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The cool stuff happens when we get more terms in there!

Pop this guy into a graphing calculator:

f( x ) = x^4 - 2x^3 - 5x^2 + 6x
 

Your graph should look about like this:


roller coaster graph with two valleys and one mountain
THINGS TO NOTICE:

Both tails shoot up.


One mountain


Two valleys

In Precalculus, you were mostly concerned with finding where these cross or touch the x-axis -- because, really, that's all you COULD easily find without Calculus.

Now, we'll be more interested in the mountains and valleys, where the graph is going up and down and what those tails are doing.

f( x ) = x^4 - 2x^3 - 5x^2 + 6x  ...  to keep things simple, I'm going to start calling this "plus some x stuff"
 
 
 
f( x ) = x^4 - 2x^3 - 5x^2 + 6x  ...  This guy rules the basic shape.  ...  This stuff rules the wobbles.