How Polynomials Behave
A polynomial looks like this:example of a polynomial 
Continuous and Smooth
There are two main things about the graphs of Polynomials:
The graphs of polynomials are continuous, which is a special term with an exact definition in calculus, but here we will use this simplified definition:
you can draw it without lifting your pen from the paper
The graphs of polynomials are also smooth. No sharp "corners" or "cusps"
How the Curves Behave
Let us graph some polynomials to see what happens ...
... and let us start with the simplest form:
f(x) = x^{n}
Which actually does interesting things:
Even values of "n" behave the same:

And:
Odd values of "n" behave the same

Power Function of Degree n
Next, by including a multiplier of a we get what is called a "Power Function":
f(x) = ax^{n}^{ }f(x) equals a times x to the "power" (ie exponent) n
The "a" changes it this way:
 Larger values of a squash the curve (inwards to yaxis)
 Smaller values of a expand it (away from yaxis)
 And negative values of a flip it upside down
Example: f(x) = ax^{2} a = 2, 1, ½, 1 
Example: f(x) = ax^{3} a = 2, 1, ½, 1 
We can use that knowledge when sketching some polynomials:
Example: Make a Sketch of y=12x^{7}
Start with the simplest "odd power" graph of x^{3}, and gradually turn it into 12x^{7}
 You know how x^{3} looks,
 x^{7} will be similar, but flatter near zero, and steeper elsewhere,
 Squash it to get 2x^{7},
 Flip it to get 2x^{7}, and
 Raise it by 1 to get 12x^{7}.
Like this:
So by doing this stepbystep we can get a good result.
Turning Points
A Turning Point is an xvalue where a local maximum or local minimum happens:
How many turning points does a polynomial have?
Never more than the Degree minus 1
The Degree of a Polynomial with one variable is the largest exponent of that variable.
Example: a polynomial of Degree 4 will have 3 turning points or less
x^{4}2x^{2}+x has 3 turning points 
x^{4}2x has only 1 turning point 
The most is 3, but there can be less.
You may not know where they are, but at least you know the most there can be!
What Happens at the Ends
And when you move far from zero:
 far to the right (large values of x), or
 far to the left (large negative values of x)
then the graph starts to resemble the graph of y = ax^{n} where ax^{n} is the term with the highest degree.
Example: f(x) = 3x^{3}4x^{2}+x
Far to the left or right, the graph will look like 3x^{3}
Near Zero, they are different 
Far From Zero, they become similar 
This makes sense, because when x is large, then x^{3} is much greater than x^{2} etc
This is officially called the "End Behavior Model".
And yes, we have come to the end!
Summary
 Graphs will be continuous and smooth
 Even exponents behave the same: above (or equal to) 0; go through (0,0), (1,1) and (1,1); larger values of n flatten out near 0, and rise more sharply.
 Odd exponents behave the same: go from negative x and y to positive x and y; go through (0,0), (1,1) and (1,1); larger values of n flatten out near 0, and fall/rise more sharply
 Factors:
 Larger values squash the curve (inwards to yaxis)
 Smaller values expand it (away from yaxis)
 And negative values flip it upside down
 Turning points: there will be "Degree1" or less.
 End Behavior: use the term with the largest exponent