How to Graph a Hyperbola- Step-by-Step Instructions

What Is a Hyperbola, Anyway?

A hyperbola is a conic section — the shape you get when you slice a cone with a plane. Visually, it looks like two curved arms that open away from each other, called branches.

In algebra, you're dealing with equations like x²/a² - y²/b² = 1 or y²/a² - x²/b² = 1. The minus sign between terms is what makes it a hyperbola, not an ellipse. Remember that.

The Two Types of Hyperbolas

Hyperbolas open either horizontally (left and right) or vertically (up and down). Which one you have depends on which variable is positive in the equation.

• Horizontal: x²/a² - y²/b² = 1 — opens left and right
• Vertical: y²/a² - x²/b² = 1 — opens up and down

The variable with the positive coefficient tells you the axis of the hyperbola.

Key Parts You Need to Find

Before you graph anything, locate these components:

• Center (h, k) — the midpoint between the two branches
• Vertices — points where the hyperbola turns (a units from center)
• Co-vertices — b units from center, perpendicular to vertices
• Foci — points inside each branch, c units from center where c² = a² + b²
• Asymptotes — diagonal lines that the branches approach but never touch

Finding the Center

Complete the square for both x and y terms. The center is at (h, k) from the standard form.

For example, if your equation is (x-3)²/16 - (y+2)²/9 = 1, the center is (3, -2).

Calculating a, b, and c

The denominators under the squared terms give you a² and b² directly.

• In (x-3)²/16 - (y+2)²/9 = 1: a² = 16, b² = 9
• So a = 4 and b = 3
• c² = a² + b² = 16 + 9 = 25, so c = 5

How to Graph a Hyperbola: Step-by-Step

Step 1: Put the Equation in Standard Form

Get your equation to look like one of these:

• (x-h)²/a² - (y-k)²/b² = 1 (horizontal)
• (y-k)²/a² - (x-h)²/b² = 1 (vertical)

Complete the square if needed. Rearrange so the positive term comes first.

Step 2: Identify the Center

From the standard form, read off (h, k). This is your starting point.

Step 3: Find the Vertices

Move a units from the center along the axis of opening.

• Horizontal hyperbola: vertices at (h±a, k)
• Vertical hyperbola: vertices at (h, k±a)

Step 4: Find the Co-vertices

Move b units perpendicular to the vertices.

• Horizontal: co-vertices at (h, k±b)
• Vertical: co-vertices at (h±b, k)

Step 5: Draw the Asymptotes

This is the trickiest part. The asymptotes pass through the center with slope ±(a/b) for horizontal hyperbolas or ±(b/a) for vertical ones.

For a horizontal hyperbola: y - k = ±(b/a)(x - h)

For a vertical hyperbola: y - k = ±(a/b)(x - h)

Draw these as dashed lines — the hyperbola gets arbitrarily close but never reaches them.

Step 6: Plot Points and Sketch the Branches

Start at each vertex. The curve approaches the asymptotes as you move away from the center. Plot a few points if you're unsure, then draw smooth curves opening toward infinity.

Horizontal vs. Vertical Hyperbola Comparison

Worked Example

Graph: (y-1)²/9 - (x+3)²/4 = 1

This is a vertical hyperbola because y² is positive.

• Center: (-3, 1)
• a² = 9, so a = 3
• b² = 4, so b = 2
• c² = 9 + 4 = 25, so c = 5

Vertices: (-3, 1±3) → (-3, 4) and (-3, -2)

Co-vertices: (-3±2, 1) → (-5, 1) and (-1, 1)

Asymptotes: y - 1 = ±(3/2)(x + 3)

Plot the center, mark vertices and co-vertices, draw dashed asymptotes, then sketch the branches through the vertices, curving toward the asymptotes.

Common Mistakes to Avoid

• Confusing a and b — a is always under the positive term
• Forgetting to complete the square when the equation isn't in standard form
• Drawing asymptotes as solid lines — they're guides, not parts of the graph
• Mixing up slopes — check which variable is positive to determine orientation

Quick Reference: Graphing Checklist

• ☐ Rewrite equation in standard form
• ☐ Identify center (h, k)
• ☐ Calculate a and b from denominators
• ☐ Plot vertices at distance a from center
• ☐ Plot co-vertices at distance b from center
• ☐ Draw asymptotes through center with slopes ±(a/b) or ±(b/a)
• ☐ Sketch branches through vertices, approaching asymptotes

That's it. Find the center, plot vertices and co-vertices, draw asymptotes, and sketch the curves. Practice a few equations and it'll click.