# Parabola Calculator | Formula, Steps Proof & Examples

Are you attempting to solve the equation for the parabola? If so, you’ve come to the correct place. From here, you’ll learn how to calculate the parabola equation and obtain values for the vertex, focus, x and y intercepts, directrix, and axis of symmetry. You will gain a good understanding of the parabola equation concept as well as a useful calculator tool that offers the result in a fraction of a second by reading the parts below. To receive the output in a short amount of time, utilise our user-friendly Parabola Calculator tool. To obtain the vertex, x intercept, y intercept, focus, axis of symmetry, and directrix, simply enter the parabola equation in the required input boxes and press the calculator button.

A graph of a quadratic function is called a parabola. A parabola, according to Pascal, is a circular projection. Projectiles falling under the influence of homogeneous gravity follow a path known as a parabolic path, according to Galileo. A curved path in the shape of a parabola is followed by many physical motions of bodies. A parabola is a mirror-symmetrical planar curve that is usually of about U shape in mathematics. Here, we’ll look at how the standard formula for a parabola is derived, as well as the various standard forms and features of a parabola.

## Parabola Calculator with steps

The following is how to use the parabola calculator:

Step 1: In the input field, type the parabola equation.

Then, Step 2: To get the graph, click the “Submit” button.

Then, Step 3: In the next window, the parabola graph will be presented.

## What is Parabola?

A parabola is a curve equation in which a point on the curve is equidistant from both a fixed point and a fixed line. The fixed point is known as the parabola’s focus, and the fixed line is known as the parabola’s directrix. It’s also worth noting that the fixed point is not located on the fixed line. A parabola is a locus of any point that is equidistant from a specified point (focus) and a certain line (directrix). The parabola is an important curve in coordinate geometry’s conic sections.Read Also:Derivative of Sin2x | Formula, Proof, Rules & Examples

The general equation for a parabola is y = a(x-h)² + k or x = a(y-k)² +h, where (h,k) represents the vertex. A normal parabola’s standard equation is y² = 4ax.

## Three Points Parabola Calculator

Given three points on the parabola graph, this calculator finds the parabola equation using the vertical axis. This calculator works by solving a three-variable system of equations.

## Parabola Calculator Equation

The widespread equation of a parabola is:

Y = a (x – h)² + k (regular)

x = a(y – k)² + h (sideways)

where, | h, k |= vertex of the parabola.

The terminology listed below will help you comprehend the characteristics and elements of a parabola.

Focus: The focus of the parabola is the point (a, 0).

The directrix of the parabola is a line drawn parallel to the y-axis and passing through the point (-a, 0). The parabola’s axis is perpendicular to the directrix.

Focal chord: The focal chord of a parabola is the chord that passes through the parabola’s focus. The parabola is cut in two places by the focal chord.

Focal Distance: The focal distance is the distance between a point (x1, y1) on the parabola and the focus. The perpendicular distance of this point from the directrix is also equal to the focal distance.

Latus Rectum: It is the focal chord, which is perpendicular to the parabola’s axis and passes through the parabola’s focus. LL’ = 4a represents the length of the latus rectum. (a, 2a) are the latus rectum’s ends (a, -2a).

Eccentricity: (e equals one). It’s the proportion of a point’s distance from the focus to its distance from the directrix. A parabola’s eccentricity is equal to 1.

## Parabola calculator vertex and focus

A quadratic equation’s standard form is y = ax2 + bx + c. You can use this vertex calculator to convert the equation into vertex form, which will allow you to get the vertex and focus of the parabola.

The vertex form of the parabola equation is y = a(x – h)2 + k, where:

a — The same as the a coefficient in standard form;

Then, h — the parabola vertex’s x-coordinate; and

Then, k — the parabola vertex’s y-coordinate.

The values of h and k can be calculated using the formulae below:

h = – b/(2a)

k = c – b²/(4a)

## Parabola Calculator with focus and directrix

The focus and directrix of the parabola are also found using the parabola vertex form calculator. All you have to do is plug in the following numbers into the equations:

Focus x-coordinate: x₀ = – b/(2a);

Focus y-coordinate: y₀ = c – (b² – 1)/(4a); and

Directrix equation: y = c – (b² + 1)/(4a).

## Standard Equations of a Parabola Calculator

There are four standard parabola equations. The four standard shapes are dependent on the parabola’s axis and orientation. Each of these parabolas has a separate transverse axis and conjugate axis. The four standard parabola equations and shapes are shown in the graphic below.

