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I know, you've worked with polynomials before, but we need to make sure that you really understand how their shapes work.  I definitely take an artist's approach to these critters!

Understanding shapes and graphs will actually help you to know what some answers in Calculus will be before you crunch them!

I like to call the graphs of polynomials "roller coaster graphs."

roller coaster graph with one mountain and one valleyroller coaster graph with two valleys and one mountainroller coaster graph with two valleys and two mountains

  

Let's review even degree polynomials first.  Remember that these
would be polynomials whose leading term isx^2,x^4,x^6
etc.

Starting us off is Standard Parabola Guy.  Let's really pay close attention to his basic shape.  (I'm just going to draw rough sketches here.)

f( x ) = x^2

Standard Parabola Guy


The tails both shoot up...




We've got one valley.
This guy touches the x-axis once...  It doesn't cross,
it just touches (a kiss). 

Remember that this is a multiplicity of 2 (

x^2

)