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 Standard Form
The standard form of a quadratic equation is y = ax² + bx + c. You can use this parabola calculator to transform that equation into the vertex form, which allows you to find the important points of the parabola — its vertex, focus, and directrix.
- a — determines the width and direction of the parabola (a > 0: opens upward; a < 0: opens downward)
- b — affects the horizontal position of the vertex
- c — the y-intercept (the value of y when x = 0)
The Parabola Equation in Vertex Form
The parabola equation in its vertex form is y = a(x − h)² + k, where:
- a — same as the a coefficient in the standard form
- h — x-coordinate of the parabola vertex: h = −b / (2a)
- k — y-coordinate of the parabola vertex: k = c − b² / (4a)
Parabola Focus and Directrix
The parabola calculator finds the focus and directrix using the following equations:
- Focus x-coordinate: x₀ = −b / (2a)
- Focus y-coordinate: y₀ = c − (b² − 1) / (4a)
- Directrix equation: y = c − (b² + 1) / (4a)
- p value (focal length): p = 1 / (4a)
The parameter p represents the distance from the vertex to both the focus and the directrix.
How to Use the Parabola Equation Calculator: An Example
Let's assume the equation is y = 2x² + 3x − 4. Enter the coefficients a = 2, b = 3, c = −4 into the calculator.
The calculator will compute:
- Vertex: h = −b / (2a) = −3 / 4 = −0.75, k = −4 − 9/8 = −5.125 → vertex (−0.75, −5.125)
- Focus: y₀ = −4 − (9 − 1) / 8 = −5 → focus (−0.75, −5)
- Directrix: y = −4 − (9 + 1) / 8 = −5.25 → directrix y = −5.25
- Axis of symmetry: x = −0.75
- Direction: Opens upward (a > 0)
FAQs
What is a parabola?
A parabola is a symmetrical U-shaped curve such that every point on the curve is equidistant from the directrix and the focus.
What is the vertex of a parabola?
The vertex is the turning point of the parabola — the highest point if the parabola opens downward (a < 0), or the lowest point if it opens upward (a > 0). Its coordinates are (h, k) where h = −b/(2a) and k = c − b²/(4a).
What is the focus of a parabola?
The focus is a special point inside the parabola. Every point on the parabola is exactly the same distance from the focus as it is from the directrix. The focus lies on the axis of symmetry at a distance p = 1/(4a) from the vertex.
What is the directrix of a parabola?
The directrix is a straight line perpendicular to the axis of symmetry, located at a distance p = 1/(4a) on the opposite side of the focus from the vertex.
How do I find the x-intercepts of a parabola?
Set y = 0 and solve the quadratic equation ax² + bx + c = 0 using the quadratic formula: x = (−b ± √(b² − 4ac)) / (2a). The discriminant b² − 4ac determines the number of real roots: two roots if positive, one root if zero, no real roots if negative.