How do you describe the end behavior for #f(x)=-x^5+4x^3-5x-4#?

1 Answer
Oct 29, 2016

Answer:

As #xrarr-oo#, #f(x)rarr oo#
As #xrarroo#, #f(x)rarr -oo#

Explanation:

#f(x)=color(blue)(-1)x^color(red)5 +4x^3-5x-4#

End behavior is determined by the degree of the polynomial and the leading coefficient (LC).

The degree of this polynomial is the greatest exponent, or #color(red)5#.

The leading coefficient is the coefficient of the term with the greatest exponent, or #color(blue)(-1)#.

For polynomials of even degree, the "ends" of the polynomial graph point in the same direction as follows.

Even degree and positive LC:
As #xrarr-oo#, #f(x)rarr oo#
As #xrarr oo#, #f(x)rarr oo#

Even degree and negative LC:
As #xrarr-oo#, #f(x)rarr -oo#
As #xrarroo#, #f(x)rarr -oo#

For polynomials of odd degree, the "ends" of the polynomial graph point in opposite directions as follows (note, there is a saying that Odd means Opposite when graphing).

Odd degree and positive LC:
As #xrarr-oo#, #f(x)rarr -oo#
As #xrarr oo#, #f(x)rarr oo#

Odd degree and negative LC:
As #xrarr-oo#, #f(x)rarr oo#
As #xrarroo#, #f(x)rarr -oo#

In this example, the degree is odd and the leading coefficient is negative. Therefore,
As #xrarr-oo#, #f(x)rarr oo#
As #xrarroo#, #f(x)rarr -oo#