Question

consider the function \\(f(x) = 2x^3 + 4x^2 - 5x - 1\\). determine which of the following is its graph, based on end behavior.

Explanation:

Identify the leading term

Using the Leading Coefficient Test and Polynomial End Behavior knowledge points

$$ f(x) = 2x^3 + 4x^2 - 5x - 1 $$

The leading term is \(2x^3\).

Analyze degree and coefficient

Using the Leading Coefficient Test and Polynomial End Behavior knowledge points

• Degree \(n = 3\) (odd).
• Leading coefficient \(a_n = 2 > 0\) (positive).

Determine the end behavior

Using the Polynomial End Behavior knowledge points

• As \(x \to \infty\), \(f(x) \to \infty\).
• As \(x \to -\infty\), \(f(x) \to -\infty\).

The graph falls to the left and rises to the right.

Match with the given options

Using the Polynomial Factors from Graphs knowledge point

• Graph A falls to the left and rises to the right.
• Graph B rises to both the left and right.
• Graph C falls to both the left and right.

Therefore, Graph A is the correct representation.

Answer:

• (A) Graph falling to the left and rising to the right (Correct answer)
• (B) Graph rising to both the left and right
• (C) Graph falling to both the left and right