Parabola Calculator

About Parabola Calculator

The Parabola Calculator finds all key properties of any parabola: vertex, focus, directrix, axis of symmetry, latus rectum length, and opening direction. It supports three equation forms — standard, vertex, and conic — for both vertical and horizontal parabolas. Results include step-by-step solutions and an interactive graph showing every component.

How to Use the Parabola Calculator

1. Choose the equation form: Select Standard Form (\(y = ax^2 + bx + c\)), Vertex Form (\(y = a(x-h)^2 + k\)), or Conic Form (\((x-h)^2 = 4p(y-k)\)).
2. Select the orientation: Choose Vertical (opens up/down) or Horizontal (opens left/right).
3. Enter the coefficients: Fill in the values for your chosen form. Use the quick examples above the form to try preset equations.
4. Click "Calculate Parabola" to see results including vertex, focus, directrix, and more.
5. Explore the interactive graph: The color-coded diagram shows the parabola curve, vertex (red), focus (amber), directrix (green dashed), and latus rectum (cyan).

What Is a Parabola?

A parabola is a U-shaped curve defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). It is one of the four conic sections, formed when a cone is cut by a plane parallel to its side. Every parabola has an eccentricity of exactly 1.

Forms of the Parabola Equation

There are three common ways to express a parabola's equation, each useful for different purposes:

• Standard Form: \(y = ax^2 + bx + c\) — Useful for finding y-intercepts and working with polynomial operations. The sign of \(a\) determines opening direction.
• Vertex Form: \(y = a(x - h)^2 + k\) — Directly reveals the vertex \((h, k)\). Best for graphing and transformations.
• Conic Form: \((x - h)^2 = 4p(y - k)\) — Directly reveals the focal distance \(p\). Best for finding focus and directrix quickly.

Key Components of a Parabola

• Vertex: The turning point of the parabola. For \(y = ax^2 + bx + c\), the vertex is at \(\left(-\frac{b}{2a},\ c - \frac{b^2}{4a}\right)\).
• Focus: A point inside the parabola at distance \(|p|\) from the vertex along the axis of symmetry. Reflective properties direct signals to this point.
• Directrix: A line perpendicular to the axis at distance \(|p|\) from the vertex on the opposite side of the focus.
• Axis of Symmetry: The line passing through the vertex and focus, dividing the parabola into two mirror-image halves.
• Latus Rectum: A chord through the focus perpendicular to the axis. Its length is \(|4p|\) and indicates the parabola's width at the focus.

Vertical vs. Horizontal Parabolas

A vertical parabola (\(y = ax^2 + bx + c\)) opens upward when \(a > 0\) and downward when \(a < 0\). A horizontal parabola (\(x = ay^2 + by + c\)) opens rightward when \(a > 0\) and leftward when \(a < 0\). The calculator handles both orientations with the toggle switch.

Real-World Applications

• Satellite dishes & telescopes: Parabolic reflectors focus incoming parallel signals to the focus point.
• Projectile motion: The trajectory of a thrown ball (ignoring air resistance) follows a parabolic path.
• Car headlights: A bulb at the focus of a parabolic reflector produces parallel light beams.
• Bridge arches & suspension cables: Many structural designs use parabolic curves for optimal load distribution.
• Solar cookers: Parabolic mirrors concentrate sunlight to a focal point to generate heat.

FAQ

What is a parabola?

A parabola is a U-shaped curve where every point is equidistant from a fixed point (the focus) and a fixed line (the directrix). It is one of the four conic sections and has an eccentricity of exactly 1.

How do you find the vertex of a parabola?

For the standard form y = ax² + bx + c, the vertex is at x = -b/(2a) and y = c - b²/(4a). For vertex form y = a(x-h)² + k, the vertex is simply the point (h, k).

What is the focus of a parabola?

The focus is a fixed point inside the parabola. For a vertical parabola with vertex (h, k), the focus is at (h, k + p) where p = 1/(4a). Every point on the parabola is equidistant from the focus and the directrix.

What is the directrix of a parabola?

The directrix is a line perpendicular to the axis of symmetry. For a vertical parabola with vertex (h, k), the directrix is the line y = k - p. The parabola is the set of all points equidistant from the focus and directrix.

What is the latus rectum?

The latus rectum is a chord through the focus perpendicular to the axis of symmetry. Its length is |4p|, where p is the distance from the vertex to the focus. It helps determine the width of the parabola at the focus.