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									Polynomial Expression and Functions
Lesson #1: Characteristics of Polynomial Functions - Part 1
A POLYNOMIAL FUNCTION is an algebraic expression                 A relation is a FUNCTION iff for every x there
where all exponents of the variables are whole numbers and no    is only 1 y value. A function must pass the VLT.
variable is contained in the denominator.
There are 3 types of functions we will study this semester:
                    Polynomial                       Exponential                         Periodic




A POLYNOMIAL FUNCTION can be written more generally as


                                    where n is a whole number and a is a real number.
                 Notice that the polynomial function is written in terms of decreasing order of powers of x.

A polynomial function can be written as components, or POWER FUNCTIONS,                        . A power function is
the simplest type of polynomial function. So, if              is the polynomial function, then the power functions
are a                                  ,c        .


Recall: Given the polynomial function,                       ,
                    the coefficients are 3, - 4, and 1, or                  that are all real numbers.
                    the degree of the function, n, which is the exponent of the greatest power of x with a non-zero
                       coefficient, is 2.
                    the leading coefficient is         , or 3.
                    the constant term is , or 1.

                Quadrants                                 Read all graphs from left to right.
                                                          x-int occur where the relation crosses the x-
                                                           axis.
                                                          y-int occur where the relation crosses the y-
                                                           axis.
                                                          Domain is the set of x values that are
                                                           defined for the function.
                                                          Range is the set of y values that are defined
                                                           for the function.
Key Features of Standard Power Functions




    Degree 1                Degree 2                Degree 3                Degree 4                Degree 5                Degree 6

  ODD degree              EVEN degree             ODD degree             EVEN degree             ODD degree               EVEN degree
 Linear Function           Quadratic             Cubic Function         Quartic Function        Quintic Function           6th Degree
                            Function                                                                                        Function


    x-int = 0               x-int = 0               x-int = 0               x-int = 0               x-int = 0               x-int = 0
    y-int = 0               y-int = 0               y-int = 0               y-int = 0               y-int = 0               y-int = 0

The END BEHAVIOUR of          is the behavior of the y-values as x increases or                                 and as x decreases or
       . NOTE: Infinity ( is not a number. It is a concept.

  As                ,    As                 ,    As                 ,    As                 ,    As                 ,    As                 ,
                .                       .                       .                       .                       .                       .
  As                ,    As                 ,    As                 ,    As                 ,    As                 ,    As                 ,
              .                       .                       .                       .                       .                       .
Starts in quadrant      Starts in quadrant      Starts in quadrant      Starts in quadrant      Starts in quadrant      Starts in quadrant
  3 and ends in           2 and ends in           3 and ends in           2 and ends in           3 and ends in           2 and ends in
   quadrant 1.             quadrant 1.             quadrant 1.             quadrant 1.             quadrant 1.             quadrant 1.

A graph has LINE SYMMETRY iff                                             A graph has POINT SYMMETRY about
divides the graph into two parts that are reflections                   iff each part of the graph on one side of can be
rotated
of each other.                                                                   to coincide with part of the graph on the other side
of




 Point symmetry          Line symmetry           Point symmetry          Line symmetry           Point symmetry          Line symmetry
   about (0,0)           along                     about (0,0)           along                     about (0,0)           along
Example 1: State the type of function each of the following equations represents.
   a)                                            b)                               c)
           Polynomial                                 Exponential                    Periodic


Example 2: Complete the table.



        Graph




      Equation

       Degree                             3                               4                                 5


Type of Function                       Cubic                           Quartic                            Quintic


       Domain


        Range

                             As               ,       .      As               ,       .         As              ,        .
  End Behaviour              As               ,       .      As               ,       .         As              ,        .


    Starts/Ends

                                  Point Symmetry at             Line Symmetry along                  Point Symmetry at
     Line/Point
     Symmetry
                            Polynomial Concept Attainment Activity

Compare and contrast the examples and non-examples of polynomial functions below.
Through reasoning, identify 3 attributes of every polynomial function that distinguish them from non-polynomial
functions:
   a. _______________________________________

   b. _______________________________________

   c. _______________________________________



                                   Examples                         Non Examples



                                       yx                           y x


                                                                                       1
                                    y  2x  1                       f  x   3x 2  x



                                        2
                                     y x                           x  6
                                        5



                                       y  x2                        x 2  y 2  16




                                 y   x  2  1                    h x 
                                                2                              3
                                                                                   x




                                 f  x    x2  x                  y  sin



                                                                           1
                            y  0.2  4 x  3  x  3             y
                                                                          x2



                              y  x 3  2 x 2  x  11               y  2x
Polynomial Concept Attainment Activity (continued)


                        Examples                Non Examples


                                                            x 1
                            y4                      y
                                                          x  x 1
                                                           2




