Find The Following for This Parabola Calculator

A parabola is a U-shaped curve that can be defined as the set of all points equidistant from a fixed point (the focus) and a fixed straight line (the directrix). This calculator helps you find the vertex, focus, directrix, and other key properties of a parabola given its equation.

What is a Parabola?

A parabola is a symmetric, U-shaped curve that appears in many natural and man-made phenomena. In mathematics, a parabola is defined as the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

The standard form of a parabola's equation is y = ax² + bx + c. The shape and position of the parabola depend on the coefficients a, b, and c. When a parabola opens upwards or downwards, it has a vertical axis of symmetry. When it opens left or right, it has a horizontal axis of symmetry.

Key Properties of a Parabola

The key properties of a parabola include:

Vertex: The highest or lowest point on the parabola, depending on its orientation.
Focus: A fixed point inside the parabola that determines its shape.
Directrix: A fixed line outside the parabola that, together with the focus, defines the parabola.
Axis of Symmetry: A line that divides the parabola into two mirror-image halves.

Vertex Form: y = a(x - h)² + k, where (h, k) is the vertex.

Standard Form: y = ax² + bx + c.

To find these properties from the standard form, you can complete the square or use the formulas below.

How to Use This Calculator

This calculator allows you to find the vertex, focus, directrix, and other properties of a parabola given its equation in standard form (y = ax² + bx + c).

Enter the coefficients a, b, and c in the input fields.
Click the "Calculate" button to compute the properties.
View the results in the result panel.
Use the chart to visualize the parabola.

Note: The calculator assumes the parabola is in standard form and opens upwards or downwards. For horizontal parabolas, use the equivalent x = ay² + by + c form.

Worked Examples

Here are some examples of how to use the calculator and interpret the results.

Example 1: Simple Parabola

Given the equation y = x² - 4x + 3:

a = 1, b = -4, c = 3
Vertex: (2, -1)
Focus: (2, -0.25)
Directrix: y = -0.75

Example 2: Complex Parabola

Given the equation y = 2x² + 8x - 5:

a = 2, b = 8, c = -5
Vertex: (-2, -13)
Focus: (-2, -12.25)
Directrix: y = -13.75

Property	Example 1	Example 2
Vertex	(2, -1)	(-2, -13)
Focus	(2, -0.25)	(-2, -12.25)
Directrix	y = -0.75	y = -13.75

FAQ

What is the difference between a parabola and a hyperbola?: A parabola is a U-shaped curve with one focus and one directrix, while a hyperbola is an S-shaped curve with two foci and two directrices.
How do I find the vertex of a parabola?: For a parabola in standard form y = ax² + bx + c, the vertex is at (-b/(2a), f(-b/(2a))).
Can this calculator handle horizontal parabolas?: No, this calculator is designed for vertical parabolas. For horizontal parabolas, use the equivalent x = ay² + by + c form.
What if my parabola is in vertex form?: You can convert it to standard form by expanding the equation. For example, y = a(x - h)² + k becomes y = ax² - 2ahx + (ah² + k).