+577 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.
+2446 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!
+2446 Maybe a diagram can do more explaining than I can rambling on ad infinitum.
+9479 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.
