2.2 Polynomial Functions of Higher Degree

Graphs of Polynomial Functions

Continuous- the graph has no breaks, holes, or gaps

End Behavior 

The exponent of the first term will determine whether the ends of the polynomial function will move in the same direction or the opposite

• An even exponent means the ends will be the same (both rise or both fall)
• An odd exponent means the ends will be opposite (one rises and one falls)

                                                                     Polynomial with a degree of 0

  

                                                                   Polynomial with a degree of 1

                                                                                      odd = opposite

                                     

                                                                        Polynomial with a degree of 2

                                                                                       even = same

                                                                        Polynomial with a degree of 3

                                                                                         odd = opposite

                                                                          Polynomial with a degree of 4 

                                                                                              even = same

                                                                          Polynomial with a degree of 5

                                                                                          odd = opposite

Leading Coefficient Test

     f (x) = 28x⁶ + 15x³ – 12x² + 87

• Leading coefficient = 28
• The leading coefficient, in this case 28, will tell us whether the right end of the graph will rise or fall
• A positive coefficient means the right end will rise/ go towards infinity
• A negative coefficient means the right end will fall/ go towards - infinity

Zeros of Polynomial Functions

A zero of a function f is a number x for which f (x) = 0

An n^th degree polynomial will have a maximum of n x-intercepts

A polynomial to the n^th degree will have a maximum of n-1 extremas (relative minimums or maximums) 

Example:    f (x) = x³- x

The degree of this polynomial is 3.  The maximum number of extremas this polynomial can have is two because 3 - 1 = 2

A polynomial with a degree of 3 can have zero extremas as well. 

Example:      y = x³ 

Section 2.1- Quadratic FUNctions

Polynomial functions

• Let n be a non negative integer and let   be real numbers with.

• The function is called a polynomial function of x with degree n.

• Polynomial functions are classified by degree.

• Constant function:

• It has a degree of 0.

• Linear function:

• It has a degree of 1.

• The type of polynomial function discussed in this section is the quadratic function.

Quadratic Functions

• Let a, b, and c be real numbers with
• The graph of a quadratic function is a special type of U shaped curve called a parabola.All parabolas are symmetric with respect to a line called the axis of symmetry.
• The point where the axis intersects the parabola is the vertex of the parabola.
• If the leading coefficient a is positive, the graph is a parabola that opens upward. If the leading coefficient a is negative, the graph is a parabola that opens downward.

• The simplest type of quadratic is
• In the graph a > 0, the vertex is the minnimum point on the graph; and if a < 0, the vertex is the maximum point on the graph.

• The Standard form
• , 
• It identifies the vertex of the parabola as (h,k)
• Ex: 

 (step 1)

 (step 2)

 (step 3)

Vertex = (-2, -9)

• How to complete the square
• Step 1: factor out any coefficient of that is different from 1.
• Step 2: addinside the parentheses and subtract multiplied by the coefficient of outside the parentheses.
• Step 3: simplify into standard form.