Parabola is a curve that represents an equation. Each point on the curve is at an equal distance from a fixed point or line. This article explains what this curve is, the equation of a parabola, and what happens when we shift the origin in a parabola.  

A parabola is the locus of a point that moves such that it always remains at the same distance from a fixed point (focus) or line (directrix). A parabola calculator helps carry out calculations related to the parabola. 

Let us learn more about the shifting of origin in a parabola.

Equation of a Parabola

A parabola is an equation of a curve where each point is equidistant from the focus or directrix. This curve is among the most important curves under conic sections of coordinate geometry. One can easily find out this equation with the help of a parabola calculator.

Typically, the equation of this curve is denoted as follows.

Y = a(x – h)2 + k

OR

X = a(y – k)2 + h

Where (h, k) is the vertex of this curve.

The standard equation of a regular curve is represented by y2 = 4ax. Alternatively, the standard parabolic equation can be written as x2 = 4ay. 

Properties of a Parabola

Let us take a look at some properties of the parabola.

• The direction of this curve is defined by the coefficient ‘a’
• Any chord whose passing takes place via the focus of this curve is known as the focal chord of this curve
• The distance of any point p(x, y) on the curve y2= 4ax is the distance between the point ‘P’ and the focus
• The eccentricity of this curve defines how circular the conic section is
• The greater the eccentricity, the less spherical this curve will be and vice-versa
• One can also define eccentricity as the ratio of the distance between the focus and a point on the Cartesian plane to the vertex
• The eccentricity is always 1 since it is a locus of all the points that are equidistant from the focus and the directrix
• The latus rectum of this curve runs parallel to the directrix while passing through the focus
• The symmetry’s axis is perpendicular to the latus rectum, whose length is 4a

Graphical Representation of a Parabola

Let us now take a look at the graphical representation of a parabola.

It is first essential to know where its focus and vertex lie to draw the graph of a parabola.

If the vertex of this curve is at its origin and the axis of symmetry lies on the x-axis, then this curve can be on the x-axis’ positive or negative side. The case is similar if the axis of symmetry lies on the y-axis.

If:

• a>0
• ‘a’ lies on the positive side of the x-axis of the equation y2= 4ax

Then, the coordinates will be:

F = (a, 0)

It indicates that the equation lies on the positive side of the x-axis. The directrix lies at x = -a.

Alternatively, if:

• a < 0 
• ‘a’ lies on the negative side of the x-axis for the equation y2 = -4ax 

Then:

F = ( -a, 0), 

Therefore, the equation lies on the negative side of the x-axis, with the directrix at x = a.

In case of the y-axis, the above coordinates will be:

• F = (0, a) for the equation x2 = 4 ay on the positive side of the y-axis with a directrix at y = a.
• F = (0, -a) for x2= -4ay on the y-axis’s negative side with y = -a. 

Shifting of Origin in Parabola

There are two cases when the origin in the parabola shifts or is not at the origin.

Case 1: x = a (y – k)2 + h

Now, (h, k) are the vertex’s coordinates. The equation proves that if a > 0, this curve opens towards the right and vice-versa when a < 0.

Case 2: y = a(x – k)2 + h

Here, h & k are the vertex coordinates, and if a > 0, this curve opens upwards and vice-versa if a < 0.

Conclusion

The article thus explains what a parabola is, its different equations, and how to graph this curve. It is also necessary to understand what happens when there is a shifting of the origin in a parabola. 

One can define a parabola as an equation of a curve where each point is equidistant from the focus or directrix. This curve has several distinct properties and can be graphically represented.