Comprehensive Analysis of Polynomial Functions: A School Assignment

This document is a comprehensive solution to a school assignment focused on exploring the graphs of polynomial functions, specifically quadratic, cubic, and quartic functions. The assignment delves into key features of parabolas, including turning points, axis of symmetry, and axial intercepts, with an emphasis on understanding the effects of parameter transformations on the graphs. The student explores how parameters impact the shape, position, and orientation of parabolas and cubics through sketching and analysis. The solution includes detailed descriptions of parameter effects, sketching exercises, and the application of concepts such as completing the square to convert quadratic equations into vertex form. Additionally, the document addresses solving quadratic equations using Carlyle's method and provides application questions involving cubic functions, such as optimizing the volume of a cardboard box. The solution demonstrates a strong understanding of polynomial functions and their graphical representations.