My Physics book has many graphs. Some are straight lines, some parabolas while others are hyperbolas. I have not studied these curves (conic sections) yet and to me parabola and hyperbola look just the same. Is there any way of knowing whether a line is a parabola or a hyperbola just by seeing the graph of the line.
How to know the form of a line in a graph without any calculation?
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conic-sections
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0Hyperbolas have asymptotes whereas parabolas aren't usually bounded in the $x$ direction (or $y$ direction if you consider those parabolas) If you want to do a little calculation, then choose three points on the graph. Since there's a unique parabola that fits those three points, find it, draw it and if it matches, it was a parabola. Otherwise, it was some other function (possibly a hyperbola). – 2017-01-13
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0It really depends how carefully and completely the graph is drawn. If you zoom in close enough on the vertex of a parabola, it's practically indistinguishable even from a circle, although if you zoom out far enough the difference will become obvious. – 2017-01-13
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0It would seem preferable to refer to lines as special kinds of curves, rather than to refer (as the title seems to) to curves as "lines". – 2017-01-14
2 Answers
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As you go out from the vertex (turning-point) of a parabola, tangents to opposite sides of the curve approach parallelism. With a hyperbola there's a limit to how small an angle the tangents can make with each other--the angle of the "asymptotes". Parabolas are more u-shaped, hyperbolas more v-shaped.
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0Can you add an illustration? – 2017-01-13
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This isn't an infallible method, but every hyperbola has two asymptotes, whereas parabolas don't have even one.