John R. answered • 12/18/12

John R: Math, Science, and History Teacher

Left and right hand behaviors of a function tell us what number the function approaches as x approaches positive and negative infinite.

You are currently looking at a polynomial function (a function in which each term is multiplied by the variable raised to a non-negative integer value). End behaviors of polynomials can be determined by looking at the term witht he largest exponant (in the case of this problem, we will look at -4n^{3})

**Rules for determining end behavior of a polynomial function**

Largest exponant is odd:

1. If the coefficient on the term is positive, the function approaches negative infinite as x approaches negative infinite and the function approaches positive infinite as the function approaches positive infinite

2. If the coefficient on the term is negative, the function approaches positive infinite as x approaches negative infinite and the function approaches negative infinite as the function approaches positive infinite

Largest exponant is even:

1. If the coefficient on the term is positive, the function approaches positive infinite as x approaches negative infinite and the function approaches positive infinite as the function approaches positive infinite

2. If the coefficient on the term is negative, the function approaches negative infinite as x approaches negative infinite and the function approaches negative infinite as the function approaches positive infinite

For your problem, we are looking at a largest exponant of 3, so we will be using the rules for odd exponants. Since the coeficient on n^{3} is -4, we will use rule 2, the rule for negative coefficients. The answer is:

The function approaches positive infinite as x approaches negative infinite and the function approaches negative infinite as the function approaches positive infinite.