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Parabola Calculator

Calculate the vertex, focus, directrix, and axis of symmetry of a parabola from its standard form coefficients.

Reviewed by Christopher FloiedPublished Updated

This free online parabola calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Coefficient of x² (determines width and direction)

Coefficient of x

Constant term (y-intercept)

Results

Vertex x

2

Vertex y

-1

Focus y

-0.75

Directrix y =

-1.25

Latus Rectum Length

1

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Parabola Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Parabola Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Parabola Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Parabola Calculator

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). When given in standard form y = ax² + bx + c, a parabola's key properties can be calculated algebraically. The coefficient a determines whether the parabola opens upward (a > 0) or downward (a < 0) and controls its width — larger |a| produces a narrower parabola. The vertex is the turning point, the focus is the point where reflected rays converge, and the directrix is a horizontal line opposite the focus. Parabolas appear everywhere in nature and engineering: the trajectory of a projectile, the shape of satellite dishes, the cross-section of suspension bridge cables, and the reflectors in car headlights all follow parabolic curves. The reflective property of parabolas — that rays parallel to the axis reflect through the focus — makes them essential in optics and telecommunications. This calculator takes the three coefficients of the standard form and computes all the key geometric properties of the parabola.

The Math Behind It

A parabola y = ax² + bx + c can be rewritten in vertex form y = a(x - h)² + k by completing the square, where (h, k) is the vertex. The vertex x-coordinate is h = -b/(2a), and the vertex y-coordinate is k = c - b²/(4a) or equivalently k = f(h). The focus lies at (h, k + 1/(4a)) for a vertical parabola, and the directrix is the line y = k - 1/(4a). The parameter p = 1/(4a) is the signed distance from the vertex to the focus; when a > 0, the focus is above the vertex, and when a < 0, it is below. The latus rectum is the chord through the focus perpendicular to the axis of symmetry, with length |1/a| = 4|p|. This chord determines the width of the parabola at the focus level. The axis of symmetry is the vertical line x = h. Every parabola is similar to every other parabola — they differ only by scaling and position. The eccentricity of a parabola is always exactly 1, placing it at the boundary between ellipses (e < 1) and hyperbolas (e > 1) in the family of conic sections. The reflective property states that a ray parallel to the axis of symmetry reflects off the parabola and passes through the focus.

Formula Reference

Standard Form

y = ax² + bx + c

Variables: a = curvature coefficient; b = linear coefficient; c = constant term

Vertex

(-b/(2a), f(-b/(2a)))

Variables: b, a = coefficients from standard form

Focus

(h, k + 1/(4a))

Variables: h, k = vertex coordinates; a = leading coefficient

Worked Examples

Example 1: Analyze y = x² - 4x + 3

Find the vertex, focus, and directrix of y = x² - 4x + 3.

Step 1:Vertex x = -(-4)/(2×1) = 4/2 = 2
Step 2:Vertex y = 1(2²) + (-4)(2) + 3 = 4 - 8 + 3 = -1
Step 3:Focus distance = 1/(4×1) = 0.25
Step 4:Focus at (2, -1 + 0.25) = (2, -0.75)
Step 5:Directrix: y = -1 - 0.25 = -1.25
Step 6:Latus rectum = |1/1| = 1

Vertex (2, -1), focus (2, -0.75), directrix y = -1.25, latus rectum = 1.

Example 2: Downward-opening parabola

Find the vertex and focus of y = -2x² + 8x - 5.

Step 1:Vertex x = -8/(2×(-2)) = -8/(-4) = 2
Step 2:Vertex y = -2(4) + 8(2) - 5 = -8 + 16 - 5 = 3
Step 3:Focus distance = 1/(4×(-2)) = -0.125
Step 4:Focus at (2, 3 + (-0.125)) = (2, 2.875)
Step 5:Directrix: y = 3 - (-0.125) = 3.125

Vertex (2, 3), focus (2, 2.875), directrix y = 3.125. The parabola opens downward.

Common Mistakes & Tips

  • !Forgetting to negate b when computing the vertex x-coordinate: h = -b/(2a), not b/(2a).
  • !Confusing the focus distance direction — when a > 0, focus is above the vertex; when a < 0, it is below.
  • !Setting a = 0, which produces a line, not a parabola.
  • !Mixing up standard form y = ax² + bx + c with vertex form y = a(x - h)² + k.

Related Concepts

Vertex Form

An alternative form for parabolas that directly reveals the vertex.

Latus Rectum

The chord through the focus perpendicular to the axis of symmetry.

Conic Sections

Parabolas are one of the four conic sections.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Catenary Curve Calculator

Conic Sections Calculator

Latus Rectum Calculator

Vertex Form Calculator

Frequently Asked Questions

How do I know if a parabola opens up or down?

If the coefficient a is positive, the parabola opens upward (U-shape). If a is negative, it opens downward (inverted U). The magnitude of a controls the width: larger |a| means narrower.

What is the axis of symmetry?

The axis of symmetry is the vertical line x = -b/(2a) that passes through the vertex. The parabola is symmetric about this line.

Can this calculator handle horizontal parabolas?

This calculator handles vertical parabolas of the form y = ax² + bx + c. Horizontal parabolas have the form x = ay² + by + c and require swapping x and y roles in all formulas.

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