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Leading Coefficient Test

The graph of the polynomial function f ( x ) = a n x n + a n 1 x n 1 + ... + a 1 x + a 0 eventually rises or falls depends on the leading coefficient ( a n ) and the degree of the polynomial function.

Case
End Behavior of graph
When n is odd and a n is positive Graph falls to the left and rises to the right
When n is odd and a n is negative Graph rises to the left and falls to the right
When n is even and a n is positive Graph rises to the left and right
When n is even and a n is negative Graph falls to the left and right

Example:

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function f ( x ) = x 3 + 5 x .

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.

Math diagram

 

 

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