Higher Degree Polynomial Functions
A Polynomial Function is a function of the form:
f(x) = a_{n }x^{n} + a_{n  1 }x^{n  1}+ a_{n
 2 }x^{n  2} + ... + a_{1 }x + a_{0}
where the a's are real numbers and n is a nonnegative integer.
Domains: (∞, ∞)
Ranges: vary
Degree of a polynomial function is n.
The GRAPHS of polynomial functions are smooth, continuous curves, with no sharp turns.
Cubic Functions are third degree polynomial functions.
The turning points are called Local Extreme Points, or Local Extrema.
Sometimes we are interested in finding the highest or lowest point on the graph of a function over a certain
interval. These points are called absolute maximum or absolute minimum points.
For example, if we graphed the cubic function f(x) = 3x^{3} + 8x^{2} over the interval [
0.5, 3], this is what we would get:
Quartic Functions are fourth degree polynomial functions.
In general, we can tell certain things about polynomial functions just from looking at their equations. We
can tell the maximum possible number of xintercepts and turning points.
For an nth degree polynomial function:
The maximum number of xintercepts is n.
The maximum number of turning points is n  1.
We can also tell about the end behavior of polynomial functions. A cubic function will have ends that go in
opposite directions, while a quartic function will have ends that either both go up or both go down. The
leading coefficient determines the end behavior.
Function 
Possible Graphs 
Degree 
Turning points 
End Behavior 
Degree Even or Odd 
possitive leading coefficient 
negative leading coefficient 
Linear 
Positive slope 
Negative slope 
1 
0 


odd 
Quadratic 
Possitive leading coefficient 
Negative leading coefficient 
2 
1 


even 
Cubic 
Positive leading coefficient
Negative leading coefficient

3 
2 or 0 


odd 
Quartic 
Positive leading coefficient
Negative leading coefficient 
4 
3 or 1 


even 