                                     1 2
                   h  x   x4      x 3
                                     2




                   y  4 x 0  4
                   y  x  x 2  4   x  2
Lesson #2: Characteristics of Power Functions - Part 2
Even degree power functions have line symmetry along the y-axis or   .
Odd degree power functions have point symmetry at the origin or    .
Example 1: Complete the table.
                       (-1.7, 4.9)                                                                                                     (2, 16)




      Graph


                                                (1.7,- 4.9)                           (-2, -128)

    Equation
     Leading                          -1                                                               +1                      +1
   Coefficient
    Constant                           0                                  0                              0                     0
     Degree                            3                                  6                              7                     4
     Type of                         Cubic                          6 th Degree                    7 th Degree               Quartic
    Function
                       As                   ,                      As             ,           As                 ,      As                ,
End Behaviour
                                                                   As             ,           As                 ,      As                ,
                              As                   ,

  Starts/Ends
   Line/Point             Point Symmetry at        Line Symmetry along                    Point Symmetry at          Line Symmetry along
   Symmetry
An interval can be expressed as an inequality or bracket interval.


    -4             0                    5                     -7          0                  4           -4            0                      6




     Domain
    Examples


     Range
    Examples
Summary
Given an EVEN DEGREE power function with a POSITIVE LEADING
COEFFICIENT, the function will start in quadrant 2 and end in quadrant 1.

This function will have line symmetry along        .


Given an EVEN DEGREE power function with a NEGATIVE LEADING
COEFFICIENT, the function will start in quadrant 3 and end in quadrant 4.

This function will have line symmetry along        .



Given an ODD DEGREE power function with a POSITIVE LEADING
COEFFICIENT,
the function will start in quadrant 3 and end in quadrant 1.

This function will have point symmetry at      .



Given an ODD DEGREE power function with a NEGATIVE LEADING
COEFFICIENT, the function will start in quadrant 2 and end in quadrant 4.

This function will have point symmetry at      .
Lesson #3: Characteristics of Polynomial Functions - Part 1


This is the graph of the function                          .                The LOCAL MAXIMUM is a “PEAK” and
                                                                            it occurs where the slope of the function is
     The (ABSOLUTE/GLOBAL) MAXIMUM is                                       zero. The local max is approximately 8 and it
     the largest value of y for the function which is defined or            occurs when x is approximately -1.8.
     there is no value.




 The (ABSOLUTE/GLOBAL) MINIMUM is the                                      The LOCAL MINIMUM is a “VALLEY”
 smallest value of y for the function which is defined or there is         and it occurs where the slope of the function is
 no value.                                                                 zero. The local min is approximately – 4 and it
                                                                           occurs when x is approximately 1.


Example 1: Given the following graph of f(x), answer the following questions:
1.
                                                     e)   The local max is _____________________ and occurs at x = ________.

                                                     f)   The local min is _____________________ and occurs at x = ________.

                                                     g) The absolute max is _______________ and it occurs when x = ________.

                                                     h) The absolute min is _______________ and it occurs when x = ________.




2.        a)   The local max is _____________________ and occurs at x = ________.

          b) The local min is _____________________ and occurs at x = ________.

          c) The absolute max is _______________ and it occurs when x = ________.
        Students are to complete on page 15 – 17 Investigate #1 during this period.
          d) The absolute min is _______________ and it occurs when x = ________.
Investigate #1 What are the key features of the graphs of polynomial functions?
A: Polynomial Functions of Odd Degree
                     Group A                                            Group B




    Leading
   Coefficient

 End Behaviour

     # of
Minimum Points

     # of
Maximum Points

   # of Local
Minimum Points

  # of Local
Maximum Points


 Similar to y = x
   or y = - x?
   Leading
  Coefficient

End Behaviour

     # of
Minimum Points

     # of
Maximum Points

   # of Local
Minimum Points

  # of Local
Maximum Points


Similar to y = x
  or y = - x?
   Leading
  Coefficient

End Behaviour

     # of
Minimum Points

     # of
Maximum Points

   # of Local
Minimum Points

  # of Local
Maximum Points


Similar to y = x
  or y = - x?
2.




