Use this user friendly Parabola Calculator tool to get the output in a short span of time. You just need to enter the parabola equation in the specified input fields and hit on the calculator button to acquire vertex, x intercept, y intercept, focus, axis of symmetry, and directrix as output.

**Parabola Calculator: **Are you trying to solve the parabola equation? If yes, this is the right spot for you. From here you will be going to learn the process of calculating parabola equation and finding vertex, focus, x and y intercepts, directrix and axis of symmetry values. By checking the below sections, you will get a good knowledge on the parabola equation concept and you will also obtain a handy calculator tool that gives result in fraction of seconds.

We can find the x intercept, y intercept, vertex, focus, directrix, axis of symmetry using any parabola equation in the form of y = ax^{2} + bx + c. In the following sections, we are providing the simple steps to find all those parameters of parabola equation. Follow them while solving the equation.

- At first, take any parabola equation.
- Find out a, b, c values in the given equation
- Substitute those values in the below formulae
- Vertex v (h, k).
- h = -b / (2a), k = c - b
^{2}/ (4a). - Focus of the x coordinate is -b/2a.
- Focus of the y coordinate is c - (b
^{2}- 1)/ (4a) - Then, focus is (x, y)
- Directrix equation y = c - (b
^{2}+ 1) / (4a) - Axis of symmetry is -b/ 2a.
- Solve the y intercept by keeping x = 0 in the parabola equation.
- Perform all mathematical operations to get the required values.

**Examples**

**Question 1: Find vertex, focus, y-intercept, x-intercept, directrix, and axis of symmetry for the parabola equation y = 5x ^{2} + 4x + 10?**

**Solution:**

Given Parabola equation is y = 5x^{2} + 4x + 10

The standard form of the equation is y = ax^{2} + bx + c

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

The parabola equation in vertex form is y = a(x-h)^{2} + k

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

= -2/5

k = c - b^{2} / (4a)

= 10 - 4^{2} / (4.5)

= 10- 16 / 20 = 10*20 - 16 / 20

= 184/ 20 = 46/5

y = 5(x-(-2/5))^{2} + 46/5

= 5(x+2/5)^{2} + 46/5

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

The focus of x coordinate = -b/ 2a = -2/5

Focus of y coordinate is = c - (b^{2} - 1)/ (4a)

= 10 - (16 - 1) / (4.5)

= 10 - 15/20

= 37/4

Focus is (-2/5, 37/4)

Directrix equation y = c - (b^{2} + 1) / (4a)

= 10 - (4^{2} + 1) / (4.5)

= 10 - 17 / 20

=200 - 17 / 20

=183/20

Axis of Symmetry = -b/ 2a = -2/5

To get y-intercept put x = 0 in the equation

y = 5(0)^{2} + 4(0) + 10

y = 10

To get x-intercept put y = 0 in the equation

0 = 5x^{2} + 4x + 10

No x-intercept.

**Question 2: Find the equation, focus, axis of symmetry, vertex, directrix, focal parameter, x-intercepts, y-intercepts of the parabola that passes through the points (1,4), (2,9), (−1,6)?**

**Solution:**

Given points are (1,4), (2,9), (−1,6)

The standard form of the equation of the parabola is y = ax^{2} + bx + c

When the parabola passes through the point (1,4) then, 4 = a+b+c

when the parabola passes through the point (2,9), then 9 = a(2)^{2} + b(2) + c = 4a + 2b + c

when the parabola passes through the point (-1,6), then 6 = a - b + c

Solve first and third equation

a + b+ c = 4

a - b + c = 6

2(a + c) = 10

a + c = 10/2 = 5

substitute a + c = 5 in first equation

5 + b = 4

b = 4-5

b = -1

Put a = 5-c, b = -1 in second equation

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

20 - 4c -2 +c = 9

18 - 3c = 9

18-9 = 3c

c = 3

Substitute b = -1 c = 3 in the third equation

a +1 + 3 = 6

a + 4 = 6

a = 2

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

y = 2x^{2} - x + 3

The parabola equation in vertex form is y = a(x-h)^{2} + k

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

k = c - b^{2} / (4a) = 3 - 1 / 8 = 23/8

y = 2(x-1/4)^{2} + 23/8

Vertex is (1/4, 23/8)

The focus of x coordinate = -b/ 2a = 1/4

Focus of y coordinate is = c - (b^{2} - 1)/ (4a)

= 3 - (1 - 1) / (4.2)

= 3/8

Focus is (1/4, 3/8)

Directrix equation y = c - (b^{2} + 1) / (4a)

= 3 - (1 + 1) / (4.2)

= 3-2/8

=24-2/8 = 22/8 = 11/4

Axis of Symmetry = -b/ 2a = 1/4

To get y-intercept put x = 0 in the equation

y = 2(0)^{2} - 0 + 3

y = 3

y intercept (0, 3)

To get x-intercept put y = 0 in the equation

0 = 2x^{2} - x + 3

No x-intercept.

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**1. What is meant by parabola?**

Parabola is a curve where any point is at an equal distance from a fixed point (focus) and a fixed straight line (directrix).

**2. Where is parabola used in real life?**

Parabola can be seen in nature or in man made items. From the paths of thrown baseballs, to fountains, even functions, to satellite, and radio waves.

**3. Where is the focus of a parabola?**

A parabola is set of all points in a plane which are equal distance from a given point and given line. The point is called parabola focus and the line is known as directrix of parabola. The focus lies on the axis of symmetry of the parabola.

**4. What is a parabola used for?**

The parabola has many important applications, from a parabolic microphone or parabolic antenna to automobile headlight reflectors and the design of ballistic missiles. They are frequently used in engineering, physics and other areas.