Question

1. which of the following is the graph of an even degree polynomial with a positive lead coefficient?

Explanation:

Analyze the degree of the polynomial

Using the Degree of Polynomial and Polynomial Classification knowledge points
An even-degree polynomial has end behaviors that point in the same direction on both the left and right sides of the graph. This means as \(x \to \infty\) and \(x \to -\infty\), the function values \(y\) will either both approach \(\infty\) or both approach \(-\infty\).

Analyze the leading coefficient

The leading coefficient is positive. When the leading coefficient of an even-degree polynomial is positive, the graph opens upwards on both ends. Mathematically, this is expressed as:

$$ \lim_{x \to \infty} f(x) = \infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = \infty $$

This corresponds to a graph where both the left-hand and right-hand tails point upwards (forming a general "U" or "W" shape).

Evaluate the given options

• First option: The left tail points down and the right tail points up. This represents an odd-degree polynomial with a positive leading coefficient.
• Second option: Both the left and right tails point up (forming a "W" shape). This matches an even-degree polynomial with a positive leading coefficient.
• Third option: The left tail points up and the right tail points down. This represents an odd-degree polynomial with a negative leading coefficient.
• Fourth option: Both the left and right tails point down (forming an "M" shape). This represents an even-degree polynomial with a negative leading coefficient.

Answer:

• (A) Graph with left tail pointing down and right tail pointing up
• (B) Graph with both tails pointing up (W-shape) (Correct answer)
• (C) Graph with left tail pointing up and right tail pointing down
• (D) Graph with both tails pointing down (M-shape)