      Leading
     Coefficient

End Behaviour

       # of
     Minimum
      Points

       # of
     Maximum
      Points

     # of Local
     Minimum
       Points

     # of Local
     Maximum
       Points


Similar to y = x
  or y = - x?
3. a) What are the similarities and differences between the graphs of linear, cubic, and quintic functions?



   b) What are the minimum and the maximum numbers of x-intercepts of graphs of cubic polynomial functions?



   c) Describe the relationship between the number of minimum and maximum points, the number of local minimum and
      local maximum points, and the degree of a polynomial function.



   d) What is the relationship between the sign of the leading coefficient and the end behavior of graphs of polynomial
      functions with odd degree?



   e) Do you think the results in part d) are true for all polynomial functions with odd degree? Justify your answer.
B: Polynomial Functions of Even Degree
                     Group A             Group B




    Leading
   Coefficient

 End Behaviour

     # of
Minimum Points

     # of
Maximum Points

   # of Local
Minimum Points

  # of Local
Maximum Points


Similar to y =
  or y = - ?
    Leading
   Coefficient

 End Behaviour

     # of
Minimum Points

     # of
Maximum Points

   # of Local
Minimum Points

  # of Local
Maximum Points


Similar to y =
  or y = - ?
    Leading
   Coefficient

 End Behaviour

     # of
Minimum Points

     # of
Maximum Points

   # of Local
Minimum Points

  # of Local
Maximum Points


Similar to y =
  or y = - ?


2. a) What are the similarities and differences between the graphs of linear, cubic, and quintic functions?



   b) What are the minimum and the maximum numbers of x-intercepts of graphs of cubic polynomial functions?



   c) Describe the relationship between the number of minimum and maximum points, the number of local minimum and
      local maximum points, and the degree of a polynomial function.
  d) What is the relationship between the sign of the leading coefficient and the end behavior of graphs of polynomial
     functions with an even degree?



   e) Do you think the results in part d) are true for all polynomial functions with an even degree? Justify your answer.



Summary:
  A polynomial function with n degree could have n – 1 local minimum and maximum points where                .
                                                  Odd Degree
                     Positive Leading Coefficient              Negative Leading Coefficient
               Starts in Quad 3 and ends in Quad 1            Starts in Quad 2 and ends in Quad 4




                                    At least 1 x-intercept, up to n x-intercepts




                                               Maximum Point and
                                                 Minimum Point
                                      Or no Maximum or Minimum Point
                                              May have Point Symmetry
                                                   Even Degree

                    Positive Leading Coefficient                 Negative Leading Coefficient
               Starts in Quad 2 and ends in Quad 1            Starts in Quad 3 and ends in Quad 4




                       At least 1 Min Point                             At least 1 Max Point

                                               Zero to n x-intercepts
                                      , range is dependent on max or min value
                                              May have Line Symmetry
Lesson #4: Characteristics of Polynomial Functions - Part 2
      Students are to complete Investigate 2 on page 17 – 18. Students will need a graphing calculator. Students are
      responsible for making notes on the punching sequence for the TI83, TI83+ and/or TI84+ in order to create
      differences and to regress.

Example 1: Use finite differences to determine:
                                                    a) the degree




                                                    b) the sign of the leading
                                                       coefficient




                                                    c) the value of the leading coefficient




Example 2: A medical researcher establishes that a patient’s reaction time, r, in minutes, to a dose of a particular drug is
                     , where d is the amount of the drub, in millimeters, that is absorbed into the patient’s blood.
   a) What type of function is                        ?




    b) Using a graphing calculator, determine the value of the constant differences.




    c) Describe the end behavior of this function if no restrictions are considered.




    d) State the restrictions for this situation.
Numerical Properties of Polynomial Functions

        1. Consider the function y = x                     x        y          First Differences
           a) What type of function is it?
           b) Complete the table of values.                –3
           c) Calculate the first differences.
                                                           –2
           d) In this case, the first differences
              were positive. How would the
              graph differ if the first differences        –1
              were negative?
                                                           0

                                                           1

                                                           2

                                                           3




        2. Consider the function y  x 2                                      Differences
                                                      x         y
           a) What type of function is it?                              First           Second
           b) Complete the table of values.           –3
           c) Calculate the first and second
              differences.                            –2

                                                      –1

                                                      0

                                                      1

                                                      2

                                                      3
Numerical Properties of Polynomial Functions (continued)

       3. Consider the function y  x 3                                             Differences
                                                      x           y
          a) What type of function is it?                                First        Second            Third
          b) Complete the table of values.
                                                     –3
          c) Calculate the first, second,
             and third differences.
                                                     –2

                                                     –1

                                                      0

                                                      1

                                                      2

                                                      3



       4. Consider the                                                    Differences
          function y  x 4             x      y
                                                          First        Second        Third              Fourth
          a) What type of
             function is it?          –3
          b) Complete the             –2
             table of values.
          c) Calculate the
                                      –1
             first, second,
             third, and fourth
             differences.              0

                                       1

                                       2

                                       3



5. a) Summarize the patterns you observe in Questions 1–4.

   b) Hypothesize as to whether or not your patterns hold when values for the b, c, d, and k parameters are
      not equal to 0 in y  ax  k , y  ax 2  bc  k , y  ax 3  bx 2  cx  k , and y  ax 4  bx 3  cx 2  dx  k .

   c) Test your hypothesis on at least 6 different examples. Explain your findings.

								
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