I explored mathematical ideas using technology when I used Desmos to graph and analyze several polynomial functions, including f(x)=(x−1)(x+2)2(x−4)f(x) = (x – 1)(x + 2)^2(x – 4)f(x)=(x−1)(x+2)2(x−4), g(x)=(x−1)(x+2)2(x−4)g(x) = (x – 1)(x + 2)^2(x – 4)g(x)=(x−1)(x+2)2(x−4), and h(x)=(x−1)(x−5)h(x) = (x – 1)(x – 5)h(x)=(x−1)(x−5). Through this exploration, I learned how to manipulate polynomials, understand their behaviour at different points, and visualize their roots, such as single, double, and triple roots, using the interactive graphing tool. I analyzed data and used criteria to conclude when I examined the behaviour of the polynomials and identified key features, such as the x-intercepts, y-intercepts, degree, and end behaviour of each graph. By setting specific constraints on the x- and y-values, I was able to create a visual “frame” around the graph and colour the regions differently, ensuring that adjacent regions did not share the same colour. I justified my conclusions with evidence when I confirmed my findings, such as the degree of the polynomials and their intercepts, by both visually inspecting the graph and using mathematical formulas. This process allowed me to validate my understanding of polynomial functions and their graphical representations, with Desmos providing the evidence to support my conclusions. Through this experience, I gained valuable insights into the intersection of mathematics and technology, particularly in visualizing complex functions.

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