The following are the results of using the conventional form of equations:

In relation to its axis, a parabola is symmetric. The axis of symmetry is along the x-axis if the equation contains a term with y2, and the axis of symmetry is along the y-axis if the equation has a term with x2.

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 x-axis.

The parabola opens upwards if the coefficient of y is positive and downwards if the coefficient of y is negative when the axis of symmetry is along the y-axis.

## Parabola Calculator Formula

The Parabola Formula is used to depict the general shape of a parabolic path in space. The formulas for calculating the parameters of a parabola are as follows.

The value of a determines the orientation of the parabola.

Vertex = (h,k), where h = -b/2a and k = f(h)

Latus Rectum = 4a

Focus: (h, k+ (1/4a))

Directrix: y = k – 1/4a

## Parabola Calculator Graph

Take the equation y = 3×2 – 6x + 5 as an example. A = 3, b = -6, and c = 5 for this parabola. The graph of the given quadratic equation, which is a parabola, is shown below.

The parabola opens up because an is positive in this case.

Vertex: (h,k)

h = -b/2a

= 6/(2 ×3) = 1

k = f(h)

= f(1) = 3(1)2 – 6 (1) + 5 = 2

Thus vertex is (1,2)

Latus Rectum = 4a = 4 × 3 =12

Focus: (h, k+ 1/4a) = (1,25/12)

Axis of symmetry is x =1

Directrix: y = k-1/4a

y = 2 – 1/12 ⇒ y – 23/12 = 0 ## Parabola Calculator Equation’s Derivation

Consider the point P on the parabola with coordinates (x, y). The distance of this point P from the Directrix is equal to the distance of this point F from the focus F, according to the definition of a parabola. In this case, we’ll use the perpendicular distance PB to calculate a point B on the directrix.

We have PF = PB (because e = PF/PB = 1) according to this definition of the eccentricity of the parabola.

The focus’s coordinates are F(a,0), and we may calculate its distance from P using the coordinate distance formula (x, y)

PF = √(x – a)2 + (y – 0)²

= √(x -a)² + y²

The directrix’s equation is x + a = 0, and we calculate PB using the perpendicular distance formula.

PB= (x+a) / (√1²+0² )

=√(x + a)²

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

√(x − a)² + y² = √(x + a)²

Squaring the equation on both sides,

(x – a)2 + y2 = (x + a)2

x2 + a2 – 2ax + y2 = x2 + a2 + 2ax

y2 – 2ax = 2ax

y2 = 4ax

We’ve now correctly deduced the typical parabola equation.

Similarly, the equations of parabolas can be deduced as:

(b): y2 = – 4ax,

(c): x2 = 4ay,

(d): x2 = – 4ay.

The Standard Equations of Parabolas are the four equations listed above.

## How to Solve the Parabola Equation using Parabola Calculator?

Any parabola equation in the form of y = ax2 + bx + c can be used to obtain the x intercept, y intercept, vertex, focus, directrix, and axis of symmetry. However, we’ll show you how to find all of the parameters of the parabola equation in the sections below. While solving the problem, keep them in mind.

Take any parabola equation as a starting point.

Then, find the values of a, b, and c in the above equation.

Then, substitute the values into the formulas below.

vertex (h, k).

Hence, h = -b / (2a), k = c – b2 / (4a).

Then, the x coordinate focus is -b/2a.

However, the y coordinate focus is c – (b2 – 1)/ (4a)

Then, focus is (x, y)

Directrix equation y = c – (b2 + 1) / (4a)

Axis of symmetry is -b/ 2a.

Maintaining x = 0 in the parabola equation, solve for the y intercept.

To get the appropriate values, perform all mathematical operations.

## Example: 1

However, for the parabola equation y = 5×2 + 4x + 10, what are the vertex, focus, y-intercept, x-intercept, directrix, and axis of symmetry?

Solution:

In the following question, y = 5×2 + 4x + 10 is the given parabola equation.

Then, y = ax2 + bx + c is the conventional version of the equation.

