Matt Wolf
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.
Example: Sketch the graph of the function [pic].
1) Apply the Leading Coefficient Test to determine the end behavior of the graph.
[pic] as [pic] [pic] as [pic]
2) Find the zeros of the function. Identify the multiplicity of each zero to determine if the graph crosses or touches the x-axis at this point.
|Zeros | | | | |
|Multiplicity | | | | |
|Graph crosses | | | | |
|or touches x-axis | | | | |
3) Arrange the zeros in order to form test intervals. Choose a test value in each test interval. Evaluate the *factored form* of the function at each test value to determine if the function is positive or negative in the interval.
|Test Interval | | | | | |
|Test Value | | | | | |
|Sign of [pic] | | | | | |
|(at Test Value) | | | | | |
|Graph is Above | | | | | |
|or Below x-axis | | | | | |
4) Sketch the graph with smooth, continuous curves using the information obtained in Steps 1-3.
-----------------------
[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]
[pic]
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- kate beckinsale and matt rife
- wolf life quizzes for girls
- john macarthur s son matt macarthur
- wolf cabinets vs kraftmaid cabinets
- wolf quizzes your life story
- wolf themed birthday party supplies
- robb wolf food matrix
- cva wolf parts diagram
- cva wolf parts manual
- cva wolf magnum parts
- cva wolf 50 cal manual
- cva wolf instruction manual