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# End Behavior Of Graphs

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What determines the end behavior of a graph, e.g. down and down, up and down, up and up. My math book gave me a really vague explanation of it. Can someone make it easy to explain? (I am turning my questions that get answers into a wealth of knowledge) Helping me would be very much appreciated.

Nov 20, 2017

### 3+0 Answers

#1
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There are few things to look for to determine whether the end behavior is "down and down, up and down, up and up."

1. Look at the Degree of the Polynomial Function

If the degree is odd, then the function will behave in an "up-down" behavior, if that is how you would like to think about it.

If the degree is even, then you will have to check one more thing.

2. If the Degree is Odd, then Look at the Leading Coefficient

The leading coefficient is the coefficient of the highest-degree term in the polynomial function.

If the leading coefficient is positive, the graph will have an "up-up" behavior.

If the leading coefficiennt is negative, then the corresponding graph will have a "down-down" behavior.

Hope this helps!

Nov 21, 2017
#3
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Maybe a diagram can do more explaining than I can rambling on ad infinitum.

TheXSquaredFactor  Nov 22, 2017
#2
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Here are some examples, too....

If a polynomial's degree is odd, and leading coefficient is positive:

For example:  y = x ,   which is   y = 1x1

degree  =  1 , odd

leading coefficient  =  1 , positive

As the  x  values get smaller, the  y values get smaller.

As the  x  values get bigger, the  y  values get bigger.

--------------------

If its degree is odd, and leading coefficient is negative:

For example:  y = -x ,   which is   y = -1x1

degree  =  1 , odd

leading coefficient  =  -1 , negative

As the  x  values get smaller, the  y values get bigger.

As the  x  values get bigger, the  y  values get smaller.

--------------------

If its degree is even, and leading coefficient is positive:

For example:  y = x2 ,   which is   y = 1x2

degree  =  2 , even

leading coefficient  =  1 , positive

As the  x  values get smaller, or more negative, the  y values get bigger.

As the  x  values get bigger, or more positive, the  y  values get bigger.

--------------------

If its degree is even, and leading coefficient is negative:

For example:  y = -x2 ,   which is   y = -1x2

degree  =  2 , even

leading coefficient  =  -1 , negative

As the  x  values get smaller, or more negative, the  y values get smaller.

As the  x  values get bigger, or more positive, the  y  values get smaller.

Nov 22, 2017
edited by hectictar  Nov 22, 2017
edited by hectictar  Nov 22, 2017

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