LEADING COEFFICIENT TEST

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.

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