So, a = 5, b = 4, c = 10

However, the parabola equation in vertex form is y = a(x-h)2 + k

Then, h = -b / (2a) = -4 / (2.5)

= -2/5

Then, k = c – b2 / (4a)

= 10 – 42 / (4.5)

= 10- 16 / 20 = 10*20 – 16 / 20

= 184/ 20 = 46/5

Then, y = 5(x-(-2/5))2 + 46/5

= 5(x+2/5)2 + 46/5

Then, Vertex is (-2/5, 46/5)

Then, the focus of x coordinate = -b/ 2a = -2/5

However, focus of y coordinate is = c – (b2 – 1)/ (4a)

= 10 – (16 – 1) / (4.5)

= 10 – 15/20

= 37/4

Then, focus is (-2/5, 37/4)

Then, directrix equation y = c – (b2 + 1) / (4a)

= 10 – (42 + 1) / (4.5)

= 10 – 17 / 20

=200 – 17 / 20

=183/20

Then, axis of Symmetry = -b/ 2a = -2/5

However, to get y-intercept put x = 0 in the equation

y = 5(0)2 + 4(0) + 10

y = 10

However, to get x-intercept put y = 0 in the equation

0 = 5×2 + 4x + 10

No x-intercept.

## Example 2:

However, find the parabola’s equation, focus, axis of symmetry, vertex, directrix, focal parameter, x-intercepts, and y-intercepts that pass through the points (1,4), (2,9), and (1,6).

Solution:

y = ax2 + bx + c is also the conventional form of the parabola equation.

4 = a+b+c

Similarly, 9 = 4a + 2b + c

In the same way, at point (-1,6), 6 = a – b + c

First of all, solve the first and third equations.

a + b+ c = 4

Then, a – b + c = 6

Then, 2(a + c) = 10

Similarly, a + c = 10/2 = 5

Then, substitute a + c = 5 in first equation

So, 5 + b = 4

Or, b = -1

Then, put a = 5-c, b = -1 in second equation

Then, 4(5- c) -2 + c = 9

So, c = 3

Then, substitute b = -1 c = 3 in the third equation

Then, a +1 + 3 = 6

Or, a + 4 = 6

So, a = 2

Then, put a =2, b = -1, c = 3 in the standard form of parabola equation

Then, y = 2×2 – x + 3

However, the parabola equation in vertex form is y = a(x-h)2 + k

Then, h = -b / (2a) = 1/4

Then, k = c – b2 / (4a) = 3 – 1 / 8 = 23/8

Likewise, y = 2(x-1/4)2 + 23/8

Then, Vertex is (1/4, 23/8)

Then, the focus of x coordinate = -b/ 2a = 1/4

Similarly, Focus of y coordinate is = c – (b2 – 1)/ (4a)

= 3 – (1 – 1) / (4.2)

= 3/8

Then, focus is (1/4, 3/8)

Then, directrix equation y = c – (b2 + 1) / (4a)

= 11/4

Then, axis of Symmetry = -b/ 2a = 1/4

However, to get y-intercept put x = 0

y = 2(0)2 – 0 + 3

Then, y = 3

Similarly, y intercept (0, 3)

However, to get x-intercept put y = 0

2×2 – x + 3 = 0

## Parabola Calculator Properties

Here, we’ll look at some of the most important properties and phrases associated with a parabola.

A tangent is a line that touches the parabola. At the point of contact (x1, y1), a tangent to the parabola y2 = 4ax has the equation yy1 = 2a (x + x1).

The normal is a line that is drawn perpendicular to the tangent and passes through the point of contact and the focus of the parabola. The equation of the normal passing through the point (x1, y1) and having a slope of m = -y1/2a is (y – y1) = -y1/ 2a (x – x1) for a parabola y2 = 4ax, the equation of the normal passing through the point (x1, y1) and having a slope of m = -y1/2a.

Chord of Contact: The chord of contact is constructed to connect the point of contact of the tangents drawn from an external point to the parabola. The chord of contact equation for a location (x1, y1) outside the parabola is yy1 = 2x (x + x1).

The locus of the points of intersection of the tangents, drawn at the ends of the chords, traced from this point is called the polar for a point located outside the parabola. This referring point is referred to as the pole. The equation of the polar is yy1 = 2x (x + x1) for a pole with coordinates (x1, y1) and a parabola with y2 =4ax.

The parametric coordinates of the equation of a parabola y2 = 4ax are as follows: (at2, 2at). All of the points on the parabola are represented by the parametric coordinates.

## Parabola Calculator Examples

Example 1: A parabola’s equation is y2 = 24x. Determine the latus rectum’s length, focus, and vertex.

Solution:

To locate: The focus and vertex of the parabola, as well as the length of the latus rectum

Given: A parabola’s equation is y2 = 24x.

