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?
“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.
Wikipedia.org: Parabola, Definition as a locus of points, Definition as a locus of points