The Parabola Calculator displays the graph of the given parabola equation online for free. STUDYQUERIES’s online parabola calculator tool makes the calculation faster and displays the graph of the parabola in a matter of seconds.

How to Use the Parabola Calculator?

To use the parabola calculator, follow these steps:

  • Step 1: Enter the parabola equation in the input box
  • Step 2: Click “Submit” to get the graph
  • Step 3: The parabola graph will appear in the new window

Parabola Calculator

What is Parabola?

Parabolas are graphs of quadratic functions. Pascal defined a parabola as a projection of a circle. According to Galileo, projectiles falling under the influence of uniform gravity follow a path called a parabolic path. Many physical motions of bodies follow a curvilinear path that resembles a parabola.

An approximate U-shaped plane curve that is mirror-symmetrical is called a parabola in mathematics. In this section, we will attempt to understand the standard formula for a parabola, the standard forms of a parabola, and the properties of a parabola.

In other words, we can say that a parabola is an equation of a curve such that a point on the curve is equidistant from a fixed point and a fixed-line. Parabolas have a fixed point and a fixed-line. The fixed-line is called the directrix of the parabola. Moreover, a crucial point to keep in mind is that the fixed point is not on the fixed line.

Parabolas are loci that are equidistant from a given point (focus) and a given line (directrix). The parabola is an important curve of the conic sections of coordinate geometry.

Parabola Equation

The general equation of a parabola is: $$y = a(x-h)^2 + k\ or\ x = a(y-k)^2 +h$$ where \((h,k)\) denotes the vertex. The standard equation of a regular parabola is $$y^2 = 4ax$$

To understand the features and parts of a parabola, the following terms are helpful.

Parabola Calculator
Parabola Calculator
  • Focus: The point \((a, 0)\) is the focus of the parabola
  • Directrix: The line drawn parallel to the y-axis and passing through the point \((-a, 0)\) is the directrix of the parabola. It is perpendicular to the parabola’s axis.
  • Focal Chord: The focal chord of a parabola is the chord passing through the focus of the parabola. There are two points of intersection on the focal chord.
  • Focal Distance: The distance of a point \((x_1,y_1)\) on the parabola, from the focus, is the focal distance. Also, the focal distance is equal to the perpendicular distance of this point to the directrix.
  • Latus Rectum: A chord that passes through the focus of a parabola and is perpendicular to its axis. The length of the latus rectum is taken as \(LL’ = 4a\). The endpoints of the latus rectum are \((a, 2a)\), \((a, -2a)\).
  • Eccentricity: \((e = 1)\). It is the ratio of the distance of a point from the focus, to the distance of the point from the directrix. The eccentricity of a parabola is equal to 1.
    Forms Of Parabola
    Forms Of Parabola

Standard Equations of a Parabola

A parabola has four standard equations. Based on the axis and the orientation of the parabola, there are four standard forms. In each of these parabolas, the transverse axis and conjugate axis are different. Below is an image displaying the four standard equations and forms of a parabola.

Standard Equations of a Parabola
Standard Equations of a Parabola

The following observations can be made from the standard form of equations:

  • Parabola is symmetric with respect to its axis. If the equation has the term with \(y^2\), then the axis of symmetry is along the x-axis and if the equation has the term with \(x^2\), then 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 the x is positive and opens to the left if the coefficient of x is negative.
  • If the axis of symmetry is along the y-axis, the parabola opens upward if the coefficient of y is positive, and downward if the coefficient of y is negative.

Read Also– Derivative Of Tangent – Slope, Derivative & More

Parabola Formula

Parabolic paths are represented in the plane using the Parabola Formula. The following formulas can be used to get the parameters of a parabola.

$$The\ direction\ of\ the\ parabola\ is\ determined\ by\ the\ value\ of\ a.$$

$$Vertex = (h,k)\ where\ h=\frac{-b}{2a}\ and\ k = f(h)$$

$$Latus\ Rectum = 4a$$

$$Focus:\ (h, k+\frac{1}{4a})$$

$$Directrix:\ y = k – \frac{1}{4a}$$

Graph of a Parabola

Consider an equation $$y = 3x^2 – 6x + 5$$ For this parabola, \(a = 3 , b = -6\ and\ c = 5\). This is a graph of the given quadratic equation, which is a parabola.

Graph of a Parabola
Graph of a Parabola

Direction: Here a is positive, and so the parabola opens up.

