Parabola Calculator

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.

• 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.

  

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.

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.

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.

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.

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.

Parabola Equation Calculator:

Description: The Parabola Equation Calculator determines the equation of a parabola given specific parameters such as the vertex, focus, or directrix.

Example: Calculate the equation of a parabola with a vertex at (2, 3) and a focus at (2, 5).
Answer: The equation of the parabola is y = (1/2)(x – 2)^2 + 3.

Parabola Graph Calculator:

Description: The Parabola Graph Calculator plots the graph of a parabola based on its equation, allowing you to visualize its shape and characteristics.

Example: Graph the parabola given by the equation y = x^2 – 4x + 3.
Answer: The graph of the parabola forms an upward-opening parabola that intersects the x-axis at (1, 0) and (3, 0) and has a vertex at (2, -1).

Vertex Of A Parabola Calculator:

Description: The Vertex Of A Parabola Calculator determines the coordinates of the vertex, which represents the highest or lowest point on the parabola.

Example: Find the vertex of the parabola with the equation y = -2x^2 + 4x – 1.
Answer: The vertex of the parabola is (1, -3).

Find Equation Of Parabola Given Vertex And Point Calculator:

Description: The Find Equation Of Parabola Given Vertex And Point Calculator calculates the equation of a parabola using its vertex and a known point on the curve.

Example: Find the equation of a parabola with a vertex at (-2, 1) and passing through the point (1, 5).
Answer: The equation of the parabola is y = -(3/4)(x + 2)^2 + 1.

Focus Of A Parabola Calculator:

Description: The Focus Of A Parabola Calculator determines the coordinates of the focus, which is a fixed point inside the parabola that is equidistant to all points on the curve.

Example: Determine the focus of the parabola given by the equation y = 4x^2 – 8x + 5.
Answer: The focus of the parabola is (1, 1).

Parabola Calculator With Steps:

Description: The Parabola Calculator With Steps provides a detailed explanation of the calculations involved in finding various properties of a parabola, such as the vertex, focus, and directrix.

Example: Calculate the vertex, focus, and directrix of the parabola with the equation y = -3x^2 + 6x – 2.
Answer:
– Step 1: To find the vertex, use the formula h = -b / (2a), where a = -3 and b = 6.
h = -6 / (2 * (-3)) = 1.
The x-coordinate of the vertex is 1.

– Step 2: Substitute the x-coordinate of the vertex into the equation to find the y-coordinate.
y = -3(1)^2 + 6(1) – 2 = 1.
The vertex is (1, 1).

– Step 3: To find the focus, use the formula (h, k + (1 / (4a))), where a = -3, h = 1, and k = 1.
Focus = (1, 1 + (1 / (4 * (-3)))) = (1, 1 – 1

/12) = (1, 11/12).

– Step 4: The equation of the directrix is given by y = k – (1 / (4a)), where a = -3 and k = 1.
Directrix: y = 1 – (1 / (4 * (-3))) = 1 – 1/12 = 11/12.

Therefore, the vertex is (1, 1), the focus is (1, 11/12), and the directrix is y = 11/12.

Latus Rectum Of Parabola Calculator:

Description: The Latus Rectum Of Parabola Calculator determines the length of the line segment perpendicular to the axis of symmetry and passing through the focus of a parabola.

Example: Find the length of the latus rectum for the parabola with the equation y^2 = 8x.
Answer: The length of the latus rectum is 4.

Directrix Of Parabola Calculator:

Description: The Directrix Of Parabola Calculator calculates the equation of the line that is equidistant to all points on the parabola but on the opposite side of the focus.

Example: Calculate the equation of the directrix for the parabola given by the equation x^2 = 12y.
Answer: The equation of the directrix is y = -3/4.

Closest Point On Parabola Calculator:

Description: The Closest Point On Parabola Calculator determines the coordinates of the point on a parabola that is closest to a given external point.

Example: Find the closest point on the parabola y = x^2 – 3x + 2 to the point (4, 5).
Answer: The closest point on the parabola is (4, 6).

Volume Of Parabola Calculator:

Description: The Volume Of Parabola Calculator calculates the volume of the solid obtained by rotating a parabolic shape around a given axis.

Example: Calculate the volume of the solid obtained by rotating the parabola y = x^2 around the x-axis between x = 0 and x = 2.
Answer: The volume of the solid is (32/15)π cubic units.

