A polynomial function extends from Quadrant III to Quadrant I.Which of the following equations could represent this function?

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The question asks us to identify which polynomial could represent a function that extends from Quadrant III to Quadrant I.
Option A: f(x) = 4x^4 − 9x^2 + 1
This is a fourth-degree polynomial (even degree) with positive leading coefficient. As x → ±∞, the x^4 term dominates and f(x) → +∞ on both ends. That means the end behavior goes to Quadrant I and Quadrant II, not from Quadrant III to Quadrant I. So this option cannot represent a curve that travels from Quadrant III to Quadrant I.
Option B: f(x) = 3x^3 − 6x + 2
This is a cubic (odd degree). For odd-degree polynomials, the end behavior is opposite on the two ends: as x → −∞, f(x) → −∞ (Quadrant III), and as x → ∞, f(x) → ∞ (Quadrant I). This matches a curve that starts in Quadrant III and ends in Quadrant I, which is consistent with the described behavior. The lower-degree terms do not change this fundamental end behavior.
Option C: f(x) = −5x^3 + 7x^2 − 4
This is also a cubic, but with a negative leading coefficient. Its end behavior is reversed: as x → −∞, f(x) → ∞ (Quadrant II or I depending on the y), and as x → ∞, f(x) → −∞ (Quadrant III or IV). That does not move from Quadrant III to Quadrant I in the standard sense, so it does not fit the description.
Option D: f(x) = −2x^4 + 5x^2 − 3
This is a fourth-degree polynomial with negative leading coefficient. As x → ±∞, f(x) → −∞, so the ends go to Quadrant III/IV, not from Quadrant III to Quadrant I. Therefore this cannot represent the required path.
In summary, among the choices, only the cubic with positive leading coefficient (Option B) has end behavior that starts in Quadrant III and ends in Quadrant I, aligning with the given description.

MHF4U - Advanced Functions 12 (2025-26) - A

A polynomial function extends from Quadrant III to Quadrant I.Which of the following equations could represent this function?

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