Therefore, 4a = 24

a = 24/4 = 6

Now, parabola formula for latus rectum is:

Length of latus rectum = 4a

= 4(6) = 24

Now, focus= (a,0) = (6,0)

Now, Vertex = (0,0)

Answer: Length of latus rectum = 24, focus = (6,0), vertex = (0,0)

Example 2: A parabola’s equation is 2(y-3)2 + 24 = x. Determine the latus rectum’s length, focus, and vertex.

Solution: To locate: The centre and vertex of a parabola are the length of the latus rectum.

Given: 2(y-3)2 + 24 = x is a parabola’s equation.

When we compare it to the general parabola equation x = a(y-k)2 + h, we get

a = 2

(h, k) = (24, 3)

Now, parabola formula for latus rectum: Length of latus rectum = 4a

= 4(2) = 8

Now, focus= (0,a) = (0,2)

Now, Vertex = (24,3)

Answer: Length of latus rectum = 8, focus = (0, 2), Vertex = (24,3)

### What is Parabola in the Conic Section?

The parabola is an essential conic section curve. The focus is the locus of a point that is equidistant from a fixed point, and the fixed-line is the directrix. A parabolic route is followed by many motions in the physical world. As a result, physicists must first grasp the properties and applications of a parabola.

### What is the Equation of Parabola Calculator?

A parabola’s conventional equation is y2 = 4ax. The x-axis, which is also the parabola’s transverse axis, is the parabola’s axis. The focus of the parabola is F(a, 0), and the directrix equation for this parabola is x + a = 0.

### What is the Vertex of the Parabola Calculator?

The point where the parabola cuts through the axis is the vertex of the parabola. Because it cuts the axis at the origin, the vertex of the parabola with the equation y2 = 4ax is (0,0).

### How to Find the Equation of a Parabola Calculator?

The basic definition of the parabola can be used to determine the equation of the parabola. The locus of a point that is equidistant from a fixed point called the focus (F), and the fixed-line is called the Directrix (x + a = 0), is called a parabola. Consider the point P(x, y) on the parabola, and we can find the equation of the parabola using the formula PF = PM. On the directrix, the point ‘M’ is the foot of the perpendicular from the point P. As a result, the parabola’s derived standard equation is y2 = 4ax.

### What is The Eccentricity of Parabola Calculator?

A parabola’s eccentricity is equal to 1 (e = 1). The eccentricity of a parabola is the ratio of the point’s distance from the focus to the point’s distance from the parabola’s directrix.

### What is the Foci of a Parabola Calculator?

There is just one point of focus on the parabola. The focus of the parabola is F for a conventional equation of the parabola y2 = 4ax (a, 0). It is a point on the parabola’s x-axis as well as the transverse axis.

### What is the Conjugate Axis of a Parabola Calculator?

The conjugate axis of the parabola is a line that is perpendicular to the parabola’s transverse axis and passes through the parabola’s vertex. However, the y-axis is the conjugate axis for a parabola y2 = 4ax.

### What are The Vertices of a Parabola Calculator?

The vertex of the parabola is the point on the axis where the parabola passes through it. F

However, for a conventional equation of a parabola, y2 = 4ax, the vertex of the parabola is equal to (0, 0). We cut the x-axis at the origin with the help of parabola.

### What is the Standard Equation of a Parabola Calculator?

However, in the coordinate plane, a parabola is represented algebraically using the standard equation of a parabola. A parabola’s general equation is y = a(x-h)2 + k or x = a(y-k)2 +h, where (h,k) represents the vertex. A normal parabola’s standard equation is y² = 4ax.

### How to Find the Transverse Axis of a Parabola Calculator?

The transverse axis of the parabola is the line that passes through the vertex and focus of the parabola. The x-axis is the axis of the parabola in the conventional equation of the parabola, y2 = 4ax.

The general equation of a parabola is:

y = a(x – h)2 + k (regular)

And, x = a(y – k)2 + h (sideways)

where,

(h,k) = vertex of the parabola

### Where is the Parabola Calculator Formula Used in Real Life?

However, the routes of ballistic missiles, the construction of automotive headlight reflectors, and other applications of parabolas are employed in physics and engineering.

### How Do You Solve Problems Using Parabola Calculator Formula?

We use the general equation of the parabola to address problems with parabolas. However, it has the general form y = ax² + bx + c (vertex form y = a(x – h) 2 + k), where (h,k) is the parabola’s vertex.

### Do all Parabolas Calculator Formula Represent a Function?

It’s not true that all parabolas are functions. Functions are parabolas that open either upwards or downwards.