- 1). Check the exponents of the variables of your conic section equation. If only one of them is raised to the second power, or squared, and the other is not, the equation is that of a parabola. A parabola is a simple arc. If the y coordinate is squared, it has a horizontal vertex, meaning the parabola opens to the side, either left or right. If the x coordinate is squared, the parabola has a vertical vertex, and opens either up or down.
- 2). Check the sign of squared variables. If both the variables are squared but one is positive and the other is negative, the curve is a hyperbola. A hyperbola looks like a pair of arcs, very much like a parabola. However, its branches have a special feature: they are asymptotes. This means as they run on, they always approach closer and closer to certain values while never actually reaching them. If the coefficient of the squared x coordinate is larger than that of the squared y coordinate, the hyperbola is horizontal, opening to the sides. If the coefficient of the squared y coordinate is larger, the hyperbola is vertical, opening up and down.
- 3). Check the sign of the squared variables and the value of their coefficients. If both of the squared variables are positive and the coefficients are different, the curve is an ellipse. An ellipse looks like a squashed circle, longer in one direction than it is in the other. If the value of the x-squared coefficient is larger than that of the y-squared coefficient, the ellipse is horizontal on its long axis. If it is the other way around, the ellipse is vertical on its long axis. If both the x-squared and y-squared variables are positive and the coefficients are equal, the equation is that of a circle.
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