Question

1. determine the end behavior for the polynomial function $f(x) = -4x^3 + 2x^2 - 6x + 1$.

Explanation:

Step1: Identify Degree and Leading Coefficient

The polynomial \( f(x) = -4x^3 + 2x^2 - 6x + 1 \) has a degree of 3 (odd) and a leading coefficient of \(-4\) (negative).

Step2: Apply End Behavior Rules

For polynomials:

• If degree is odd and leading coefficient is positive: As \( x \to +\infty \), \( f(x) \to +\infty \); as \( x \to -\infty \), \( f(x) \to -\infty \).
• If degree is odd and leading coefficient is negative: As \( x \to +\infty \), \( f(x) \to -\infty \); as \( x \to -\infty \), \( f(x) \to +\infty \).

Here, degree is odd (3) and leading coefficient is negative (\(-4\)). So:

• As \( x \to +\infty \), \( f(x) \to -\infty \) (right end down).
• As \( x \to -\infty \), \( f(x) \to +\infty \) (left end up).

Looking at the options, the second option (with left arrow up, right arrow down) matches this behavior.

Answer:

The second option (with left arrow up, right arrow down)