Understanding Polynomial Functions and Their Transformations

This page provides a detailed exploration of polynomial functions and their graphical behaviors. The content focuses on essential definitions, end behaviors, and symmetry properties.

Definition: A polynomial function consists of one or more terms, with at least one containing a variable.

Vocabulary:

• Cubic Function: A polynomial function of degree three $f(x)=x³$
• Quartic Function: A polynomial function of degree four $f(x)=x⁴$

Highlight: End behavior of polynomial functions depends on two key factors:

1. Degree of polynomial (odd vs even)
2. Sign of leading coefficient (positive vs negative)

Example: For odd degree polynomials:

• Positive leading coefficient: Left side goes down, right side goes up
• Negative leading coefficient: Left side goes up, right side goes down

Example: For even degree polynomials:

• Positive leading coefficient: Both sides continue upward
• Negative leading coefficient: Both sides continue downward

Highlight: Function symmetry can be determined by checking if:

• f$-x$ = f$x$ for even functions (symmetrical about y-axis)
• f$-x$ = -f$x$ for odd functions (symmetrical about origin)
• Neither condition met for neither even nor odd functions

The page concludes with practical examples of graphing cubic functions and determining function symmetry through coordinate plotting and analysis.