\(Vertex: (h,k)\)

$$h = \frac{-b}{2a}$$

$$= \frac{6}{2\times 3} = 1$$

$$k = f(h)$$

$$= f(1) = 3\times 1\times 2 – 6\times 1\times+5 = 2$$

$$Thus\ vertex\ is\ (1,2)$$

$$Latus Rectum = 4a = 4\times 3 =12$$

$$Focus: (h, k+\frac{1}{4a}) = (1,\frac{25}{12})$$

$$Axis of symmetry is x =1$$

$$Directrix: y = k-\frac{1}{4a}$$

$$y = 2-\frac{1}{12}\Rightarrow y-\frac{23}{12} = 0$$

Derivation of Parabola Equation

Let us consider a point P with coordinates \((x, y)\) on the parabola. According to the definition of a parabola, the distance of this point from the focus F is equal to the distance of this point P from the Directrix. Calculations are performed based on the perpendicular distance PB from point B on the directrix.

Derivation of Parabola Equation
Derivation of Parabola Equation

As per this definition of the eccentricity of the parabola, we have PF = PB (Since \(e = PF/PB = 1)\)

The coordinates of the focus are \(F(a,0)\) and we can use the coordinate distance formula to find its distance from \(P(x, y)\)

$$PF = \sqrt{(x-a)^2+(y-0)^2}$$

$$=\sqrt{(x-a)^2+(y)^2}$$

The equation of the directrix is \(x + a = 0\) and we use the perpendicular distance formula to find PB.

$$PB={\sqrt{(x-(-a))^2+(y-y)^2}}$$

$$PB=\sqrt{(x+a)^2}$$

We need to derive the equation of parabola using \(PF = PB\)

$$\sqrt{(x-a)^2+(y)^2}=\sqrt{(x+a)^2}$$

Squaring the equation on both sides,

$$(\sqrt{(x-a)^2+(y)^2})^2=(\sqrt{(x+a)^2})^2$$

$$(x-a)^2+(y)^2=(x+a)^2$$

$$x^2+a^2-2ax+y^2 = x^2+a^2+2ax$$

$$y^2-2ax = 2ax$$

$$y^2 = 4ax$$

As a result, we have successfully derived the standard equation of a parabola.

Similarly, we can derive the equations of the parabolas as follows:

$$y^2 = -4ax$$

$$x^2 = 4ay$$

$$x^2 = -4ay$$

These four equations are the Standard Equations of Parabolas.

Read Also– Point-Slope Form Calculator

Properties of a Parabola

In this section, we will look at some of the important properties and terms associated with parabolas.

Tangent: The tangent is a line touching the parabola. The equation of a tangent to the parabola \(y^2 = 4ax\) at the point of contact \((x_1,y_1)\) is $$yy_1=2a(x+x_1)$$

Normal: A line drawn perpendicular to the tangent and passing through both the point of contact and the focus of a parabola is called the normal. For a parabola \(y^2 = 4ax\), the equation of the normal passing through the point \((x_1,y_1)\) is and having a slope of \(m =\frac{-y_1}{2a}\), the equation of the normal is

$$y-y_1=\frac{-y_1}{2a}(x-x_1)$$

Chord of Contact: The chord connecting the points of contact of the tangents drawn from an external point to the parabola is the chord of contact. For a point \((x_1,y_1)\) outside the parabola, the equation of the chord of contact is $$yy_1=2x(x+x_1)$$

Pole and Polar: For a point lying outside the parabola, the locus of the points of intersection of the tangents, draw at the ends of the chords, drawn from this point is called the polar. And this referred point is called the pole. For a pole having the coordinates \((x_1,y_1)\) for a parabola \(y^2 =4ax\) the equation of the polar is $$yy_1=2x(x+x_1)$$

Parametric Coordinates: The parametric coordinates of the equation of a parabola \(y^2 = 4ax\) are \((at^2, 2at)\) The parametric coordinates represent all the points on the parabola.

FAQs

How do you find the equation of a parabola in standard form?

Parabolas that open either up or down have the standard form equation (x – h)^2 = 4p(y – k). If a parabola opens sideways, the standard form equation is (y – k)^2 = 4p(x – h). The vertex of our parabola is the point (h, k).

What is the parabola equation?

The general equation of a parabola is y = x² in which x-squared is a parabola. Work up its side it becomes y² = x or mathematically expressed as y = √x. The formula for Equation of a Parabola. Taken as known the focus (h, k) and the directrix y = mx+b, parabola equation is y−mx–b² / m²+1 = (x – h)² + (y – k)² .

How do you find the equation of a parabola given two points?

Using the vertex form of a parabola f(x) = a(x – h)^2 + k where (h,k) is the vertex of the parabola. The axis of symmetry is x = 0 so h also equals 0.

How do you find c in a parabola?

The c-value is the point where the graph intersects the y-axis. This graph has a c-value of -1, and its vertex is the highest point on the graph, known as the maximum. This is the graph of a parabola that opens up. Where the graph intersects the y-axis is known as the c-value.