What do the left and right behaviors (arrows of graph)of the function f(x)=-4n^3--5n^2+7n-4 look like?

Question

Can someone show me how to do this?

Answer 1

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 -4n³)

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 n³ 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.

 

Answer 2

the highest powered term dominates with large or small n

as n gets very large, -4n^3 approaches negative infinity

as n gets very small -4n^3 approaches positive infinity