## Parabola Calculator

### Solution

**Your input: find the equation, focus, axis of symmetry, eccentricity, latus rectum, length of the latus rectum, vertex, directrix, focal parameter, x-intercepts, y-intercepts of the parabola that passes through the points $$$\left(1, 4\right)$$$, $$$\left(2, 9\right)$$$, $$$\left(-1, 6\right)$$$.**

Assume that the equation of the parabola is $$$y=a x^{2} + b x + c$$$.

Since the parabola passes through the point $$$\left(1, 4\right)$$$, then $$$4=a + b + c$$$.

Since the parabola passes through the point $$$\left(2, 9\right)$$$, then $$$9=4 a + 2 b + c$$$.

Since the parabola passes through the point $$$\left(-1, 6\right)$$$, then $$$6=a - b + c$$$.

Thus, we have obtained the following system: \begin{cases}a + b + c=4\\4 a + 2 b + c=9\\a - b + c=6\end{cases}

Solving it (for steps, see system of linear equations calculator), we get that $$$a=2$$$, $$$b=-1$$$, $$$c=3$$$.

Thus, the equation of the parabola is $$$y=2 x^{2} - x + 3$$$.

### Answer

**Equation of the parabola: $$$y=2 x^{2} - x + 3$$$.**

**Vertex form: $$$y=2 \left(x - \frac{1}{4}\right)^{2} + \frac{23}{8}$$$.**

**No intercept form.**

**Vertex: $$$\left(\frac{1}{4},\frac{23}{8}\right)$$$.**

**Focus: $$$\left(\frac{1}{4},3\right)$$$.**

**Eccentricity: $$$1$$$.**

**Directrix: $$$y=\frac{11}{4}$$$.**

**Latus rectum: $$$y=3$$$.**

**The length of the latus rectum: $$$\frac{1}{2}$$$.**

**Axis of symmetry: $$$x=\frac{1}{4}$$$.**

**Focal parameter: $$$\frac{1}{4}$$$.**

**No x-intercepts.**

**y-intercept: $$$\left(0, 3\right)$$$.**

**Graph:** to graph the parabola, visit the parabola graphing calculator (choose the "Implicit" option).

## Parabola Calculator

Any time you come across a quadratic formula you want to analyze, you'll find this parabola calculator to be the perfect tool for you. Not only will it provide you with the **parabola equation** in both the **standard form** and the **vertex form**, but also calculate the parabola vertex, focus, and directrix for you.

### What is a parabola?

A parabola is a **U-shaped symmetrical curve**. Its main property is that every point lying on the parabola is equidistant from both a certain point, called the focus of a parabola, and a line, called its directrix. It is also the curve that corresponds to quadratic equations.

The axis of symmetry of a parabola is always perpendicular to the directrix and goes through the focus point. The vertex of a parabola is the point at which the parabola makes its sharpest turn; it lies halfway between the focus and the directrix.

A real-life example of a parabola is the path traced by an object in projectile motion.

### The parabola equation in vertex form

The standard form of a quadratic equation is . You can use this vertex calculator to transform that equation into the vertex form, which allows you to find the important points of the parabola – its vertex and focus.

The parabola equation in its vertex form is , where:

**a**— Same as the**a**coefficient in the standard form;**h**— x-coordinate of the parabola vertex; and**k**— y-coordinate of the parabola vertex.

You can calculate the values of **h** and **k** from the equations below:

### Parabola focus and directrix

The parabola vertex form calculator also finds the focus and directrix of the parabola. All you have to do is to use the following equations:

- Focus x-coordinate: ;
- Focus y-coordinate: ; and
- Directrix equation: .

### How to use the parabola equation calculator: an example

Enter the coefficients a, b and c of the standard form of your quadratic equation. Let's assume that the equation is , what means that a = 2, b = 3 and c = -4.

Calculate the coordinates of the vertex, using the formulas listed above:

Find the coordinates of the focus of the parabola. The x-coordinate of the focus is the same as the vertex's (x₀ = -0.75), and the y-coordinate is:

Find the directrix of the parabola. You can either use the parabola calculator to do it for you, or you can use the equation:

### FAQ

### What is a parabola?

A parabola is a symmetrical U shaped curve such that every point on the curve is equidistant to the directrix and the focus.

### How do I define a parabola?

A parabola is defined by the equation such that every point on the curve satisfies it. Mathematically, .

### How do I calculate the vertex of a parabola?

To calculate the vertex of a parabola defined by coordinates (x, y):

Find x coordinate using the axis of symmetry formula:

Find y coordinate using the equation of parabola:

### How to calculate the focus of a parabola?

To calculate the focus of a parabola defined by coordinates (x, y):

- Find y coordinate using the formula
- Find x coordinate using the equation of parabola.

### Most Used Actions

\mathrm{simplify} | \mathrm{solve\:for} | \mathrm{inverse} | \mathrm{tangent} | \mathrm{line} |

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### Related

### Number Line

### Graph

### Examples

ellipse-function-foci-calculator

en

### Most Used Actions

\mathrm{simplify} | \mathrm{solve\:for} | \mathrm{inverse} | \mathrm{tangent} | \mathrm{line} |

Our online expert tutors can answer this problem

Get step-by-step solutions from expert tutors as fast as 15-30 minutes. Your first 5 questions are on us!

