Hyperbolas in Standard Form

Learn to write and graph hyperbolas in standard form, identify key features like center, vertices, foci, and asymptotes.

Advanced30 minLesson
Printable worksheet

Definition

A hyperbola is the set of all points where the absolute difference of distances from two fixed points (foci) is constant.

Standard Forms

Horizontal transverse axis (opens left and right):
(x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1
Vertical transverse axis (opens up and down):
(y−k)2a2−(x−h)2b2=1\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1
Where:
  • (h,k)(h, k) is the center
  • aa is the distance from center to each vertex
  • bb is used to find the asymptotes
  • c=a2+b2c = \sqrt{a^2 + b^2} is the distance from center to each focus
Key relationship: For hyperbolas, c2=a2+b2c^2 = a^2 + b^2 (unlike ellipses where c2=a2−b2c^2 = a^2 - b^2)

Try it now

Which equation represents a hyperbola with a horizontal transverse axis?

Worked Examples

Find the center, vertices, foci, and asymptotes of (x−2)216−(y+3)29=1\frac{(x-2)^2}{16} - \frac{(y+3)^2}{9} = 1
1

Identify the form

The xx term is positive, so this has a horizontal transverse axis (opens left/right) → Horizontal hyperbola
2

Find the center (h,k)(h, k)

From (x−2)2(x-2)^2 and (y−(−3))2(y-(-3))^2: h=2h = 2, k=−3k = -3 → Center: (2,−3)(2, -3)
3

Find aa and bb

a2=16a^2 = 16, so a=4a = 4; b2=9b^2 = 9, so b=3b = 3 → a=4a = 4, b=3b = 3
4

Find the vertices

Vertices are aa units left and right of center: (2±4,−3)(2 \pm 4, -3) → Vertices: (−2,−3)(-2, -3) and (6,−3)(6, -3)
5

Find cc for the foci

c=a2+b2=16+9=25=5c = \sqrt{a^2 + b^2} = \sqrt{16 + 9} = \sqrt{25} = 5 → c=5c = 5
6

Find the foci

Foci are cc units from center along transverse axis: (2±5,−3)(2 \pm 5, -3) → Foci: (−3,−3)(-3, -3) and (7,−3)(7, -3)
7

Find the asymptotes

For horizontal: y−k=±ba(x−h)y - k = \pm\frac{b}{a}(x - h), so y+3=±34(x−2)y + 3 = \pm\frac{3}{4}(x - 2) → Asymptotes: y=34x−92y = \frac{3}{4}x - \frac{9}{2} and y=−34x−32y = -\frac{3}{4}x - \frac{3}{2}

Common Mistakes

Confusing a2a^2 and b2b^2 positions in horizontal vs vertical forms

Why it's wrong: In standard form, a2a^2 is always under the POSITIVE term, regardless of whether that's xx or yy.

Correct: Identify which variable has the positive term first. That tells you the orientation, and a2a^2 is under that term.

Using c2=a2−b2c^2 = a^2 - b^2 (the ellipse formula) instead of c2=a2+b2c^2 = a^2 + b^2

Why it's wrong: Hyperbolas and ellipses have different relationships between aa, bb, and cc.

Correct: For hyperbolas, c>ac > a and c2=a2+b2c^2 = a^2 + b^2. For ellipses, c<ac < a and c2=a2−b2c^2 = a^2 - b^2.

Getting asymptote slopes backwards for vertical hyperbolas

Why it's wrong: The asymptote formula changes depending on orientation.

Correct: Horizontal: slope =±ba= \pm\frac{b}{a}. Vertical: slope =±ab= \pm\frac{a}{b}. Remember: aa is always the denominator in the direction the hyperbola opens.

Forgetting the ±\pm when locating vertices and foci

Why it's wrong: Hyperbolas have two branches, so there are two vertices and two foci.

