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:
- Degree of polynomial (odd vs even)
- 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.
