Equations of Hyperbolas — Quiz

Question 5 • Precalculus

Which graph represents the hyperbola the fraction with numerator x squared and denominator 5 squared minus the fraction with numerator y squared and denominator 4 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 4 comma 0) and (4 comma 0). The branches of the hyperbola open left and right. A dashed box is shown around the points (negative 4 comma 0), (0 comma 5), (4 comma 0), and (0 comma negative 5).

Image with description A four-quadrant coordinate plane. A hyperbola is shown centered at the origin. The transverse axis has endpoints at (negative 5 comma 0) and (5 comma 0). The branches of the hyperbola open left and right. A dashed box is shown around the points (negative 5 comma 0), (0 comma 4), (5 comma 0), and (0 comma negative 4).

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 5) and (0 comma 5). The branches of the hyperbola open up and down. A dashed box is shown around the points (negative 4 comma 0), (0 comma 5), (4 comma 0), and (0 comma negative 5).

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 4) and (0 comma 4). The branches of the hyperbola open up and down. A dashed box is shown around the points (negative 5 comma 0), (0 comma 4), (5 comma 0), and (0 comma negative 4).

Explanation

The equation is in the form x²/a² − y²/b² = 1, so the hyperbola is centered at the origin and opens left and right. Here a = 5, giving vertices at (−5, 0) and (5, 0), while b = 4 gives the other corners of the dashed box at (0, −4) and (0, 4). Choice D shows those vertices and opens in the correct direction.

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