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Parabola Calculator

Choose the input form and enter coefficients in the designated fields. The parabola calculator will instantly determine parabola-related parameters and display the graph of the parabolic expressions.

Choose What to Input:

Standard Form: y = ax² + bx + c

Related

The online Parabola Calculator helps you quickly find the standard and vertex forms of a parabola equation. It also calculates key properties such as the focus, vertex, directrix, axis of symmetry, and intercepts, while displaying a graph of the parabola for easy visualization.

What is a Parabola?

A parabola is a symmetric, U-shaped curve that is one of the conic sections, formed by the intersection of a right circular cone with a plane. It has a unique property: any point on the parabola is equidistant from:

A fixed point called the focus
A fixed line called the directrix

Parabola equations describe this curve mathematically, and a parabola calculator simplifies these computations efficiently.

Parabola Formulas

Simple form: \(y = x^2\)
General form: \(y^2 = 4ax\)

Parabola in Standard Form

Standard form: \(x = ay^2 + by + c\)

Parabola in Vertex Form

Vertex form: \(x = a(y-k)^2 + h\)

The vertex form makes it easier to identify the vertex, focus, and other essential points.

How to Find the Equation of a Parabola

Using the standard form \(x = ay^2 + by + c\), you can determine the axis of symmetry, vertex, focus, directrix, and intercepts:

Identify coefficients \(a\), \(b\), and \(c\)
Vertex: \(h = -\frac{b}{2a}, k = c - \frac{b^2}{4a}\)
Focus: \(F = (h, c - \frac{b^2-1}{4a})\)
Directrix: \(y = c - \frac{b^2+1}{4a}\)
Axis of symmetry: \(x = h = -\frac{b}{2a}\)
Y-intercept: set \(x = 0\)
X-intercept: set \(y = 0\)

Parabola Calculator Graph

Example

Find the axis of symmetry, y-intercept, x-intercept, vertex, directrix, and focus for \(x = 11y^2 + 10y + 16\).

Given: a = 11, b = 10, c = 16

Vertex:

h = -b/2a = -10 / (2*11) = -5/11

k = c - b²/(4a) = 16 - 100/44 = 604/44 = 151/11

Vertex = (-5/11, 151/11)

Focus:

x-coordinate = h = -5/11

y-coordinate = c - (b² - 1)/(4a) = 16 - 99/44 = 605/44 ≈ 13.75

Focus = (-5/11, 605/44)

Directrix: y = c - (b² + 1)/(4a) = 16 - 101/44 = 603/44

Axis of symmetry: x = -5/11

Y-intercept: x = 0 → y = 16

X-intercept: y = 0 → 0 = 11(0)² + 10(0) + 16 → No x-intercept

Finding the Directrix of a Parabola

For a parabola in standard form:

Vertical: \((x - h)^2 = 4p(y - k)\) → Focus: (h, k+p), Directrix: y = k-p
Horizontal: \((y - k)^2 = 4p(x - h)\) → Focus: (h+p, k), Directrix: x = h-p

How the Parabola Calculator Works

Input:

Select the parabola type: standard form, vertex form, three points, or vertex and points.
Enter the corresponding values in the input fields.
Click "Calculate".

Output:

Equation in standard and vertex forms
Vertex, Focus, Directrix, Axis of Symmetry, Latus Rectum, Eccentricity
X-intercept and Y-intercept
Step-by-step calculations
Graph of the parabola

FAQs

How does the distance between the focus and directrix affect the parabola's shape?

The wider the distance, the broader the parabola. Mathematically, |a| decreases as the distance between focus and directrix increases.

What are the steps to graph a parabola manually?

Determine the vertex, x-intercepts, and y-intercept
Find additional points for a minimum of five points
Plot points and sketch the parabola

What transformations can be applied to a parabola?

1. Translation – shifts the parabola along axes
2. Rotation – rotates the parabola around a pivot point

How is a parabola’s transformation described?

Vertical or horizontal translations move the parabola along the corresponding axis without changing its shape.

Conclusion

The Parabola Calculator delivers accurate, step-by-step solutions and visual graphs for any parabolic equation. It simplifies finding vertices, intercepts, focus, directrix, and other critical properties, making calculations faster and error-free.