Equations of Hyperbolas — Quiz

Question 7 • Precalculus

Which graph represents the hyperbola the fraction with numerator y squared and denominator 6 squared minus the fraction with numerator x squared and denominator 2 squared is equal to 1 ?

Answer

Image with description A four-quadrant coordinate plane. A hyperbola is shown centered at the origin. The transverse axis has endpoints at (negative 6 comma 0) and (6 comma 0). The branches of the hyperbola open left and right. A dashed box is shown around the points (negative 6 comma 0), (0 comma 2), (6 comma 0), and (0 comma negative 2).

Image with description A four-quadrant coordinate plane. A hyperbola is shown centered at the origin. The transverse axis has endpoints at (0 comma negative 6) and (0 comma 6). The branches of the hyperbola open up and down. A dashed box is shown around the points (negative 2 comma 0), (0 comma 6), (2 comma 0), and (0 comma negative 6).

Image with description A four-quadrant coordinate plane. A hyperbola is shown centered at the origin. The transverse axis has endpoints at (negative 2 comma 0) and (2 comma 0). The branches of the hyperbola open left and right. A dashed box is shown around the points (negative 2 comma 0), (0 comma 6), (2 comma 0), and (0 comma negative 6).

Image with description A four-quadrant coordinate plane. A hyperbola is shown centered at the origin. The transverse axis has endpoints at (0 comma negative 2) and (0 comma 2). The branches of the hyperbola open up and down. A dashed box is shown around the points (negative 6 comma 0), (0 comma 2), (6 comma 0), and (0 comma negative 2).

Explanation

The equation is y²/6² − x²/2² = 1, which matches the vertical hyperbola form y²/a² − x²/b² = 1. Here a = 6, so the vertices are (0, 6) and (0, −6), and the branches open upward and downward. Choice D shows those vertices and that orientation.

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