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Higher Degree Polynomial Functions

A Polynomial Function is a function of the form:

f(x) = an xn + an - 1 xn - 1+ an - 2 xn - 2 + ... + a1 x + a0

where the a's are real numbers and n is a non-negative 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) = -3x3 + 8x2 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 x-intercepts and turning points.

For an nth degree polynomial function:

The maximum number of x-intercepts 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
 
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