Parabola Graph: Standard Form of Parabola Equation

Parabola is a quadratic function graph. Various mathematicians explained the concept of a parabola through their study. Parabola, according to Pascal, is a projection of a circle. Galileo explained that projectiles starting to fall under the influence of uniform gravity follow a path known as a parabolic path. Many bodily motions follow a curvilinear path in the shape of a parabola. Here, in this section, we will discuss the concept of the parabola, the properties, and the graph of a parabola.

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What is Parabola?

The words "para" and "bola" mean "for" and "throwing," respectively. i.e., the shape described when a ball is thrown in the air. In mathematics, a parabola is any mirror-symmetrical plane curve with an approximate U shape.

A series of points in a plane that is equally distant from both a fixed-line and a fixed point (not on the line) is called a parabola.

• Directrix is a fixed-line of the parabola. 
• Focus is the fixed point.
• The Axis of the parabola is the line through the focus and the perpendicular to the directrix.
• The vertex of the parabola is the point where the parabola intersects with the axis.

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Standard Equations of Parabola

A parabola's equation is the simplest. If the vertex is at the origin and the axis of symmetry is either the x-axis or the y-axis, A parabola can be oriented in four different ways, as shown below:

With the focus at (a,0) a>0 and directrix x= -a, we can derive the equation for the parabola shown in fig. 1.

The above equations are the standard equations of a parabola.

Let F represent the focus and l represent the directrix; FM is perpendicular to the directrix and bisects FM at point O. Extend MO to X.

The mid-point O is on the parabola and is known as the vertex of the parabola, according to the definition of a parabola.

Take O as the origin, OX as the x-axis, and OY as the y-axis perpendicular to it.

Let 2a be the distance between the directrix and the focus. The coordinates of the focus are (a,0), and the directrix's equation is x+a = 0.

Let P(x,y) be a point on the parabola in such a way that,

PF = PB, -----------------------------------(1)

Where PB is perpendicular to l and the coordinates of B are (-a,y).

Let us calculate the distance using the distance formula.

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So, as per the distance formula, we have,

Also, PF = PB ------------(from 1)

Thus, any point on the parabola satisfies the eq. Y² = 4ax ----------------(2)

Conversely, let us assume p(x,y) satisfy the eq.(2) then,

Thus, the point p(x,y) lies on the parabola.

The following observations can be drawn from the standard parabola equations:

• The parabola is symmetric with its axis. If the equation contains a y² term, the axis of symmetry is along the x-axis; otherwise, the axis of symmetry is along the y-axis.
• When the axis of symmetry is along the x-axis: the parabola opens to the Right If the coefficient of x is positive and to the Left If the coefficient of x is negative.
• When the axis of symmetry is along the y-axis: the parabola opens upwards if the coefficient of y is positive and downwards if the coefficient of y is negative.

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Latus Rectum

Latus Rectum of the parabola is a line segment perpendicular to the parabola's axis that passes through the focus and has endpoints that tie on the parabola.

Example

Let us take y² = 4ax.

 Now, let us compute the length of the latus rectum of a parabola y² = 4ax.

According to the parabola definition, AF = FM

However, AC = FM = 2a

As a result, AF = 2a. Furthermore, because the parabola is symmetrical to the x-axis, AF = FB and thus AB = length of the latus rectum = 4a.

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Things to Remember

• Parabola is a Quadratic Function Graph (part of Conic Sections), formed by the intersection of a right circular cone to a plane surface.
• Parabola forms a symmetrical plane curve and a u-shaped curve. The Standard Form of a Parabola can be plotted with the following equation: f(x) = ax²+bx+c.
• The Vertex to plot a parabola Graph can be derived using  x=-b/2a and y = f(-b/2a). The quadratic equation can be presented as f(x) = a(x-h)2 + k, where (h,k) is the vertex of the parabola, its vertex form. 
• Latus Rectum of the parabola is a line segment perpendicular to the parabola's axis that passes through the focus and has endpoints that tie on the parabola.

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