Parabola Calculator
Complete parabola analysis and graphing properties
Understanding Parabolas
A parabola is the graph of a quadratic function y = ax² + bx + c. It is a symmetric U-shaped (or inverted U-shaped) curve with many important geometric properties. Parabolas appear in nature, physics, engineering, and everyday life.
Vertex
The vertex is the turning point of the parabola—its highest or lowest point. For y = ax² + bx + c, the vertex is at (h, k) where h = -b/(2a) and k = c - b²/(4a). When a > 0, the vertex is a minimum; when a < 0, it's a maximum.
Axis of Symmetry
The vertical line x = h passing through the vertex divides the parabola into two mirror-image halves. Any point (x, y) on the parabola has a symmetric counterpart (2h - x, y).
Focus and Directrix
The focus is a special point inside the parabola, and the directrix is a line outside it. Every point on the parabola is equidistant from the focus and directrix. This property defines the parabola geometrically and is crucial in applications like satellite dishes and headlights.
Intercepts
Y-intercept: The point where the parabola crosses the y-axis, found by setting x = 0. It's always (0, c).
X-intercepts: The points where the parabola crosses the x-axis, found by solving ax² + bx + c = 0. There can be 0, 1, or 2 x-intercepts depending on the discriminant.
Coefficient Effects
- a: Controls opening direction and width. Larger |a| makes a narrower parabola
- b: Affects horizontal position of vertex and axis of symmetry
- c: Determines the y-intercept and vertical shift
Frequently Asked Questions
What makes a parabola different from other curves?
A parabola has a unique property: every point is equidistant from a fixed point (focus) and a fixed line (directrix). It's also the only conic section that represents a quadratic function.
Why is the vertex important?
The vertex represents the extreme value (maximum or minimum) of the quadratic function. In applications, this often represents optimal values like maximum profit, minimum cost, or highest trajectory point.
What are real-world applications of parabolas?
Parabolas describe projectile motion, satellite dish shapes, bridge cables, headlight reflectors, water fountain arcs, profit/cost curves in economics, and the path of comets around the sun.
How do I find the focus and directrix?
First find the vertex (h, k). Then calculate p = 1/(4a). The focus is at (h, k + p) and the directrix is the line y = k - p. These are measured from the vertex.
Can a parabola open sideways?
Yes, but that would be x = ay² + by + c, not a function of x. Such parabolas have horizontal axes of symmetry. The parabola y = ax² + bx + c always opens vertically (up or down).
What if my parabola has no x-intercepts?
This means the discriminant is negative and the parabola doesn't cross the x-axis. If a > 0, the entire parabola is above the x-axis. If a < 0, it's entirely below.
How is the focus used in technology?
Parabolic reflectors use the focus property: rays parallel to the axis reflect through the focus. This is used in satellite dishes (collecting signals), headlights (projecting light), and solar collectors (concentrating energy).
Why do all parabolas look similar?
All parabolas are similar shapes—they differ only in size, orientation, and position. You can transform any parabola into y = x² through scaling, reflection, and translation. This makes them easier to study and understand.