How to Graph a Parabola – Formula, Properties, and Equation!

What is Parabola?

A parabola is a U-shaped plane curve in which a point on the curve is equidistant between a fixed point and a fixed line. The fixed point is the focus of the parabola and the fixed line is the directrix. 

Parabola Equation

When the directrix is parallel to the y-axis, the most straightforward equation of a parabola graph is \( y^2 = x \).

If the directrix is parallel to the y-axis, the parabola equation is: \( y^2 = 4ax \)

1- Standard Equation

When the vertex is at the origin, and the axis of symmetry is along the x and y axes, the parabola equation is the simplest. The Standard Equations of Parabola are:

\( y^2 = 4ax \)

\( y^2 = -4ax \)

\( x^2 = 4ay \)

\( x^2 = -4ay \)

2- Directrix

A parabola’s directrix is a line that is perpendicular to the parabola’s axis. The parabola’s directrix aids in the definition of the parabola.

3- Axis of Symmetry

A parabola’s axis of symmetry is a vertical line that divides the parabola into two congruent halves. 

If the equation contains a \( y^2 \) term, the symmetry axis is along the x-axis. The parabola, in this case, opens to the.

(a) true if the x coefficient is positive

(b) to the left, if the x coefficient is negative

If the equation contains an \( x^2 \) term, the symmetry axis is along the y-axis. The parabola in this situation opens-

(c) if the y coefficient is positive, upwards

(d) if the coefficient of y is negative, downwards

4- Vertex

A parabola’s vertex is the point at which the parabola makes its sharpest turn. A parabolic function has a maximum value (if it has the shape “∩”) or a minimum value (if it has the shape “U”). The vertex of a parabola is also the point where the parabola and its axis of symmetry intersect.

5- Focal Chord

The focal chord of a parabola is any chord that goes through its center. The Focal Chord divides the parabola into two parts.

6- Focal Distance

The focal distance is the distance from the focus of any point p(x, y) on the parabola.

7- Latus Rectum of Parabola

A parabola’s latus rectum is the chord that runs across the focus and is perpendicular to the parabola’s axis.

8- Eccentricity

Eccentricity is defined as the ratio of a point on a parabola’s distance from a fixed point to its perpendicular distance from a fixed-line.

Parabola Formula

The Parabola Formula is useful in describing the general shape of a parabolic path in the plane. 

Here a determines the direction of the parabola.

(h, k) = vertex, where \( h = \dfrac{-b}{2a} \) and k = f(h)

4a = Latus Rectum

(h, k + \( \dfrac{1}{4a} \)) Focus

y = k – \( \dfrac{1}{4a} \) is the directrix.

Parabola Graph

When a value is less than zero, the graph of the parabola is downward (or opens down), when the value of a is greater than zero, the parabola graph rises (or opens up).

Concave up and Concave down

A parabola \( y = ax^2 + bx + c \) will be concave-up or concave-down depending on the sign of a and the \( x^2 \) coefficient:

The parabola will be concave-up if a > 0.

a < 0: the parabola is concave downward.

Parabola Properties

1- Tangent 

The tangent to a parabola is a line intersecting the parabola exactly at one point.

Point Form: Tangent to parabola \( y^2 = 4ax \) at (\(x_1\), \(y_1\)) equals \(yy_1\) = \(2a(x + x_1)\).

The tangent’s point of contact is (\(x_1\), \(y_1\)).

Parametric Form: The tangent to the parabola \( y^2 = 4ax \) at (\(at^2\), \(2at\)) has the equation: \(ty = x + at^2\).

Tangent’s point of contact is (\(at^2\), \(2at\))

Slope Form: If m is the slope of the tangent to the parabola \( y^2 = 4ax \), then \( y = mx + \frac{a}{m} \) is the equation of tangent.

Tangent’s point of contact is (\( \frac{a}{m_2} \), \( \frac{2a}{m} \)).

2- Normal 

The normal of a parabola is the line perpendicular to the tangent of the parabola at the point of contact.

The normal to a parabola equation can be written in point form, parametric form, or slope form.

Point Form: The normal to the parabola equation \( y^2 = 4ax \) at (\(x_1\), \(y_1\)) is given by \( y – y_1 = (\frac{-y_1}{2a}) (x – x_1) \).

Slope Form: The normal to the parabola equation \( y^2 = 4ax \) at (\(am_2\), \(-2am\)) is given by \( y = mx – 2am – am_3 \).

The contact person is (\( am_2 \), \(-2am\)).

Parametric Form: The normal to the parabola equation \( y^2 = 4ax \) at position (\(at^2\), \(2at\)) is given by \( y = -tx + 2at + at_3\).

Chord of Contact

If two tangents are formed from an external point to a conic, the secant line connecting the contact points is called the “chord of contact of that point.” T = 0 gives the chord of contact of Tangents from a point p(\(x_1\), \(y_1\)) to the parabola \( y^2 = 4ax \), i.e., \(yy_1 – 2a(x + x_1) = 0\).

Pole and Polar

The straight line comprising the points of intersection of the tangents formed at the extremities of chords flowing through the point is the “polar of a point” with respect to a conic.

The locus of the point of intersection of the tangents to the parabola is referred to as the polar of the given point P with respect to the parabola, and the point P itself is referred to as the pole of the polar.

Parametric Coordinates

The parametric coordinates of the parabola equation \( y^2 = 4ax \) are (\(at^2\), \(2at\)). All of the points on the parabola are represented by the parametric coordinates.

Parabolic Reflector or Parabola Dish

Parabolic reflectors are parabolas that have been rotated about their axis of symmetry to form a bowl-like shape known as a paraboloid. Because of its capacity to efficiently capture incoming information, this shape is used in many current applications. TV dishes, for example, reflect incoming television signals toward a receiver centered on the dish’s focus. In addition, parabolic reflectors are used in satellite dishes, telescopes, and other applications.

Parabola in real life

Parabolas can be found in both natural and artificial artifacts. Consider a fountain that sprays water into the air before returning in a parabolic path. Galileo demonstrated that a ball tossed into the air follows a parabolic path. Likewise, any roller coaster rider is familiar with the rise and fall created by the track’s parabolas.