Equation of a Parabola Calculator – Find Parabola Equation

Equation of a Parabola Calculator

Calculate Parabola Equation

Enter the vertex and another point on the parabola, or the focus and directrix.

Calculation Method:

Vertex (h):

Vertex (k):

Point (x):

Point (y):

Orientation:

Results

Enter values to see the equation.

Graphical representation of the parabola.

Parameter	Value
'a' value	–
Vertex (h, k)	–
Focus	–
Directrix	–
Axis of Symmetry	–

Calculated parameters of the parabola.

What is an Equation of a Parabola Calculator?

An equation of a parabola calculator is a tool designed to determine the standard or vertex form equation of a parabola based on certain given geometric properties. A parabola is a U-shaped curve, and its equation describes the relationship between the x and y coordinates of every point on that curve. This calculator typically requires inputs such as the coordinates of the vertex and another point on the parabola, or the focus and the equation of the directrix.

Anyone studying algebra, geometry, physics (e.g., projectile motion), or engineering might use an equation of a parabola calculator. It simplifies the process of finding the equation, which can then be used for graphing, analysis, or solving related problems. Common misconceptions include thinking all U-shaped curves are parabolas or that only one form of the equation exists.

Equation of a Parabola Formula and Mathematical Explanation

A parabola is defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

1. Vertex Form

If the vertex of the parabola is at (h, k):

If the parabola opens up or down (vertical axis of symmetry): The equation is y = a(x - h)² + k.
If the parabola opens left or right (horizontal axis of symmetry): The equation is x = a(y - k)² + h.

The value 'a' determines the width and direction of the parabola. If you know the vertex (h, k) and another point (x, y) on the parabola, you can find 'a':

For vertical axis: a = (y - k) / (x - h)² (provided x ≠ h)
For horizontal axis: a = (x - h) / (y - k)² (provided y ≠ k)

2. Standard Form (from Focus and Directrix)

Let the focus be (f, g) and the directrix be a line.

If the directrix is horizontal (y = d), the parabola opens up or down. The vertex is at (f, (g+d)/2), and the equation relates to (x-f)². The distance 'p' from vertex to focus/directrix is |g-d|/2. The equation is (x - f)² = 4p(y - (g+d)/2) or y = (1/4p)(x-f)² + (g+d)/2. 'a' = 1/(4p).
If the directrix is vertical (x = d), the parabola opens left or right. The vertex is at ((f+d)/2, g), and the equation relates to (y-g)². The distance 'p' is |f-d|/2. The equation is (y - g)² = 4p(x - (f+d)/2) or x = (1/4p)(y-g)² + (f+d)/2. 'a' = 1/(4p).

The equation of a parabola calculator uses these formulas based on your inputs.

Variables Used
Variable	Meaning	Unit	Typical Range
(h, k)	Coordinates of the vertex	Length units	Any real numbers
(x, y)	Coordinates of a point on the parabola	Length units	Any real numbers
a	Scaling factor, determines width and direction	Inverse length units (for vertical/horizontal)	Any non-zero real number
(f, g)	Coordinates of the focus	Length units	Any real numbers
d	Value defining the directrix line (y=d or x=d)	Length units	Any real number
p	Distance from vertex to focus/directrix	Length units	Any non-zero real number

Practical Examples (Real-World Use Cases)

Example 1: Parabolic Reflector

A satellite dish is shaped like a paraboloid. If the vertex is at (0, 0) and the dish is 4 units wide at a depth of 1 unit (meaning a point (2, 1) is on the parabola if it opens up), find its equation.

Vertex (h, k) = (0, 0)
Point (x, y) = (2, 1)
Orientation: Opens up (y = a(x-h)² + k)
1 = a(2 - 0)² + 0 => 1 = 4a => a = 1/4
Equation: y = (1/4)x²

The equation of a parabola calculator would quickly give y = 0.25x².

Example 2: Projectile Motion (Simplified)

The path of a ball thrown can be modeled by a parabola (ignoring air resistance). If it reaches a maximum height (vertex) of 10 meters at a horizontal distance of 8 meters, and it started at (0, 0), what's the equation?

Vertex (h, k) = (8, 10)
Point (x, y) = (0, 0) (starting point, or landing point 16,0)
Orientation: Opens down (y = a(x-h)² + k)
0 = a(0 - 8)² + 10 => 0 = 64a + 10 => a = -10/64 = -5/32
Equation: y = (-5/32)(x - 8)² + 10

Using our equation of a parabola calculator with h=8, k=10, x=0, y=0, and opens up/down, we get a=-0.15625, so y = -0.15625(x-8)²+10.

