Matt Wolf - Central Bucks School District
|Learning Goals: |At the completion of the lesson, you will be able to… |
| |Identify the degree and leading coefficient of polynomial functions |
| |Calculate the zeros of polynomial functions algebraically |
| |Define and apply multiplicity to zeros of polynomial functions |
| |Apply Leading Coefficient Test to describe end behavior of graphs |
Properties of Polynomial Functions
degree – leading coefficient –
Zeros of Polynomial Functions
If[pic] is a zero of the polynomial function[pic], then the following statements are true:
• (c, 0) is an x-intercept of the graph of the function.
Multiplicity ()
If [pic] is a polynomial function with factor[pic], then a is called a repeated zero if the factor [pic] occurs more than once in the linear factorization of[pic]. The number of times the factor [pic] occurs is called the multiplicity of [pic].
• If the multiplicity of [pic]is odd, the graph of [pic]crosses the x-axis at[pic].
• If the multiplicity of [pic]is even, the graph of [pic]touches (but does not cross) the x-axis at[pic].
Examples: Find all zeros of the following functions. Determine the multiplicity of each zero to determine whether the graph of the function crosses or touches the x-axis at each.
-----------------------
[pic] [pic] [pic]
The Leading Coefficient Test
The end behavior of a graph of a polynomial function can be determined using the Leading Coefficient Test which identifies four basic cases.
|degree is even |degree is even |degree is odd |degree is odd |
|leading coefficient is pos. |leading coefficient is neg. |leading coefficient is pos. |leading coefficient is neg. |
|Graph rises to left and right |Graph falls to left and right |Graph falls to left and rises to right |Graph rises to left and falls to right |
|[pic] |[pic] |[pic] |[pic] |
|Trick: Think of [pic] |Trick: Think of [pic] |Trick: Think of [pic] |Trick: Think of [pic] |
|[pic] |[pic] |[pic] |[pic] |
Examples: Apply the Leading Coefficient Test to describe the end behavior of a graph of the functions.
[pic] [pic]
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