In partnership with

You are being redirected to Course Hero

Correct Answer :)

Let's Try Again :(

Try to further simplify

### Related

### Number Line

### Graph

### Examples

parabola-function-foci-calculator

en

## Calculator focus

## Depth of Field Calculator

Simply put, depth-of-field is how much of a photograph is in sharp focus from front to back.

Digital Landscape Photography: In the Footsteps of Ansel Adams and the Masters, Michael Frye, 2010.

We can achieve critical focus for only one plane in front of the camera, and all objects in this plane will be sharp. In addition, there will be an area just in front of and behind this plane that will appear reasonably sharp (according to the standards of sharpness required for the particular photograph and the degree of enlargement of the negative). This total region of adequate focus represents the

depth of field.

The Camera (Ansel Adams Photography, Book 1), Ansel Adams, Tenth Edition, 1995

### Hyperfocal Distance Definition

... the

hyperfocal distance setting... is simply a fancy term that means thedistance settingat any aperture that produces thegreatest depth of field.How to Use Your Camera,New York Institute of Photography, 2000.

If you set the camera's focus to the hyperfocal distance, your depth of field will extend from half of the hyperfocal distance to infinity—a much deeper depth of field.

Complete Digital Photography, Ben Long, 2012.

Another important control for landscape photography is depth of field, the amount of sharpness in a scene, from close to the camera into the distance away from the camera. It's sharpness in depth.

Landscape Photography: From Snapshots to Great Shots, Rob Sheppard, 2012.

Parabola Calculator

__Parabola Grapher Online__)

Parabola Vertex Focus Calculator Formulas

^{2}+ bX + c, a≠0)

• Focus X = -b/2a

• Focus Y = c - (b

^{2}- 1)/4a

• Vertex X = -b/2a

• Directrix Y = c - (b

^{2}+ 1)/4a

• X Intercept = -b/2a ± √(b * b - 4ac)/2a,0

Parabola equation and graph with major axis parallel to y axis. If a>0, parabola is upward, a

Segment of a Parabola Calculator

Segment of a Parabola Formulas

• Arc Length = sqrt(b

^{2}+ 16 * h

^{2})/2 + b

^{2}* ln((4 * h + sqrt(b

^{2}+ 16 * h

^{2}))/b)/(8*h)

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### How to use the DoF Calculator

This calculator will help you assess what camera settings are required to achieve a desired level of sharpness.

Depth of field is one of the most powerful creative tools in photography and, to help you master it, we've prepared a DoF guide with lots of love. Read it and you'll become a truly story teller, I promise:

Depth of Field: The Definitve Guide

Sometimes, you’ll want to maximize depth of field in order to keep everything sharp. A classic example is when you’re photographing the Milky Way, where you typically want to capture detail from the foreground to the horizon while capturing stars as big bright spots. Commonly, you’ll use deep depth of field when photographing landscapes (daytime and at night), cityscapes and architecture.

Other times, you’ll prefer to use a shallow depth of field to direct viewers’ attention to a specific place, for example, to separate a subject from a busy background. It’s the typical case of portrait photography, but it also comes very handy in landscape, street, product, event and close-up photography.

Have a look at the following picture, we use shallow depth of field to lure the viewer's attention to the model. We used this photo when our t-shirt shop went live.

In the calculator, just introduce your camera type (sensor size), aperture, focus distance, focal length (*the real one! not the equivalent in 35mm*) and teleconverter to calculate the depth of field:

**Hyperfocal distance:**The closest distance at which a lens can be focused while keeping objects at infinity acceptably sharp. When the lens is focused at this distance, all objects at distances from half of the hyperfocal distance out to infinity will be acceptably sharp.**Hyperfocal near limit:**The distance between the camera and the first element that is considered to be acceptably sharp when focusing at the hyperfocal distance.**Depth of field (DOF):**The distance between the farthest and nearest points which are in acceptable focus. This can also be identified as the zone of acceptable sharpness in front of and behind the subject to which the lens is focused on.**DOF near limit:**The distance between the camera and the first element that is considered to be acceptably sharp.**DOF far limit:**The distance between the camera and the furthest element that is considered to be acceptably sharp.**Depth of Field (DOF) In Front:**Distance between the DoF Near Limit and the focus plane.**Depth of Field (DOF) Behind:**Distance between the focus plane and the DoF Far Limit.

### Depth of Field Calculator in PhotoPills app

This calculator is also available in PhotoPills app, extended with an augmented reality view to help you visualize where to focus. Besides, you’ll also find an advanced DOF calculator to let you set the Circle of Confusion (CoC). Both classic and advanced calculators include an inverse mode (DoF to settings).

Note: given the sensor size, the circle of confusion is calculated assuming a print size of 8''×10'' (20cm×25cm), a viewing distance of 10" (25cm) and the manufacturers standard visual acuity.

Finally, If you're interested in improving your photography, check our detailed photography guides on:

And also check these fundamental photography guides:

### How to embed the DoF Calculator on your website

Take the power of PhotoPills’ Depth of Field (DOF) calculator with you. Just copy the following lines and paste them within the code of your website, right in the place where you want to embed it:

The code will run asynchronously, without penalizing the loading time of your website.

**261**262 263