Algebra 2 · Unit 8 · Topic 8.5
Hyperbolas and Asymptotes
For (x − h)²/a² − (y − k)²/b² = 1 with a, b > 0, a hyperbola opens left and right and has asymptotes y − k = ±(b/a)(x − h).
Definition and explanation
For (x − h)²/a² − (y − k)²/b² = 1 with a, b > 0, a hyperbola opens left and right and has asymptotes y − k = ±(b/a)(x − h). Exchanging the positive squared term changes its orientation.
Worked example
For x²/16 − y²/9 = 1, the center is (0, 0), the vertices are (±4, 0), and the asymptotes are y = ±3x/4. The point (5, 9/4) satisfies the equation because 25/16 − 9/16 = 1.
Common mistake
The difference of squared terms distinguishes a hyperbola from an ellipse; its asymptotes are not part of the curve.
Key takeaway
For (x − h)²/a² − (y − k)²/b² = 1 with a, b > 0, a hyperbola opens left and right and has asymptotes y − k = ±(b/a)(x − h).
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.