## Topic 10

## Graphing Polynomials

A **polynomial **is a monomial or sum or terms that are
all monomials. Polynomials can be classified by **degree**,
the highest exponent of any individual term in the polynomial.
The degree tells us about the general shape of the graph.

As the degree increases above 1, the graph gets points of inflection
where it changes direction. This change of direction causes a
U-turn. These points of inflection give relative mins or maxes.
Look at the chart below for some extra clarification.

**Degree** |
**Name** |
**Shape** |
**U-Turns** |

0 |
Constant |
Point |
0 |

1 |
Linear |
Line |
0 |

2 |
Quadratic |
Parabola |
1 |

3 |
Cubic |
S-Shape |
2 |

4 |
Quartic |
W-Shape |
3 |

Do you notice a pattern? You should, after the degree goes
above 1, the number of turns in the graph is one less than the
degree.

**10 Point Bonus: **Is the number of
turns always one less than the degree? For example, is it possible
to have a quartic equation with only two U-turns? If so when?
What would it look like? What is that called? Give a specific
example.

Let's look at a few examples of each of these.

**Degree
0**
These are constants. In other words, numbers without variables.

**Examples: **5, -100, .75, -3.6

If we graphed these, they would just be points on the number
line.

**Degree 1**
These are lines with the general form: y
= mx + b. They have no inflection points and some examples
of these are:

**Examples: **

If you need a further review of lines click **here**
to link back to topic 1.

**Degree 2**
These are parabolas with the general form:

They have one point of inflection and one U-turn. Examples
of this type of equation would be:

For further review of these see solving quadratic equations,
**topic
7**.

**Degree Three**
These equations will have two U-turns and look like an S. They
have two points of inflection and some examples of these would
be:

Notice how all of the different colored graphs have an S shape
and in order to do that hav 2 U -turns.

**Degree 4**
These equations will have three U-turns and look like a W.
They have three points of inflections and some examples of these
would be:

Notice how the different colored graphs have the shape of a
W and 3 U-turns.

Now, you should have a good enough understanding of how to
graph polynomials that you should be able to make predictions
about the shape of the graph based on a given equation.

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