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My question is somewhat dumb but for some reason I haven't been able to come up with an answer.

Why is the hyperbola round at the bottom? Namely, I'm thinking of any of these two equations x^2-y^2=1 or y^2-x^2=1

These two behave like a line as you approach infinity but then becomes round at the bottom. Why do they behave roundy at the bottom?

edit: I completely understand why this must be the case if I consider the hyperbola as a conic section. But I'm specifically talking about the equation for a hyperbola.

Why is the hyperbola round at the bottom? Namely, I'm thinking of any of these two equations x^2-y^2=1 or y^2-x^2=1

These two behave like a line as you approach infinity but then becomes round at the bottom. Why do they behave roundy at the bottom?

edit: I completely understand why this must be the case if I consider the hyperbola as a conic section. But I'm specifically talking about the equation for a hyperbola.

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