Question

Consider the polynomial function p(x)=-5x^6-3x^5+4x^2+6xp(x)=−5x

6
−3x
5
+4x
2
+6xp, left parenthesis, x, right parenthesis, equals, minus, 5, x, start superscript, 6, end superscript, minus, 3, x, start superscript, 5, end superscript, plus, 4, x, squared, plus, 6, x.
What is the end behavior of the graph of p\p p, space?

Answer 1

|x| → ∞, p(x) → -∞

Explanation:

You want the end behavior of p(x)=-5x^6-3x^5+4x^2+6x.

Polynomial end behavior

The magnitude of a polynomial function will tend to infinity when the magnitude of the independent variable tends to infinity. The sign of the infinity may or may not match the sign of the independent variable. The match is determined by the sign of the leading coefficient, and the degree of the polynomial.

The leading term of p(x) is -5x^6.

Odd degree

The sign of the leading term tells you the sign of the overall slope of the polynomial. As |x| gets large, the sign of f(x) will match the sign of x for a positive leading coefficient, and will match the sign of -x for a negative leading coefficient.

Even degree

The sign of the leading term tells you the sign of f(x) as |x| gets large. For a positive leading coefficient, the overall shape is ∪. For a negative leading coefficient, the overall shape is ∩.

The given polynomial has a negative leading coefficient, and is of even degree.

|x| → ∞, p(x) → -∞