Explore the fundamentals of graphing polynomials in this 38-minute mathematics lecture. Learn about plotting integer and rational points, using straightedge lines to connect them, and delve into a detailed examination of a quartic polynomial. Discover the Grid plane and Cartesian coordinates, investigate integral and rational graphs, and practice joining dots to create polynomial curves. Analyze various tangents and their integer graphs through practical exercises. Part of the MathFoundations series, this lecture aims to establish solid mathematical foundations while addressing logical challenges in modern pure mathematics.