Question

1 answer: $y = -(x + 2)(x + 1)^2$ determine the end behavior:

Explanation:

Step1: Identify Degree and Leading Coefficient

First, expand the function \( y = -(x + 2)(x + 1)^2 \). Expand \( (x + 1)^2 = x^2 + 2x + 1 \), then multiply by \( (x + 2) \): \( (x + 2)(x^2 + 2x + 1) = x^3 + 2x^2 + x + 2x^2 + 4x + 2 = x^3 + 4x^2 + 5x + 2 \). Now multiply by -1: \( y = -x^3 - 4x^2 - 5x - 2 \). The degree is 3 (odd), and the leading coefficient is -1 (negative).

Step2: Apply End Behavior Rules

For a polynomial, if the degree is odd:

• If leading coefficient is positive: as \( x \to +\infty \), \( y \to +\infty \); as \( x \to -\infty \), \( y \to -\infty \).
• If leading coefficient is negative: as \( x \to +\infty \), \( y \to -\infty \); as \( x \to -\infty \), \( y \to +\infty \).

Here, degree 3 (odd), leading coefficient -1 (negative). So:

• As \( x \to +\infty \), \( y \to -\infty \)
• As \( x \to -\infty \), \( y \to +\infty \)

Answer:

As \( x \to +\infty \), \( y \to -\infty \); as \( x \to -\infty \), \( y \to +\infty \)