Question

he corresponding number on the picture.

d
$f(x) = -2x^5 - 3x^3 + 2x - 8$
yellow
as $x \to -\infty, f(x) \to \infty$
as $x \to \infty, f(x) \to \infty$
brown
as $x \to -\infty, f(x) \to -\infty$
as $x \to \infty, f(x) \to \infty$
green
as $x \to -\infty, f(x) \to \infty$
as $x \to \infty, f(x) \to -\infty$
blue
as $x \to -\infty, f(x) \to -\infty$
as $x \to \infty, f(x) \to -\infty$

Explanation:

Step1: Identify the leading term

The function is \( f(x) = -2x^5 - 3x^3 + 2x - 8 \). The leading term is \( -2x^5 \), which has a degree of 5 (odd) and a leading coefficient of -2 (negative).

Step2: Analyze end - behavior for odd - degree functions

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

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

ightarrow-\infty \), \( f(x)
ightarrow-\infty \) and as \( x
ightarrow\infty \), \( f(x)
ightarrow\infty \).

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

ightarrow-\infty \), \( f(x)
ightarrow\infty \) and as \( x
ightarrow\infty \), \( f(x)
ightarrow-\infty \).

For our function, \( n = 5 \) (odd) and \( a_n=-2<0 \). So:

• As \( x

ightarrow-\infty \), we consider the behavior of \( -2x^5 \). When \( x
ightarrow-\infty \), \( x^5
ightarrow-\infty \), and multiplying by - 2 (a negative number) gives \( -2x^5
ightarrow\infty \) (since negative times negative is positive). The other terms \( - 3x^3,2x,-8 \) become negligible as \( |x| \) becomes very large. So as \( x
ightarrow-\infty \), \( f(x)
ightarrow\infty \).

• As \( x

ightarrow\infty \), \( x^5
ightarrow\infty \), and multiplying by - 2 gives \( -2x^5
ightarrow-\infty \). The other terms become negligible as \( |x| \) becomes very large. So as \( x
ightarrow\infty \), \( f(x)
ightarrow-\infty \).

Looking at the color - coded end - behaviors:

• YELLOW: as \( x

ightarrow-\infty,f(x)
ightarrow\infty \); as \( x
ightarrow\infty,f(x)
ightarrow\infty \) (does not match)

• BROWN: as \( x

ightarrow-\infty,f(x)
ightarrow-\infty \); as \( x
ightarrow\infty,f(x)
ightarrow\infty \) (does not match)

• GREEN: as \( x

ightarrow-\infty,f(x)
ightarrow\infty \); as \( x
ightarrow\infty,f(x)
ightarrow-\infty \) (matches our calculated end - behavior)

• BLUE: as \( x

ightarrow-\infty,f(x)
ightarrow-\infty \); as \( x
ightarrow\infty,f(x)
ightarrow-\infty \) (does not match)

Answer:

The function \( f(x)=-2x^5 - 3x^3+2x - 8 \) has the end - behavior that matches the GREEN category. So the answer is GREEN.