Polynomial Functions Study Notes

Polynomial Functions: Finding Equations

Section 8: Finding Equations for Polynomial Functions

• Task: Find equations for the given polynomial functions in factored form, including the vertical stretch factor, ( k ).
  • Example format for answer:
  • ( f(x) = k (x - 4)^2 (x + 1)^7 )
  • where ( k = -13 )

Part A

• Given critical points:
  • Roots of the polynomial:
  • ( -1 )
  • ( 4 )
  • ( 2 )
  • ( -4 )
  • ( -6 )
  • ( -8 )

Part B

• Given points:
  • ( (-5/2, 0) ): This indicates a root at ( x = -5/2 ).
  • ( (-1, 0) ): This indicates a root at ( x = -1 ).
  • ( (0, -2.4) ): This is a point on the graph, not a root. It informs about the value of the function at ( x = 0 ).

Section 9: Properties of the Polynomial Functions

• For each graph characterized in Section 8, provide additional information:

Part A

Degree of the Polynomial Function

• The degree can be determined by counting the number of roots taking into account their multiplicities.
  • Example Calculation:
  • If roots are counted as (( -1 ), ( 4 ), ( 2 ), ( -4 ), ( -6 ), ( -8 )), the degree is likely 6 if all are counted as single roots.

Part B

Description of the End Behavior

• Polynomial functions have specific end behaviors depending on the degree and leading coefficient:
  • Odd Degree Polynomials:
  • As ( x \to - ), ( f(x) \to - ) or ( ) depending on the sign of leading coefficient ( k ).
  • As ( x \to + ), ( f(x) \to + ) or ( - ).
  • Even Degree Polynomials:
  • As ( x \to - ), ( f(x) \to + ) (if leading coefficient ( k > 0 )) or ( - ) (if ( k < 0 )).
  • As ( x \to + ), follows the same behavior as ( x \to - ).

Example Data for Reference

• Additional given coordinates for the analysis include ( (0, 25) ) and ( (5/4, 0) ).
• Points such as these can help establish the overall graph shape and corresponding polynomial equation through vertical shifts and stretches.