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Parabola Calculator

Standard Form: x = ay² + by + c




Vertex P(h,k)

Point P₁(x₁,y₁)

Focus Q(xₒ,yₒ)

Focus Q(xₒ,yₒ) , Directrix (y)

Vertex P(h,k) , Directrix (y)


A parabola calculator helps you in solving, graphing, and analyzing equations of parabolas that are U-shaped curves. It can also calculate key properties of a parabola, such as its vertex, axis of symmetry, directrix, and x and y intercepts.

What is Parabola?

In Mathematics,

“A parabola is a U-shaped symmetrical curve formed by a set of moving points so that its distance from a fixed point (known as focus) and a fixed line (known as directrix) are equal”.

Parabolas are also known as the graphs of quadratic functions that can be easily calculated through a parabola equation calculator.

What is Quadratic Function?

The basic form of quadratic function is as follows:

f(x) = ax2 + bx + c,

Where a, b, and c represent the real numbers that are not equal to zero. The basic shape of a parabola is defined by the U-shape. Parabolas can open upwards or downwards and have varying widths or steepnesses, but all have the same basic U-shape.

Types of Parabola:

Parabolas can be classified into two forms:

1. Standard Form:

If you are curious about how to find the equation of a parabola, you have to follow the standard form of parabola equation below:

y  =  ax^2  +  bx  +  c


  • a  =  Constant
  • b  =  Constant
  • c  =  Constant


  • x  =  Variable
  • y  =  Variable

2. Vertex Form:

Here is the vertex form of the equation for parabola below:

y  =  a(x - h)²  +  k


  • a  =  Coefficient
  • h  =  x-coordinate of the parabola vertex
  • k  =  y-coordinate of the parabola vertex

How to Calculate Parabola?

You can manually calculate the parabola of a quadratic function with the help of the equation of parabola. Here’s an example that describes the step-by-step calculations.


Let’s say if a = 2, b = 4, and c = 6 in parabola’s equation that is y = ax^2 + bx + c

Let's put the values into the equation.

y = 2x^2 + 4x + 6

To simplify, let's rewrite it as a standard form quadratic equation:

y = ax^2 + bx + c

Here, a = 2, b = 4, and c = 6. 

We can use these above-mentioned values to rewrite the equation in parabola’s standard form:

2x^2 + 4x + 6 = 0

Now, to complete the square and put it in standard form, follow these steps:

Factor out the common factor (in this case, 2) from the x^2 and x terms:

2(x^2 + 2x) + 6 = 0

In order to complete the square, we must add and subtract the square of half the coefficient of x (which is 2/2 = 1) inside the parentheses:

2(x^2 + 2x + 1 - 1) + 6 = 0

Simplify the equation:
2(x^2 + 2x + 1) - 2 + 6 = 0

Now, rewrite the perfect square trinomial:
2((x + 1)^2 - 1) + 6 = 0

Distribute the 2 on the left side:
2(x + 1)^2 - 2 + 6 = 0

The constants on the left should be combined as follows:
2(x + 1)^2 + 4 = 0

Move the constant term to the right side to isolate the squared term:
2(x + 1)^2 = -4

Divide both sides by 2 to isolate the squared term:
(x + 1)^2 = -2

By taking the square root of both sides, we get:
x + 1 = ±√(-2)

Now subtract 1 from the both sides in this equation to solve for x:
x = -1 ± i√2

Thus, the standard form of the parabola equation y = 2x^2 + 4x + 6 is:

(x + 1)^2 = -2

The solutions for x are complex numbers:

x₁ = -1 + i√2
x₂ = -1 - i√2

You can also confirm these calculations through an equation of parabola calculator.

Working of Parabola Calculator:

The working of a parabola calculator is very easy and simple as you need to provide some inputs to get instant results such as:

What to do:

  • First of all, select the parabola equation type you want to calculate from the drop-down menu.
  • Put the values into their designated fields.
  • Press the Calculate button.

What you get:

  • Parabola equations for Standard and Vertex forms.
  • All the parameters such as Focus, Vertex, Directrix, Eccentricity, Latus rectum, Axis of symmetry, x-intercept, and y-intercept.
  • Displays the graph of the parabola.


What are the real-life examples of parabolas?

The Parabolic dish antennas are an example of the shape of a parabola similarly, the Rainbow in the sky takes the shape of a parabola.

References: Parabola, Definition as a locus of points, Definition as a locus of points