Question

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

f(x)=9x^5 - 18x^4 - 9216 + 2016x^2 - 504x^3 + 1152x

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 and leading coefficient

The degree of the polynomial $f(x)=9x^{5}-18x^{4}-9216 + 2016x^{2}-504x^{3}+1152x$ is 5 (the highest - power of $x$), and the leading coefficient is 9.

Step2: Determine end - behavior

For a polynomial $y = a_nx^n+\cdots+a_0$ with $n$ odd and $a_n>0$, as $x\to+\infty$, $y\to+\infty$ and as $x\to-\infty$, $y\to-\infty$. Here, $n = 5$ (odd) and $a_n=9>0$. So, $\lim_{x\to+\infty}f(x)=+\infty$ and $\lim_{x\to-\infty}f(x)=-\infty$.

Answer:

The degree of the polynomial is odd, and the leading coefficient is 9. The pair of limits that represents the end - behavior is $\lim_{x\to+\infty}f(x)=\infty$, $\lim_{x\to-\infty}f(x)=-\infty$