Polynomial Functions

                                                           Polynomial Functions

Hey guys! It's April and im going to talk about polynomial functions. This blog will include the

characteristics of a polynomial functions, the special names for its degrees and how to graph the 

function.

Characteristic of Polynomial Functions: 

     Polynomial functions are written in the form f(x)=anxn + an - 1xn - 1 + ... + a1x + a0, where (n) cannot be a negative integer. Each part of the expression is referred to as a "term", and a term has coefficient associated to with it. In polynomial functions, the term that has the highest power of x is called "leading term", while its coefficient is called the "leading coefficient". The power of x is called the "degree" of polynomial.

Special names: Degree of Polynomials

   Degree                        Functions                             Example
   
       0                             Constant                                    f(x) = 2
       1                             Linear                                        f(x) = 2x + 2
       2                             Quadratic                                  f(x) = 2x^2 - 4
       3                             Cubic                                        f(x) = x^3 +3x^2 + x - 8
       4                             Quartic                                      f(x) = x^4 = 6x^3 - 4x^2 + 2x + 6
       5                             Quintic                                      f(x) = x^5 - 7x^4 + 2x^3 - 3x^2 + 2x - 4

Constant Functions example:


Linear Functions example:


Quadratic Functions example:


Cubic Functions example:


Quartic Functions example:


Quintic Functions example:


How to graph Polynomial Functions:
In graphing ODD degree polynomials: -when the leading coefficient is positive (+) ;
left arm will go down and the right arm will go up
- when the leading coefficient is negative (-) ;
left arm will go up and right arm will go down

In graphing EVEN degree polynomials:
-when leading coefficient is positive (+) ;
both arms will go up
-when leading coefficient is negative (-) ;
both arms will go down

How to match Polynomial Function with its graph:
-Define the type of the function, as well as the degree; is it even or odd?
-The end behavior
-Determine the number of possible x-intercepts
-Find its maximum or minimum value
-What is its y-intercept

Important Graphing Rules!!!!
1. If each root of the given equation is repeated or the same, the curve crosses at the x-axis.
2. If you have 2 or more of the same root...
a) even number of the same root - curve bounces at the x-axis
b) odd number of the same root - curve crosses at the x-axis
3. The higher the degree of the roots, the more the graph will flatten out.