Whether the graph of a polynomial rises or falls can be determined by the Leading Coefficient Tests.

P(x)  =  anxn + an-1xn-1 +............. a1x + a0

In the above polynomial, n is the degree and an is the leading coefficient.

 Case End Behavior of Graph

When n is odd and an is positive

Graph falls to the left and rises to the right

When n is odd and an is negative

Graph rises to the left and falls to the right

When n is even and an is positive

Graph rises to the left and right

When n is even and an is negative

Graph falls to the left and right

## Examples

Example 1 :

Find the right-hand and left-hand behaviors of the graph of

f(x)  =  x5 + 2x3 - 3x + 5

Solution :

Because the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right as shown in the figure. Example 2 :

Determine the end behavior of the graph of the polynomial function below using Leading Coefficient Test.

P(x)  =  -x3 + 5x

Solution :

Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right as shown in the figure. Example 3 :

Determine the end behavior of the graph of the polynomial function below using Leading Coefficient Test.

P(x)  =  2x2 - 2

Solution :

Because the degree is even and the leading coefficient is positive, the graph rises to the left and right as shown in the figure. Example 4 :

Determine the end behavior of the graph of the polynomial function below using Leading Coefficient Test.

P(x)  =  -x2 + 1

Solution :

Because the degree is even and the leading coefficient is negative, the graph falls to the left and right as shown in the figure.  Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

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