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.

OfficialBubbleTanks
Nov 20, 2017

#1**+3 **

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!

TheXSquaredFactor
Nov 21, 2017

#3**+2 **

Maybe a diagram can do more explaining than I can rambling on ad infinitum.

TheXSquaredFactor
Nov 22, 2017

#2**+3 **

Here are some examples, too....

If a polynomial's degree is __odd__, and leading coefficient is __positive__:

For example: ** y = x **, which is **y = 1x ^{1}**

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 = -1x ^{1}**

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 = x ^{2}** , which is

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 = -x ^{2}** , which is

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.

hectictar
Nov 22, 2017