Polynomial Functions
What are polynomial functions?


First of all, let's look at the equation,



-The value of n must be a nonnegative integer (this means, it must be a whole number, it must be equal to zero, or positive).

-The coefficients,

,

are real numbers.

-The degree of a polynomial function is the highest possible value for n

For example:




Degree: 2 (because it is x square)

When is it not a polynomial function??
Let's look at some examples,





(This is not a polynomial function, since n cannot be negative)


Another example,



(this isn't a polynomial function either because n cannot be a fraction)

Naming:

-If the degree=0 (f(x)=3) it would be called a constant.
-If the degree=1 (g(x)=3x+5) it would be called linear.
-If the degree=2 (h(x)=2x^2+3x+5) it would be called quadratic.
-If the degree=3 (j(x)=x^3) it would be called cubic.
-If the degree=4 (k(x)=5x^4-1) it would be called quatric.

Quadratic functions:
Quadratic functions have either 2, 1, or 0 x-intercepts.

To find the intercepts, you either substitute in a 0 for the y-value, or x-values.

If this doesn't work, you use the quadratic equation



Side note: the part under the square root tells you how many solutions there are.

To find the vertex of an equation, one must put it into standard form:
So,
from:



to standard form:


Vertex= (h,k)

To do this, one must use a method called Completing the Square

Example:



to complete the sqaure, ignore the c, divide b by 2 and square it, then add it inside the paranthesis, and to even it out, subtract that number on the outside.

Vertex=(3,-8)