Find The Equation Of A Parabola Calculator

This calculator helps you find the equation of a parabola given certain information like the vertex and a point, three points, or the focus and directrix. Our find the equation of a parabola calculator is easy to use.

Parabola Equation Calculator

Parameter	Value

Summary of inputs and calculated parabola parameters.

What is a Parabola and its Equation?

A parabola is a U-shaped curve that is a graph of a quadratic function (like y = x²), or more generally, it's a conic section formed by the intersection of a cone with a plane parallel to its side. Every point on a parabola is equidistant from a fixed point (the focus) and a fixed straight line (the directrix). The find the equation of a parabola calculator helps determine this equation based on given geometric properties.

This find the equation of a parabola calculator is useful for students learning algebra and analytic geometry, engineers designing parabolic reflectors or antennas, and physicists studying projectile motion under gravity (which often follows a parabolic path, ignoring air resistance).

Common misconceptions include thinking all U-shaped curves are parabolas or that the 'a' value only affects how 'wide' the parabola is, when it also determines the direction of opening.

Parabola Formula and Mathematical Explanation

The equation of a parabola can be expressed in several forms:

1. Vertex Form (Axis Vertical): y = a(x – h)² + k
2. Vertex Form (Axis Horizontal): x = a(y – k)² + h
3. Standard Form (Axis Vertical): y = Ax² + Bx + C
4. Standard Form (Axis Horizontal): x = Ay² + By + C
5. From Focus and Directrix (Vertical Axis): (x – f)² = 4p(y – k), where p is distance from vertex to focus/directrix, vertex (f, k), focus (f, k+p), directrix y=k-p. If focus (f,g) and directrix y=d, then k=(g+d)/2, p=(g-d)/2, a=1/(4p)=1/(2(g-d)), equation is y = (1/(2(g-d)))(x-f)² + (g+d)/2.
6. From Focus and Directrix (Horizontal Axis): (y – g)² = 4p(x – k), where vertex (k, g), focus (k+p, g), directrix x=k-p. If focus (f,g) and directrix x=d, then k=(f+d)/2, p=(f-d)/2, a=1/(4p)=1/(2(f-d)), equation is x = (1/(2(f-d)))(y-g)² + (f+d)/2.

Our find the equation of a parabola calculator uses these formulas based on your inputs.

When given three points (x1, y1), (x2, y2), (x3, y3) and assuming the form y = Ax² + Bx + C, we solve the system of linear equations:

y1 = Ax1² + Bx1 + C
y2 = Ax2² + Bx2 + C
y3 = Ax3² + Bx3 + C

to find A, B, and C. A similar system is solved for x = Ay² + By + C.

Variables in Parabola Equations

Variable	Meaning	Unit	Typical Range
(h, k) or (k, g)	Coordinates of the Vertex	Length	Any real numbers
(x, y)	Coordinates of a point on the parabola	Length	Any real numbers
a or A	Coefficient determining width and direction of opening	1/Length or None	Any non-zero real number
(f, g)	Coordinates of the Focus	Length	Any real numbers
d	Value defining the Directrix line (y=d or x=d)	Length	Any real number
p	Distance from vertex to focus/directrix	Length	Any non-zero real number
B, C	Coefficients in standard form	Varies	Any real numbers

Practical Examples (Real-World Use Cases)

Using a find the equation of a parabola calculator is helpful in many scenarios.

Example 1: Vertex and a Point

Suppose a parabolic satellite dish has its vertex at (0, 0) and passes through the point (2, 1), with a vertical axis. We want to find its equation.

• Vertex (h, k) = (0, 0)
• Point (x, y) = (2, 1)
• Axis: Vertical
• Using y = a(x-h)² + k: 1 = a(2-0)² + 0 => 1 = 4a => a = 1/4
• Equation: y = (1/4)x²

Example 2: Three Points

A bridge's cable is in the shape of a parabola (y=Ax²+Bx+C). It's supported at two towers 100m apart, and the cable touches the road midway between them. The towers are 20m high, and we set the origin at the lowest point of the cable on the road. The points are (-50, 20), (0, 0), and (50, 20).

