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Exploring Higher-Degree Polynomial Functions: Graphs and Extrema

In this lesson, we delve into the graphing of higher-degree polynomial functions, focusing on cubic and quartic polynomials. We will explore how to identify key features such as local and absolute extrema, and understand the significance of turning points. The session will cover modeling techniques for these functions and provide examples highlighting their critical characteristics. By the end, students will be equipped to analyze polynomial graphs, estimate x-intercepts, and determine extrema within specific intervals.

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Exploring Higher-Degree Polynomial Functions: Graphs and Extrema

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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 higher than 2. • 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 at most n x-intercepts. • Local extremaPoints - Turning Points on these graphs • Local minimum point- where the curve changes from decreasing to increasing • Local Maximum point – Where the curve changes from increasing to decreasing • Absolute Maximum Point – the highest point on the graph over an interval • absolute minimum point – The lowest Point 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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