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

October 25, 2022

When you discharge an arrow or throw a stone, it arcs into the air and descends like a parabola. According to Pascal, A parabola is the projection of a circle. Galileo demonstrated that projectiles falling under the influence of uniform gravity have a path known as a parabolic path. Likewise, many bodily motions follow a curvilinear route shaped like a parabola. In mathematics, A parabola is a quadratic function graph.

## 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.

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