Question

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

f(x)=315x^{2}+77x + 91x^{3}+7x^{4}-490

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 \(f(x)=7x^{4}+91x^{3}+315x^{2}+77x - 490\), the degree \(n = 4\).

Step2: Identify leading coefficient

The leading coefficient is the coefficient of the term with the highest - power of the variable. For \(f(x)=7x^{4}+91x^{3}+315x^{2}+77x - 490\), the leading coefficient \(a = 7\).

Step3: Determine end - behavior

For a polynomial \(y = a x^{n}\), when \(n\) is even and \(a>0\), \(\lim_{x
ightarrow-\infty}f(x)=\infty\) and \(\lim_{x
ightarrow\infty}f(x)=\infty\). Since \(n = 4\) (even) and \(a = 7>0\), the end - behavior is \(\lim_{x
ightarrow-\infty}f(x)=\infty\) and \(\lim_{x
ightarrow\infty}f(x)=\infty\).

Answer:

The degree of the polynomial is \(4\), and the leading coefficient is \(7\). The pair of limits that represents the end - behavior is \(\lim_{x
ightarrow-\infty}f(x)=\infty,\lim_{x
ightarrow\infty}f(x)=\infty\)