Correct: Always find BOTH vertices and BOTH foci by adding and subtracting the appropriate distance from the center.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Hyperbolas Explained | Vertices, Foci, Asymptotes, and Equations

by Understand The Math

Watch on YouTube

Interactive Visual

Function graph

-10-50510-10-505xyCF₁F₂
  • c = √(a² + b²) = 5
  • b/a = 0.75
Asymptote(x − h)²/a² − (y − k)²/b² = 1
  • Center: (2, -3)
  • Foci: (-3, -3), (7, -3)
  • Vertices: (-2, -3), (6, -3)
  • Asymptotes: y = 0.75x - 4.5, y = -0.75x - 1.5

Function graph

-15-10-505-10-50510xyCF₁F₂
Asymptote(y − 1)²/25 − (x + 4)²/16 = 1
  • Center: (-4, 1)
  • Foci: (-4, -5.4), (-4, 7.4)
  • Vertices: (-4, -4), (-4, 6)
  • Asymptotes: y = 1.25x + 6, y = -1.25x - 4

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = -3 + 0.75*sqrt((x - 2)^2 - 16)

y = -3 - 0.75*sqrt((x - 2)^2 - 16)

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Practice Problems

15 problems
Problem 1 of 15
Easy
Which equation represents a hyperbola with a horizontal transverse axis?

Why It Matters

Hyperbolas appear throughout science and engineering:
  • Navigation: LORAN (Long Range Navigation) uses hyperbolic curves from radio signals to determine ship and aircraft positions
  • Astronomy: Some comets follow hyperbolic orbits, passing the sun once and never returning
  • Acoustics: Whispering galleries and sonic boom patterns follow hyperbolic shapes
  • Physics: The path of alpha particles scattered by atomic nuclei traces hyperbolic curves
  • Architecture: Hyperbolic cooling towers are structurally efficient and iconic landmarks
Understanding standard form lets you extract all key information about a hyperbola directly from its equation!

Real World Applications

LORAN Navigation

Long Range Navigation uses the time difference of radio signals from two stations. All points with the same time difference form a hyperbola with the stations as foci.

Example:

If two LORAN stations are 300 miles apart and a ship receives signals with a time difference corresponding to 100 miles, the ship lies on a hyperbola with 2a=1002a = 100 miles.

1Try It Yourself

Two radio stations are at (−150,0)(-150, 0) and (150,0)(150, 0). A ship measures that it is 100 miles closer to one station than the other.

What is the equation of the hyperbola the ship lies on?

Step 1: Write the mathematical expression

The difference in distances is 2a=1002a = 100, so a=50a = 50. With c=150c = 150:

Cooling Tower Design

Hyperbolic cooling towers use the shape's structural strength. The hyperbolic profile allows thin shells to support themselves.

Example:

A cooling tower has a waist (narrowest point) diameter of 60 meters at height 50 meters, with the hyperboloid equation based on standard form.

2Try It Yourself

A cooling tower cross-section follows x2900−(y−50)21600=1\frac{x^2}{900} - \frac{(y-50)^2}{1600} = 1 (in meters). Find the waist width.

What is the diameter at the narrowest point (the vertices)?

Step 1: Write the mathematical expression

The vertices occur at y=50y = 50 (center height). Find xx at y=50y = 50:

Key Takeaways

  • 1Hyperbolas have two standard forms: horizontal (x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 and vertical (y−k)2a2−(x−h)2b2=1\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1
  • 2The positive term determines the transverse axis direction (horizontal or vertical)
  • 3aa is the distance from center to vertex, and a2a^2 is always under the positive term
  • 4For hyperbolas, c2=a2+b2c^2 = a^2 + b^2 where cc is the distance from center to focus
  • 5Asymptotes pass through the center with slopes ±ba\pm\frac{b}{a} (horizontal) or ±ab\pm\frac{a}{b} (vertical)

Frequently Asked Questions

Look at which variable has the POSITIVE coefficient. If x2x^2 is positive, the hyperbola opens left/right (horizontal). If y2y^2 is positive, it opens up/down (vertical).
Look at which variable has the POSITIVE coefficient. If x2x^2 is positive, the hyperbola opens left/right (horizontal). If y2y^2 is positive, it opens up/down (vertical).
In an ellipse, the foci are between the vertices (c<ac < a). In a hyperbola, the foci are beyond the vertices (c>ac > a). The different formulas reflect this geometric relationship.
Asymptotes are lines the hyperbola approaches but never touches as it extends to infinity. They help you sketch the hyperbola and define the 'box' that contains the vertices.

Glossary

Hyperbola
The set of all points where the absolute difference of distances from two fixed points (foci) is constant
Transverse axis
The line segment connecting the two vertices, passing through the center
Conjugate axis
The line segment perpendicular to the transverse axis at the center, with length 2b2b
Asymptote
A line that the hyperbola approaches but never intersects as it extends to infinity
Focus (plural: foci)
One of two fixed points used to define the hyperbola; distance from center is c=a2+b2c = \sqrt{a^2 + b^2}
Today's challengeOne problem, three tries

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