Question

use the given graph of the polynomial function to state whether the leading coefficient is positive or negative and whether the polynomial function is cubic or quartic.

options:

• positive, cubic
• negative, quartic
• negative, cubic
• positive, quartic

Explanation:

Step1: Analyze End Behavior

For a polynomial, the end behavior is determined by the leading term \(a_nx^n\). If the graph rises on both ends (as \(x \to \pm\infty\)), the leading coefficient \(a_n\) is positive and the degree \(n\) is even. If it falls on both ends, \(a_n\) is negative and \(n\) is even. If one end rises and the other falls, \(n\) is odd. Here, the graph rises on both ends (as \(x \to \infty\) and \(x \to -\infty\)), so degree is even (quartic, since cubic is odd) and leading coefficient is positive.

Step2: Determine Degree Type

Cubic functions (degree 3, odd) have opposite end behaviors (one end up, one end down). Quartic functions (degree 4, even) have same end behaviors. The graph here has same end behaviors (both up), so it's quartic. And since it rises on both ends, leading coefficient is positive.

Answer:

Positive, Quartic