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:
𝑓 𝑥 =3 𝑥 3 −54
𝑦=− 1 3 𝑥 4 −3 𝑥 3 +3 𝑥 2 −12𝑥+110

3. Example of Higher-Degree Polynomial

4. Two Important Higher-Degree Polynomials
Cubic Function
𝑓 𝑥 =𝑎 𝑥 3 +𝑏 𝑥 2 +𝑐𝑥+𝑑 , (𝑎≠0)
Quartic Function
𝑓 𝑥 =𝑎 𝑥 4 +𝑏 𝑥 3 +𝑐 𝑥 2 +𝑑𝑥+𝑒 , (𝑎≠0)

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 extrema Points - 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 𝑓 𝑥 = 𝑥 3 −37𝑥.
Find the local maximum and local minimum, if possible.
Where is the absolute maximum of this function on the interval [0, 6]?

7. Positive Leading Negative Leading Coefficient Coefficient
Type of Function
Degree
Possible Graphs
Positive Negative
End Behavior
Positive Leading Negative Leading
Coefficient Coefficient
Odd
Or Even
Linear
Quadratic
Cubic
Quartic
Types of Polynomials

8. Match the Function to the Graph1 2
𝒚=𝟐 𝒙 𝟑 −𝟐𝒙
𝒚= 𝒙 𝟒 −𝟑 𝒙 𝟐
𝒚=𝟐𝒙+𝟏
𝒚=𝟔 𝒙 𝟐 −𝟒 𝒙 𝟑 +𝟏
𝒚= 𝒙 𝟑 −𝟐 𝒙 𝟒 +𝟑 𝒙 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 #1-2 #5-8 all #11-16 all #33-34 #41, 44, 46