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Understanding Higher-Degree Polynomial Functions: Cubic and Quartic Graphs

This lesson focuses on identifying, analyzing, and modeling graphs of higher-degree polynomial functions, specifically cubic and quartic functions. Students will learn how to find local and absolute extrema, as well as explore the key features of these polynomials, including x-intercepts and turning points. The session includes practical examples and assignments to apply concepts learned. By the end, students will be able to distinguish between various polynomial characteristics and effectively graph these functions within defined intervals.

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Understanding Higher-Degree Polynomial Functions: Cubic and Quartic Graphs

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  1. MAT 150 – Class #20 • Topics: • Identify Graphs of Higher-Degree Polynomials Functions • Graph Cubic and Quartic Functions • Find Local Extrema and Absolute Extrema • Modeling Cubic and Quartic Functions

  2. Higher-Degree Polynomial Functions • Higher-degree polynomials functions with degree ______________. • Examples:

  3. Example of Higher-Degree Polynomial

  4. Two Important Higher-Degree Polynomials • Cubic Function • Quartic Function

  5. Key Features of Higher-Degree Polynomials • In general, the graph of a polynomial function of degree n has _____________n x-intercepts. • Local extremaPoints - ____________on these graphs • Local minimum point- where the curve changes from ________________________ • Local Maximum point – Where the curve changes from _________________________ • Absolute Maximum Point – the ______________on the graph over an interval • absolute minimum point – The ______________on the graph over an interval

  6. Graph • Using the appropriate window, graph . • Find the local maximum and local minimum, if possible. • Where is the absolute maximum of this function on the interval [0, 6]?

  7. Types of Polynomials

  8. Match the Function to the Graph 1 2 5 3 4

  9. Graph Questions Use the given graph to graph to • Estimate the x-intercepts • Turning Points • Positive or negative Coefficient • Cubic or Quartic

  10. Assignment Pg. 424-429 #1-2 #5-8 all #11-16 all #33-34 #41, 44, 46

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