or a sum of monomials.

The page presents different polynomial degrees and their characteristics:

Example: Various polynomial types include:

• Constant functions (degree 0): f(x) = -14
• Linear functions (degree 1): f(x) = 4x - 8
• Quadratic functions (degree 2): f(x) = 2x² + x - 9
• Cubic functions (degree 3): f(x) = 2x³ + 5x + 8
• Quartic functions (degree 4): f(x) = x⁴ + 2x - 1

Highlight: End behavior of polynomials depends on two key factors:

1. The degree of the polynomial (odd or even)
2. The sign of the leading coefficient (positive or negative)

Example: For a cubic function with a negative leading coefficient:

• As x approaches positive infinity, f(x) approaches negative infinity
• As x approaches negative infinity, f(x) approaches positive infinity

The page concludes with detailed explanations of end behavior patterns for both odd and even degree polynomials, providing a solid foundation for understanding polynomial end behavior.