Question

find the correct end behavior diagram for the given polynomial function. \\(f(x)=-2x^3+5x^2-8x+1\\)
\\(\circ\\) left - negative
right - positive
\\(\circ\\) left - positive
right - negative
\\(\circ\\) left - positive
right - positive
\\(\circ\\) left - negative
right - negative
clear my selection

Explanation:

Step1: Identify Degree and Leading Coefficient

The polynomial is \( f(x) = -2x^3 + 5x^2 - 8x + 1 \). The degree is 3 (odd), and the leading coefficient is -2 (negative).

Step2: Determine End Behavior

For odd - degree polynomials:

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

ightarrow+\infty \), \( f(x)
ightarrow+\infty \); as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \).

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

ightarrow+\infty \), \( f(x)
ightarrow-\infty \); as \( x
ightarrow-\infty \), \( f(x)
ightarrow+\infty \).

Here, degree is odd (3) and leading coefficient \( - 2<0 \). So:

• As \( x

ightarrow-\infty \) (left - hand side), \( f(x)
ightarrow+\infty \) (Positive).

• As \( x

ightarrow+\infty \) (right - hand side), \( f(x)
ightarrow-\infty \) (Negative).

Answer:

Left - Positive
Right - Negative