Topics: Polynomial functions: continuous, turning points, roots, multiplicity, end behavior.
If looking for Functions: Intercepts, Positive/Negative, and Increasing/Decreasing, see Function Features.
If looking for
Functions: Maximum/Minimum, Symmetry, End Behavior, see More Function Features.
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reminder
A polynomial function is a function which is defined by a polynomial.
(In plain English, it's a polynomial expression set equal to f(x), or some other notation.)
Examples:   f(x) = x2 + x - 6;      P(x) = x3 - x2 - 12x;      y1 = x2 + 4x + 4

Basic information about the graphs of polynomial functions:

1. Continuous: The graphs of polynomial functions are continuous. You can draw the entire function without lifting your pencil from your paper.
                   graphcontinuous
2. Smooth: The graphs of polynomial functions are always nice smooth graphs.  The graphs do not contain holes, breaks or sharp point corners. 
                   graph donothave
3. Turning Points: The "hills" or "valleys" where the graph changes direction from increasing to decreasing, or from decreasing to increasing, are often called turning points.

bullet The number of turning points is never more than the degree of the polynomial minus one.
For example, the graph of a 3rd degree polynomial function can have 2 turning points or fewer. [If the degree is n, the number of turning points is at most n - 1. There could be fewer.]

graphaturning2
Other terms for turning points can be relative maximum (or minimum) and local maximum (or minimum). The designation of "relative" (or "local") tells you that while this point may be a turning point, it may not be the maximum (or minimum) value reached by this function. It is only a maximum (or minimum) "relative" to a small section of the function surrounding this point. The one, true, largest (or smallest) value reached by the entire function is called the absolute maximum (or minimum), or the global maximum (or minimum).
FYI: The plural of maximum is maxima. The plural of minimum is minima. Discussions of maxima and minima are referred to as extrema.

The connection between factoring and graphing polynomial functions:
4. Roots (or Zeros):
def If you plug in r (some real number) for x in a polynomial function, P(x), and get an answer of 0, the number, r, is called a root, or zero, of the polynomial.

It can be quite tedious to repeatedly test random numbers looking for those numbers that give an answer of zero. A faster method for finding these roots (or zeros), is to factor the polynomial, and then set the factors equal to zero (the desired answer).

This factoring method for finding roots, or zeros, utilizes the zero factor principle which states that "if a • b = 0, then either a = 0 and/or b = 0."

bullet The real numbers that create the roots (or zeros) of a polynomial correspond to the x-intercepts of the graph of the polynomial function.

This is valuable information when it comes to creating the graph of a polynomial (without a graphing calculator). It is also valuable if you are given the graph and are attempting to create a possible equation.

bullet Note: A polynomial of degree 2 will have two roots (zeros), a polynomial of degree 3 will have three roots (zeros), and so on.
Example:
factorroot
factorrootgraph

What happens to the graph when a factor repeats?
5. Multiplicity of Roots (or Zeros): We have seen in section 4 that the roots (zeros) of a polynomial correspond with the x-intercepts of the polynomial graph. In some situations, the graph will "cross" the x-axis at these points. In other situations, the graph may simply "touch" (be tangent to) the x-axis at these points. Let's see if we can determine, before we draw the graph, whether it will "cross" the x-axis at each root, or simply "touch" (be tangent to) the x-axis at each root.

Consider the example at the right. The polynomial is of degree two, so there will be two roots (zeros). The factor of (x + 3) is repeated twice, and can also be written as (x + 3)2.

def The number of times a factor appears in a polynomial is referred to as its multiplicity.

In the example at the right, the factor (x + 3) has a multiplicity of 2, since it appears twice. It creates a "repeated root".

bullet Multiplicity EVEN: When the multiplicity (the number of times a factor repeats) is two (or even), the graph will just "touch" (be tangent to) the x-axis at that point.

Why?
"EVEN" multiplicities are factors that occur an even number of times, and form squares. Since squares are always positive, the graph near the root (zero) will not change signs from positive (above the x-axis) to negative (below the x-axis), or vice versa. The graph will "touch" the x-axis at the root (zero) but remain on the same side of the x-axis.
Example degree 2:
factorrepeated2

factorrepeatedgraph

What if the multiplicity is ODD?

Consider the example at the right. The polynomial is of degree three, so there will be three roots (zeros). The factor of (x - 1) appears three times, and can be written as (x - 1)3.

bullet Multiplicity ODD: When the multiplicity (the number of times a factor repeats) is three (or odd), the graph will "cross" the x-axis at that point.

 

Did you notice?
If you see a factor such as (x - 1)3, the multiplicity is 3.
If you see a factor such as (x + 2)2, the multiplicity is 2.
If you see a factor such as (x + 3), the multiplicity is 1.

bullet Note: When you factor a polynomial, the sum of the multiplicities equals the degree of the polynomial.

x3 + x2 - 5x + 3 = (x - 1)2(x + 3) = (x - 1)2(x + 3)1
degree =
3
sum of multiplicities =
2 + 1 = 3
Example degree 3:
factorrepeat3
factrepeat3

The end behavior of polynomial functions:
6. End Behavior: Examining the DEGREE of the polynomial and the SIGN of the leading coefficient will indicate what is happening at the ends of a polynomial function graph. Remember that end behavior refers to the appearance of a graph as it is followed indefinitely in either direction.
bullet Even Degree Polynomial
("ends" behave similar to a quadratic)

Leading coefficient POSITIVE: both "ends" are UP.

endsUP

Leading coefficient NEGATIVE: both "ends" are DOWN.

graphdown

bullet Odd Degree Polynomial
("ends" behave similar to a cubic)

Leading coefficient POSITIVE: left end is DOWN and right end is UP.

graphdownup

Leading coefficient NEGATIVE: left end is UP and right end is DOWN.

graphupdown


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The graphing calculator can prove to be helpful for checking your answer when factoring and graphing.
ti84c

For help with factoring on your calculator,
Click Here!

 

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