How to Use This Equation of a Parabola Calculator

Select Method: Choose whether you have the "Vertex and a Point" or "Focus and Directrix".
Enter Vertex and Point: If selected, input the h and k coordinates of the vertex, and the x and y coordinates of another point on the parabola. Then select the orientation (opens up/down or left/right).
Enter Focus and Directrix: If selected, input the x and y coordinates of the focus, choose the directrix form (y=d or x=d), and enter the value of 'd'.
Calculate: Click the "Calculate" button or observe real-time updates if you change values after an initial calculation.
Read Results: The calculator will display:
- The equation of the parabola in vertex form or standard form.
- The value of 'a'.
- The coordinates of the vertex (if calculated from focus/directrix).
- The coordinates of the focus.
- The equation of the directrix.
- The equation of the axis of symmetry.
View Chart and Table: A graph of the parabola and a table summarizing its properties are also shown.
Reset or Copy: Use "Reset" to clear inputs or "Copy Results" to copy the main findings.

This equation of a parabola calculator is a powerful tool for quickly finding the equation and key features of a parabola.

Key Factors That Affect Parabola Equation Results

Vertex Coordinates (h, k): The location of the vertex directly shifts the parabola horizontally and vertically, appearing as (x-h) and + k in the equation.
Point Coordinates (x, y): The position of the other point, relative to the vertex, determines the 'a' value – how wide or narrow the parabola is and its direction if orientation is fixed.
Orientation: Whether the parabola opens up/down (y = a(x-h)² + k) or left/right (x = a(y-k)² + h) changes the basic form of the equation and which variable is squared.
Focus Coordinates (f, g): The focus location is fundamental to the definition of a parabola and directly influences its shape and position.
Directrix (y=d or x=d): The directrix line, along with the focus, defines the parabola. Its position and orientation (horizontal or vertical) are crucial.
Value of 'a': This coefficient dictates the parabola's "steepness" and direction. A larger |a| means a narrower parabola, a smaller |a| means a wider one. The sign of 'a' determines the opening direction (positive 'a' for up/right, negative for down/left, depending on orientation).

Understanding these factors helps in interpreting the results from the equation of a parabola calculator.

Frequently Asked Questions (FAQ)

What is a parabola?: A parabola is a U-shaped curve defined as the set of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
What is the vertex form of a parabola's equation?: For a vertical axis, it's y = a(x - h)² + k, and for a horizontal axis, it's x = a(y - k)² + h, where (h, k) is the vertex.
What does 'a' represent in the parabola equation?: 'a' is a non-zero constant that determines the parabola's width and opening direction. If |a| is large, the parabola is narrow; if |a| is small, it's wide. The sign determines direction.
How do I find the equation if I have three points?: If you have three non-collinear points, you can substitute their coordinates into y = Ax² + Bx + C (for vertical) or x = Ay² + By + C (for horizontal) to get a system of three linear equations in A, B, and C, then solve for A, B, and C. Our equation of a parabola calculator currently uses vertex/point or focus/directrix methods.
What is the focus of a parabola?: The focus is a fixed point inside the parabola used in its geometric definition. Rays parallel to the axis of symmetry reflect off the parabola and pass through the focus.
What is the directrix of a parabola?: The directrix is a fixed line outside the parabola used in its definition. Each point on the parabola is the same distance from the focus and the directrix.
What is the axis of symmetry?: It's a line that divides the parabola into two mirror images. It passes through the vertex and the focus.
Can 'a' be zero?: No, if 'a' were zero, the equation would become linear (y = k or x = h), not quadratic, so it wouldn't represent a parabola.

Related Tools and Internal Resources

{related_keywords[0]}: Explore quadratic equations, which are closely related to parabolas.
{related_keywords[1]}: Calculate the distance between two points, useful for verifying parabola properties.
{related_keywords[2]}: Find the midpoint, which can be helpful when dealing with the vertex and focus/directrix.
{related_keywords[3]}: Understand the slope of lines, including the axis of symmetry or directrix.
{related_keywords[4]}: Solve systems of equations, useful for finding the parabola through three points.
{related_keywords[5]}: Graph various functions, including parabolas.

Finding Equation Of A Parabola Calculator