Question

compare the end behavior of $y = x^3 + 4x^2 - x - 4$ versus $y = x^4 - 10x^2 + 9$. which statement is true?
yellow box: odd degree: opposite ends, even degree: same ends
purple box: odd degree: same ends, even degree: opposite ends
orange box: end behavior depends only on zeros
cyan box: neither has predictable end behavior
blue box: both graphs have same - end behavior

Explanation:

Step1: Analyze the first function \( y = x^3 + 4x^2 - x - 4 \)

The leading term is \( x^3 \), so the degree is 3 (odd) and the leading coefficient is positive. For a polynomial with odd degree and positive leading coefficient, as \( x \to +\infty \), \( y \to +\infty \); as \( x \to -\infty \), \( y \to -\infty \) (opposite ends).

Step2: Analyze the second function \( y = x^4 - 10x^2 + 9 \)

The leading term is \( x^4 \), so the degree is 4 (even) and the leading coefficient is positive. For a polynomial with even degree and positive leading coefficient, as \( x \to +\infty \) and \( x \to -\infty \), \( y \to +\infty \) (same ends).

Step3: Evaluate the options

• The green option: "Odd degree: opposite ends, even degree: same ends" matches our analysis.
• The purple option swaps the end behaviors for odd and even degrees, so it's incorrect.
• The orange option says end behavior depends only on zeros, which is wrong (it depends on leading term).
• The cyan option says neither has predictable end behavior, which is wrong (polynomial end behavior is predictable from leading term).
• The blue option says both have same - end behavior, which is wrong (first has opposite, second has same).

Answer:

The correct option is the green - colored one with the text "Odd degree: opposite ends, even degree: same ends"