Question

question
the function $f(x)$ is defined below. what is the end behavior of $f(x)$?
$f(x) = -84x^3 - 450 - 6x^4 - 780x - 408x^2$
answer
\\(\circ\\) as $x \to \infty, y \to -\infty$ and
as $x \to -\infty, y \to -\infty$
\\(\circ\\) as $x \to \infty, y \to \infty$ and
as $x \to -\infty, y \to \infty$
\\(\circ\\) as $x \to \infty, y \to \infty$ and
as $x \to -\infty, y \to -\infty$
\\(\circ\\) as $x \to \infty, y \to -\infty$ and
as $x \to -\infty, y \to \infty$

Explanation:

Step1: Identify the leading term

The function is \( f(x) = -84x^3 - 450 - 6x^4 - 780x - 408x^2 \). To find the end - behavior, we look at the leading term (the term with the highest degree). First, we re - order the terms of the polynomial in descending order of degrees: \( f(x)=- 6x^4-84x^3 - 408x^2-780x - 450 \). The leading term is \( -6x^4 \), where the degree \( n = 4 \) (even) and the leading coefficient \( a=-6 \) (negative).

Step2: Analyze the end - behavior based on the leading term

For a polynomial function of the form \( f(x)=a_nx^n + a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0 \):

• If the degree \( n \) is even:
• If the leading coefficient \( a_n>0 \), as \( x

ightarrow\infty \), \( f(x)
ightarrow\infty \) and as \( x
ightarrow-\infty \), \( f(x)
ightarrow\infty \) (because \( x^n \) is positive when \( n \) is even, whether \( x \) is positive or negative).

• If the leading coefficient \( a_n<0 \), as \( x

ightarrow\infty \), \( f(x)
ightarrow-\infty \) and as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \) (because \( x^n \) is positive when \( n \) is even, and multiplying by a negative coefficient makes the whole term negative).

Since our leading term is \( -6x^4 \) ( \( n = 4 \) even, \( a=-6<0 \) ), as \( x
ightarrow\infty \), \( f(x)
ightarrow-\infty \) and as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \).

Answer:

as \( x \to \infty,y \to -\infty \) and as \( x \to -\infty,y \to -\infty \) (the first option in the given choices)