Parabola Calculator – Guide & Formulas
Calculate parabola properties from standard or vertex form. Get vertex, focus, directrix, axis of symmetry, and latus rectum with step-by-step solutions.
Calculate all key properties of a parabola from its equation. Enter the coefficients in standard form (y = ax² + bx + c) or vertex form (y = a(x-h)² + k) to instantly find the vertex, focus, directrix, axis of symmetry, and latus rectum length.
Key Takeaway
Use the free Parabola Calculator to calculate parabola properties from standard or vertex form. get vertex, focus, directrix, axis of symmetry, and latus rectum with step-by-step solutions. Get instant results with step-by-step explanations.
How to Use the Parabola Calculator
- Select the equation form: Standard (ax² + bx + c) or Vertex (a(x-h)² + k).
- Enter the coefficients a, b, c (or a, h, k) from your parabola equation.
- Review the vertex coordinates, focus point, and directrix equation.
- Examine the axis of symmetry and latus rectum length.
The Formula
Variable Definitions
- a: Leading coefficient; determines opening direction and width
- b: Linear coefficient
- c: Constant term (y-intercept when x = 0)
- h, k: Vertex coordinates in vertex form y = a(x-h)² + k
- p: Focal length = 1/(4a); distance from vertex to focus
Parabola y = 2x² - 8x + 6
Find all properties of the parabola defined by y = 2x² - 8x + 6.
- Step 1: Identify coefficients: a = 2, b = -8, c = 6.
- Step 2: Vertex x-coordinate: h = -b/(2a) = -(-8)/(2×2) = 8/4 = 2.
- Step 3: Vertex y-coordinate: k = c - b²/(4a) = 6 - 64/8 = 6 - 8 = -2. Vertex is (2, -2).
- Step 4: Focal length: p = 1/(4a) = 1/8 = 0.125.
- Step 5: Focus: (2, -2 + 0.125) = (2, -1.875). Directrix: y = -2 - 0.125 = -2.125.
Frequently Asked Questions
How do I find the vertex of a parabola?
For y = ax² + bx + c, the vertex x-coordinate is h = -b/(2a). Substitute h back into the equation to get k = f(h). The vertex is the point (h, k), which is the minimum or maximum of the parabola.
What does the value of a tell me about the parabola?
If a > 0, the parabola opens upward (vertex is a minimum). If a < 0, it opens downward (vertex is a maximum). Larger |a| values make the parabola narrower; smaller |a| values make it wider.
How is the focus different from the vertex?
The vertex is the turning point of the parabola. The focus is a point inside the parabola at distance p = 1/(4a) from the vertex along the axis of symmetry. The focus defines the parabola as the set of points equidistant from the focus and the directrix.
What is the directrix of a parabola?
The directrix is a line perpendicular to the axis of symmetry, located at distance p = 1/(4a) from the vertex on the opposite side of the focus. Every point on the parabola is equidistant from the focus and the directrix.
How do I convert from standard form to vertex form?
Complete the square: factor a from the first two terms, add and subtract (b/(2a))² inside, then simplify. For y = ax² + bx + c, vertex form is y = a(x - h)² + k where h = -b/(2a) and k = c - b²/(4a).
What is the axis of symmetry?
The axis of symmetry is the vertical line x = h (the x-coordinate of the vertex). It divides the parabola into two mirror-image halves. Any point on the parabola has a corresponding point on the other side of this line.
Can a parabola open sideways?
Yes. A horizontal parabola has the form x = ay² + by + c or x = a(y-k)² + h. Its axis of symmetry is horizontal. This calculator handles vertical parabolas (y as a function of x).
What is the latus rectum of a parabola?
The latus rectum is the line segment through the focus perpendicular to the axis of symmetry. Its total length is |4p| = |1/a|. The endpoints of the latus rectum lie on the parabola and help define its width at the focus.