Centroid Of Parabola Calculator:

Description: The Centroid Of Parabola Calculator determines the coordinates of the centroid, which represents the center of mass of the parabolic shape.

Example: Find the coordinates of the centroid for the parabolic shape given by the equation y = -2x^2 + 3x – 1.
Answer: The coordinates of the centroid are (3/4, -13/16).

Find Equation Of Parabola Given Vertex And Directrix Calculator:

Description: The Find Equation Of Parabola Given Vertex And Directrix Calculator calculates the equation of a parabola using its vertex and the equation of the directrix.

Example: Find the equation of a parabola with a vertex at (2, -1) and a directrix given by the equation y = 3.
Answer: The equation of the parabola is y = -1/4(x – 2)^2 – 1.

Find Equation Of Parabola Given Focus And Directrix Calculator:

Description: The Find Equation Of Parabola Given Focus And Directrix Calculator determines the equation of a parabola using its focus and the equation of the directrix.

Example: Determine the equation of a parabola with a focus at (4, -2) and a directrix given by the equation y = -6.
Answer: The equation of the parabola is (y +

2)^2 = 8(x – 4).

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.

How To Find The Vertex Of A Parabola Calculator?

To find the vertex of a parabola, you can use a calculator specifically designed for this purpose. These calculators typically require the input of the parabola’s equation in the form y = ax^2 + bx + c. By inputting the coefficients a, b, and c into the calculator, it will calculate the coordinates of the vertex. The vertex represents the highest or lowest point on the parabola, depending on whether it opens upwards or downwards.

How To Find Focus Of Parabola Calculator?

To find the focus of a parabola, you can use a calculator designed for this task. These calculators typically require the input of the parabola’s equation in standard form, y^2 = 4ax or x^2 = 4ay, where ‘a’ is a constant. By inputting the value of ‘a’ into the calculator, it will determine the coordinates of the focus. The focus is a point on the axis of symmetry of the parabola and plays a crucial role in its geometric properties.

How To Find The Focus Of A Parabola Calculator?

Finding the focus of a parabola involves using a calculator specifically designed for this purpose. These calculators usually require inputting the equation of the parabola in standard form, either y^2 = 4ax or x^2 = 4ay. By providing the value of ‘a’ into the calculator, it will calculate the coordinates of the focus. The focus is a point on the axis of symmetry of the parabola and is equidistant from the vertex and the directrix.

How To Find Vertex Of Parabola Calculator?

To find the vertex of a parabola, you can utilize a calculator specifically created for this task. These calculators typically require inputting the equation of the parabola in the form y = ax^2 + bx + c. By providing the coefficients a, b, and c into the calculator, it will determine the coordinates of the vertex. The vertex is the highest or lowest point on the parabola and is an essential element in understanding its shape and behavior.

How To Find The Equation Of A Parabola Calculator?

Finding the equation of a parabola can be made easier by using a calculator designed for this purpose. These calculators typically require inputting specific points on the parabola or other key information, such as the vertex, focus, or directrix. By providing the necessary input into the calculator, it will generate the equation of the parabola. The equation describes the relationship between the x and y coordinates and allows for a deeper understanding of the parabola’s properties.

What Are The 4 Types Of Parabola?

The four types of parabolas are determined by the orientation of the parabola and the sign of the coefficient ‘a’ in its equation. They include:

– Vertical parabolas: These parabolas open either upwards (a > 0) or downwards (a < 0) and have their axis of symmetry parallel to the y-axis.
– Horizontal parabolas: These parabolas open either to the right (a > 0) or to the left (a < 0) and have their axis of symmetry parallel to the x-axis.
– Right-facing parabolas: These parabolas open to the right and have their vertex on the y-axis.
– Left-facing parabolas: These parabolas open to the left and have their vertex on the y-axis.

Understanding the different types of parabolas helps in analyzing their characteristics and graphing them accurately.

How Do You Find The Equation Of A Parabola?

To find the equation of a parabola, you need to know key information such as the vertex, focus, directrix, or points on the parabola. The specific approach may vary based on the available information. Methods include completing the square, using the vertex form, or using the focus and directrix. By applying the appropriate method and utilizing the given information, you can determine the equation of the parabola.