Hyperbola Calculator — Vertices, Foci & Asymptotes

Use this free hyperbola calculator to analyze a hyperbola in standard form. It returns the center, vertices, foci, asymptotes, and eccentricity, with a diagram and steps.

Key parts of a hyperbola

The vertices are a units from the center along the transverse axis; the foci are c = √(a² + b²) units away; the asymptotes have slope ±b/a (horizontal) and eccentricity is e = c/a.

How to use it

1. Choose horizontal or vertical.
2. Enter the center and a, b.
3. Press Analyze.

Related: the Ellipse Calculator and more free math calculators.

Frequently asked questions

What is the equation of a hyperbola?

In standard form, a hyperbola centered at (h, k) that opens left and right is (x − h)²/a² − (y − k)²/b² = 1. One that opens up and down is (y − k)²/a² − (x − h)²/b² = 1. The positive term tells you the direction of the transverse axis. For x²/9 − y²/16 = 1, the x-term is positive, so the hyperbola opens left and right.

How do you calculate a hyperbola?

Start from standard form and read off the center (h, k), a and b. The vertices are a units from the center along the transverse axis, c = √(a² + b²) gives the foci, and the asymptotes pass through the center with slopes ±b/a for a horizontal hyperbola. For x²/9 − y²/16 = 1, a = 3 and b = 4, so the vertices are (±3, 0).

How do you calculate hyperbola foci?

Use c² = a² + b², which is different from the ellipse rule because here you add. Then place the foci c units from the center along the transverse axis. For x²/9 − y²/16 = 1, c = √(9 + 16) = √25 = 5, so the foci are at (−5, 0) and (5, 0). The foci always lie farther from the center than the vertices.

How do you calculate a hyperbola asymptote?

For a horizontal hyperbola, the asymptotes are y − k = ±(b/a)(x − h). For a vertical one, they are y − k = ±(a/b)(x − h). With x²/9 − y²/16 = 1, the center is the origin and b/a = 4/3, so the asymptotes are y = (4/3)x and y = −(4/3)x. A quick check: set the right side to 0 and solve for y.

How do you find the eccentricity of a hyperbola?

Divide c by a: e = c/a, where c = √(a² + b²). Since c is always larger than a for a hyperbola, the eccentricity is always greater than 1. For x²/9 − y²/16 = 1, c = 5 and a = 3, so e = 5/3 ≈ 1.67. A larger eccentricity means the branches open more widely.

How do you graph a hyperbola on a graphing calculator?

Most graphing calculators need y by itself, so solve for y and enter both branches. For x²/9 − y²/16 = 1, that gives y = 4√(x²/9 − 1) and y = −4√(x²/9 − 1) as two separate entries. Tools like Desmos let you type the equation directly as written. Adding the asymptotes as extra lines helps you check the shape.

How to use the Hyperbola Calculator for homework

The Hyperbola Calculator is most useful when you treat it as a learning check, not just a shortcut to the final answer. Start by copying the original problem carefully, including signs, exponents, decimal points, fractions, parentheses, and units. Then enter the values in the same order the problem gives them. A small typing change can completely change the result, especially in algebra, statistics, geometry, and probability problems.

Before you press the button to calculate, make a quick estimate or prediction. The estimate does not need to be exact. Its job is to help you notice impossible answers. If a distance becomes negative, a probability is bigger than 1, an angle looks too large, or a decimal point seems misplaced, go back and check the input before trusting the final result.

Before you enter the problem

• Rewrite the problem in a clean line so every value is easy to see.
• Use parentheses around grouped expressions, especially in fractions and exponents.
• Keep units with the numbers while you work, even if the calculator only asks for the numbers.
• Check whether the problem wants an exact value, a decimal approximation, or a rounded answer.
• Look for restrictions such as positive values only, a chosen interval, or a required domain.

How to read the result

After the calculator gives a result, read more than the final line. If steps, tables, graphs, or intermediate values are shown, use them to understand how the answer was built. That is especially important when you are studying for a quiz or test, because teachers often give more credit for a correct process than for an unsupported number.

Try to identify the main idea behind the result. For example, ask yourself which formula was used, which operation changed the expression, which value controlled the graph, or which assumption made the answer possible. When you can explain that idea in your own words, the tool has helped you learn the skill instead of only checking one problem.

Common mistakes to avoid

Turn the answer into practice

One good way to study is to solve the problem by hand first, then use the Hyperbola Calculator to check your work. If your answer is wrong, do not erase everything immediately. Find the first line where your work stops matching the calculator’s logic. That line is usually where the real misunderstanding happened.

1. Work the problem on paper and circle your final answer.
2. Use the calculator to check the result.
3. If the answers match, write one sentence explaining the method.
4. If they do not match, compare each step until you find the first difference.
5. Redo a similar problem without the tool to make sure the correction sticks.

When to use a calculator and when to work by hand

Use the Hyperbola Calculator when you want to check a long calculation, explore a pattern, test a graph, or confirm a result after practicing. Work by hand when the assignment asks for steps, when you are learning a new method, or when the test will not allow a digital tool. The strongest students use both: hand work to build understanding and calculators to check accuracy.

If you are preparing for a timed test, practice some problems without the tool and some with it. That balance helps you build speed while still understanding the math. Over time, the goal is to need the calculator less often for routine steps and use it more strategically for checking, exploring, and confirming your reasoning.

Quick accuracy checklist

Before you leave the page, use the Hyperbola Calculator result to answer three quick questions: What was the input? What rule, formula, or method connects the input to the result? What would change if one number in the problem changed? These questions turn a single calculator answer into a reusable math skill.

For homework, write the original problem, the calculator’s result, and one short explanation of the method in your notebook. For test review, cover the result and try to reproduce the setup on your own. If you can set up a similar problem without looking back at the calculator, you are much closer to owning the skill.