Question

1. which of the following statements about the function, $y = -\frac{1}{2}x^4 + 2x^2 - 14$, must be true?

• the leading coefficient is negative, so the graph of the function must rise to the right.
• the degree of the polynomial is even, so the left side of the graph must have the same behavior as the right side of the graph.
• the leading coefficient is positive, so the graph of the function must rise to the right.
• the degree of the polynomial is odd, so the left side of the graph must have the same behavior as the right side of the graph.

Explanation:

Step1: Identify the degree and leading coefficient

The function is \( y = -\frac{1}{2}x^4 + 2x^2 - 14 \). The degree of a polynomial is the highest power of \( x \), so here the degree is 4 (which is even). The leading coefficient is the coefficient of the term with the highest power, so that's \( -\frac{1}{2} \) (negative).

Step2: Analyze end - behavior rules

For polynomial functions, the end - behavior is determined by the degree (even or odd) and the sign of the leading coefficient.

• If the degree is even:
• If the leading coefficient is positive, both ends of the graph rise (as \( x

ightarrow+\infty \) and \( x
ightarrow-\infty \), \( y
ightarrow+\infty \)).

• If the leading coefficient is negative, both ends of the graph fall (as \( x

ightarrow+\infty \) and \( x
ightarrow-\infty \), \( y
ightarrow-\infty \)). In other words, the left - hand and right - hand behavior of the graph is the same.

• If the degree is odd:
• If the leading coefficient is positive, the graph falls to the left and rises to the right (as \( x

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

• If the leading coefficient is negative, the graph rises to the left and falls to the right (as \( x

ightarrow-\infty \), \( y
ightarrow+\infty \); as \( x
ightarrow+\infty \), \( y
ightarrow-\infty \)). In other words, the left - hand and right - hand behavior of the graph is different.

Step3: Evaluate each option

• Option 1: The leading coefficient is negative, but for an even - degree polynomial with a negative leading coefficient, the graph falls on both ends, not rises to the right. So this option is false.
• Option 2: The degree is 4 (even), and for even - degree polynomials, the left - hand and right - hand behavior of the graph is the same (both rise or both fall depending on the sign of the leading coefficient). So this option is true.
• Option 3: The leading coefficient is \( -\frac{1}{2} \), which is negative, not positive. So this option is false.
• Option 4: The degree is 4, which is even, not odd. So this option is false.

Answer:

The second option (The degree of the polynomial is even, so the left side of the graph must have the same behavior as the right side of the graph.)