• Point 1: (-50, 20)
• Point 2: (0, 0)
• Point 3: (50, 20)
• Using (0,0): C=0
• Using (-50,20): 20 = A(-50)² + B(-50) => 20 = 2500A – 50B
• Using (50,20): 20 = A(50)² + B(50) => 20 = 2500A + 50B
• Adding the last two: 40 = 5000A => A = 40/5000 = 1/125. Then B=0.
• Equation: y = (1/125)x²

The find the equation of a parabola calculator quickly solves these.

How to Use This Find the Equation of a Parabola Calculator

1. Select Method: Choose how you know the parabola: "Vertex and a Point", "Three Points", or "Focus and Directrix".
2. Enter Data:
   • For "Vertex and a Point": Input vertex (h, k) and point (x, y) coordinates, and select axis orientation.
   • For "Three Points": Input coordinates (x1, y1), (x2, y2), (x3, y3), and select assumed form.
   • For "Focus and Directrix": Input focus (f, g) and directrix value (d), and select directrix type (y=d or x=d).
3. Calculate: Click "Calculate Equation".
4. View Results: The calculator will display the equation of the parabola, key parameters like 'a', vertex, focus, directrix, and axis of symmetry, and a graph. The table also summarizes results. Our find the equation of a parabola calculator provides comprehensive output.

Use the "Reset" button to clear inputs and "Copy Results" to copy the output.

Key Factors That Affect Parabola Equation Results

Several factors influence the equation derived by the find the equation of a parabola calculator:

• The value of 'a' or 'A': This determines how wide or narrow the parabola is and whether it opens upwards/downwards (for vertical axis) or right/left (for horizontal axis). A smaller |a| means a wider parabola.
• The coordinates of the vertex (h, k): This point dictates the location of the parabola's minimum or maximum point (or leftmost/rightmost for horizontal axis).
• The coordinates of the focus: The focus is a key point in the geometric definition of a parabola, influencing 'a' and the vertex location relative to the directrix.
• The equation of the directrix: This line, along with the focus, defines the parabola. Its position and orientation are crucial.
• The coordinates of points on the parabola: The specific points used constrain the possible equations. If using three points, make sure they are not collinear, or the calculator might not find a standard parabolic form. The find the equation of a parabola calculator handles these inputs.
• The orientation of the axis of symmetry: Whether the parabola opens up/down (vertical axis) or left/right (horizontal axis) fundamentally changes the form of the equation (y=… or x=…).

Frequently Asked Questions (FAQ)

Q1: What are the different forms of a parabola's equation?

A1: The main forms are the vertex form (y = a(x-h)² + k or x = a(y-k)² + h) and the standard form (y = Ax² + Bx + C or x = Ay² + By + C). The find the equation of a parabola calculator can output these.

Q2: How do I find the focus and directrix from the vertex form y = a(x-h)² + k?

A2: The vertex is (h, k). The distance p from vertex to focus/directrix is |1/(4a)|. If a>0, focus is (h, k+p), directrix y=k-p. If a<0, focus is (h, k-p), directrix y=k+p.

Q3: What if the three points I have are collinear?

A3: If the three points lie on a straight line, they do not define a unique parabola of the form y=Ax²+Bx+C or x=Ay²+By+C. The calculator may indicate an error or that A=0 (resulting in a line).

Q4: Can a parabola have a slanted axis of symmetry?

A4: Yes, but those parabolas have equations involving an 'xy' term (rotated conics), which this basic find the equation of a parabola calculator does not handle. It assumes axes are vertical or horizontal.

Q5: How does the 'a' value affect the graph?

A5: If 'a' is positive, the parabola opens upwards (vertical axis) or to the right (horizontal axis). If 'a' is negative, it opens downwards or to the left. The larger the absolute value of 'a', the narrower the parabola.

Q6: What is the latus rectum of a parabola?

A6: The latus rectum is a line segment passing through the focus, perpendicular to the axis of symmetry, with endpoints on the parabola. Its length is |4p| = |1/a|.

Q7: Can I use this calculator for horizontal parabolas?

A7: Yes, for "Vertex and a Point" and "Focus and Directrix" methods, you can select the orientation. For "Three Points", select the x=Ay²+By+C form.

Q8: Where are parabolas used in real life?

A8: Parabolic shapes are used in satellite dishes, car headlights, telescope mirrors, suspension bridge cables, and describe the path of projectiles under gravity. Using a find the equation of a parabola calculator is useful in these fields.