This document is an investigation into polynomial functions, focusing on the relationships between a polynomial's degree and the characteristics of its graph, specifically roots, turning points, and inflection points. The study examines polynomials of various forms, including those with distinct linear factors, squared factors, and real quadratic factors. Using graphing tools, the investigation explores patterns and derives conjectures regarding these characteristics, concluding that while general patterns hold, certain polynomial forms exhibit deviations. Limitations such as sample size and tool precision are noted, suggesting areas for further research.