Question 1204299
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The two things we look for are: the <font color=red>leading coefficient</font> and the <font color=red>degree of the polynomial</font>.
Both of those help form the leading term.
The other terms won't affect the end behavior.


Something like y = 6x^3 + 10x^2 + 5x + 7 has the same end behavior as y = 6x^3.
This is because as x gets really really large, the leading term has the most effect compared to the other terms. 
Cubing a large number makes it much bigger compared to squaring it for example.


Consider the monomial y = a*x^n
If a > 0 and n is even, then the two endpoints rise up together.
Example: y = 3x^2
{{{graph(300,300,-5,5,-5,5,-1000,3x^2)}}}


If a < 0 and n is even, then the endpoints fall together
Example: y = -3x^2
{{{graph(300,300,-5,5,-5,5,-1000,-3x^2)}}}


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If n is odd, then the endpoints point in opposite directions.
a > 0 has the end behavior "fall to the left, rise to the right"
Example: y = 2x^5
{{{graph(300,300,-5,5,-5,5,-1000,2x^5)}}}


a < 0 has the end behavior "rise to the left, fall to the right" when n is odd.
Example: y = -5x^3
{{{graph(300,300,-5,5,-5,5,-1000,-5x^3)}}}
I encourage you to try out other examples. Use Desmos or GeoGebra as a graphing tool.

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Summary chart
<table border = "1" cellpadding = "5"><tr><td></td><td>n is even</td><td>n is odd</td></tr><tr><td>a > 0</td><td>rise together</td><td>fall to the left, rise to the right</td></tr><tr><td>a < 0</td><td>fall together</td><td>rise to the left, fall to the right</td></tr></table>
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