A landscape architect is designing a park and wants to incorporate curves modeled by polynomial functions. The architect is particularly interested in using a cubic polynomial to create a smooth transition between two flat sections of the park. The cubic polynomial, $P(x)$, will be used to model the height of the terrain along a section of the park, where $x$ represents the horizontal distance in meters.
a. The architect wants the terrain to be level at $x = 0$ and $x = 4$ meters. This means $P(0) = 0$ and $P(4) = 0$. Also, the architect wants the terrain to smoothly transition from the level ground, meaning the gradient of the terrain should be zero at these points, thus $P’(0) = 0$ and $P’(4) = 0$. Explain why a cubic polynomial of the form $P(x) = ax^2(x-4)$ would satisfy the conditions $P(0) = 0$, $P(4) = 0$ and $P’(0) = 0$.
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Create Free Account Log inThis is a free VCE Units 3 & 4 Mathematical Methods practice question worth 3 marks, testing your understanding of Polynomial Function Graphs. It falls under Functions, relations and graphs in Unit 3: Mathematical Methods Unit 3. Submit your answer above to receive instant AI-powered marking and personalised feedback.
Extend introductory study of functions, algebra, calculus. Focus on functions, relations, graphs, algebra, and applications of derivatives.
Covers transformations, polynomial functions, power functions, exponential functions, logarithmic functions, circular functions, and combinations of these.
graphs of polynomial functions and their key features
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