Cartesian Equation of Parabola

A parabola refers to an open plane curve that is formed by a right circular cone junction with a plane parallel to it. Each of them has a set of points that are equidistant from a fixed-line. It is a quadratic equation’s illustration graphically.  You can notice parabolic examples in several things around you like suspension bridges, antennae, telescopes, etc. Let us learn more about the parabolas, including the Cartesian Equation of Parabola. Using a parabola calculator can make its calculation easy. 

What are Cartesian Coordinates?

Cartesian coordinates help identify the exact spot of a point in a 3D plane. The Cartesian coordinates of a point are a pair of numbers that are either in 3 dimensions or 2 dimensions. These numbers give us the signed distances, which are particularized, from the coordinate axis.

The Cartesian coordinates are in terms of two types of coordinate axes- 

•     x-coordinates axis 
•     y-coordinates axis

The origin is the intersection of the x and y-axes. These coordinates are written down in the plane as (x, y). The x-coordinate is the signed distance from the origin along the x-axis. This coordinate identifies the distance to the right or the left of the y-axis. When the distance is to the right, x is positive, while it shall be negative if the distance is to the left. 

Explanation of Cartesian Equation of Parabola

The general equation of parabola is of the following two forms:

•     General equation: y = a(x-h)2 + k or x = a(y-k)2 +h
•     The standard equation: y2 = 4ax

Consider the placing of the parabolas in the Cartesian coordinate system (O,x,y) such that the location of the vertex V is at the origin O. Also, the location of the focus F lies on the coordinate axis y positive portion.  In such a place, we can derive an analytic equation.

In the equation of the parabola, the focus at F=(0,p2),

The directrix y=−p2 can be represented in the following two ways:

•     x 2 = 4 p2 y 

Here,  p>0 is the parameter 

In order to prove this, consider P = (xP, yP) to be any point.

Furthermore, consider Q = (xq,-p2) to be the point that is located at the perpendicular’s foot. This perpendicular is from directrix d to the point P. 

So, the equation becomes:

| F P ¯ | = | P Q ¯ |

This in turn gives rise to the following:

xp2 + ( y p − p 2 ) 2, which in turn becomes = ( y p + p 2 ) 2

Now, after doing the squaring of the above:

xp2 + y2p  − 2 yp p2+ p 22, which in turn becomes = y P 2 + p y P + p 2 2

so, we get x P 2 = 4yp

Finally, we get the equation of parabola  x 2 = 4 p2 y. The whole process of finding this equation becomes easier with the help of a parabola calculator.

Properties of Parabolas in Cartesian Plane

The site of points in the Cartesian plane is at an equal distance from a fixed-line and fixed point. Here, the directrix is the fixed-line, while the focus is the fixed point. 

Assuming that S is vertex, F is focus, and d is the directrix, the properties of parabolas in the Cartesian plane are as follows:

• The axis is an axis of symmetry and it has the focus and vertex. 

• We can say that a point P on the parabola shall have a point P if the condition: d(P, F) = d(P, d), is satisfied. 
• The vertex is the point of intersection that is located between the axis and the parabola.
• The chords go via the parabola’s focus. These chords join two points which experts call the focal chords. Latus rectum are those focal chords that are perpendicular to the parabola’s focal axis.
• Due to the process of rotation, every parabola boils down to a form in which the focus is on a Cartesian plane axis. Furthermore, in this form, you will find the vertex at the origin.  
• The equation of basic parabolas is y2 = 4px.

How to Graph Using a Cartesian Coordinate System

Below are the steps that help to graph parabola by using a Cartesian Coordinate System:

•     First of all, find out the parabolic equation’s concavity. For this purpose, you need to look towards the direction of the curve’s opening. This opening could be in any direction: upwards, downwards, right, or left.  
•     Now, identify the location of the vertex. You will find it either at (0, 0) or (h, k).
•     Identify the location of the focus.
•     Now, identify the latus rectum.
•     Afterwards, identify the location of the parabolic curve’s directrix. 
•     Finally, graph the parabola by making a curve such that it joins the latus rectum’s coordinates and the vertex. Before you go, make sure that all the important points are labelled.

Conclusion

A parabola is an open plane curve whose formation is by a right circular cone junction with a plane parallel to it. You will have a set of points in it that are equidistant from a fixed-line. It is simply a quadratic equation representation in a graphical manner. You need to understand the Cartesian coordinates well enough to make sense of this topic. The most important part is the explanation of the Cartesian Equation of Parabola, which you need to learn by heart. Learn all the properties of parabolas in the Cartesian plane to grasp this topic in a formidable manner. Finally, try to graph this open curve using a Cartesian coordinate system.