Question

the function f(x) is defined below. what is the end behavior of f(x)?

f(x)=78x² - 2x⁴+84 - 6x³+166x

answer
attempt 1 out of 2

the degree of the polynomial is
, and the leading coefficient is
.

choose which pair of limits below represents the end behavior:

lim(x→ - ∞) f(x)=∞, lim(x→∞) f(x)=∞

lim(x→ - ∞) f(x)= - ∞, lim(x→∞) f(x)=∞

lim(x→ - ∞) f(x)=∞, lim(x→∞) f(x)= - ∞

lim(x→ - ∞) f(x)= - ∞,
lim(x→∞) f(x)= - ∞

Explanation:

Step1: Identify degree of polynomial

The degree of a polynomial is the highest - power of the variable. For \(f(x)=78x^{2}-2x^{4}+84 - 6x^{3}+166x\), the highest - power of \(x\) is \(4\), so the degree \(n = 4\).

Step2: Identify leading coefficient

The leading coefficient is the coefficient of the term with the highest degree. For \(f(x)=78x^{2}-2x^{4}+84 - 6x^{3}+166x\), the term with the highest degree is \(-2x^{4}\), and the leading coefficient \(a=-2\).

Step3: Determine end - behavior

For a polynomial \(y = a x^{n}\), when \(n\) is even and \(a\lt0\), \(\lim_{x
ightarrow-\infty}f(x)=-\infty\) and \(\lim_{x
ightarrow\infty}f(x)=-\infty\).

Answer:

The degree of the polynomial is \(4\), and the leading coefficient is \(-2\).
The pair of limits representing the end - behavior: \(\lim_{x
ightarrow-\infty}f(x)=-\infty,\lim_{x
ightarrow\infty}f(x)=-\infty\)