Final answer:
The given equation y² - 4x + 6y + 1 =0 is a conic section that represents a parabola because it has a y² term but no x² term.
Step-by-step explanation:
The equation provided by the student, y² - 4x + 6y + 1 =0, can be identified as a conic section. To determine which type of conic section it is, we need to rearrange the equation and complete the square if necessary. In this case, the equation does not include an x² term and has a y² term, suggesting it is most likely a parabola. The general form of a parabolic equation is y = ax + bx², but our provided equation has only linear terms in x and a square term in y, which also suggests it is a parabola. Therefore, the equation y² - 4x + 6y + 1 =0 represents a parabola.