4.1 Polynomial Functions of Degree Greater Than 2

A polynomial function can be written in this form:

-a is a coefficient

-n is a whole number (non negative integer)

Any polynomial function has domain with all real numbers. This is because a polynomial function doesn't have any denominators or radicals that contain variables.

A polynomial function always has a y-intercept but not necessarily an x-intercept(s).

Linear Function (degree of 1)

Quadratic Function (degree of 2)

Cubic Function (degree of 3)

-The degree of a function is equal to the maximum number of x-intercepts the function can have.

Intermediate Value Theorem- If f is a polynomial function and f(a) doesn't equal f(b) for a < b, then f takes on every value between f(a) and f(b) in the interval [a,b]

The bottom or top of a curve in a polynomial function's graph is the relative minimum or maximum. A value f(a) is a relative maximum of f if on the interval (x1,x2) (that includes a), f(a)>f(b) for any x2>b>x1.

-These points are called extrema

-The maximum number of extrema a polynomial function can have is equal to one less than the degree. (n-1)

End Behavior

As the you go to the left, the x values approach - ∞

As you go to the right, the x values approach ∞

From left to right, the behavior of the y (f(x)) values is dependent on the degree of the function.

-an even degree means that the right and left ends will be going in the same direction

-an odd degree means that the right and left ends will be going in opposite directions.

The leading coefficient tells what direction the right end goes (negative coefficient: negative infinity, positive coefficient: infinity)

ex.

left: x --> -∞ f(x) --> -∞

right: x--> ∞ f(x) --> ∞

Zeros of Multiplicity

if

find zeroes and its

The zero of multiplicity is 2, equal to to the power of the factor. A function with zeroes of multiplicity equaling 2, the extrema of the curve is tangent to the x-axis.

If the zero of multiplicity is 3, the graph levels out before passing through the x-axis and then continues.

Function #1 has a degree of 3, because it passes through the x-axis 3 times and none of those times is the graph tangent to the x-axis or does the graph level out then continue.

Function #2 has a degree of 4, because it is tangent (zero of multiplicity 2) to the graph once and then passes through 2 more times. 2+1+1=4

Function #3 has a degree of 6, because it is tangent once, passes through once, and levels out and continues once. 2+1+3=6

Thats it!