Parabola Calculator
Find the vertex, focus, directrix, and axis of symmetry of any parabola y=ax^2+bx+c. Free online parabola calculator with interactive graph for students.
About This Calculator
The Parabola Calculator helps you analyze any quadratic equation of the form y = ax^2 + bx + c by computing its key geometric properties. This tool is designed for algebra and precalculus students, teachers, and professionals who need to quickly find the vertex, focus, directrix, and axis of symmetry of a parabola.
A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The standard quadratic form y = ax^2 + bx + c describes a vertical parabola. The vertex is the turning point located at h = -b/(2a) and k = c - b^2/(4a). The focus coordinates are (h, c - (b^2 - 1)/(4a)) and the directrix is the horizontal line y = c - (b^2 + 1)/(4a). The axis of symmetry is the vertical line x = h that passes through both the vertex and focus, dividing the parabola into two mirror-image halves.
This calculator generates an interactive graph of the parabola centered on the vertex, allowing you to visualize the curve's shape and its key features. Whether you are studying conic sections, solving quadratic equations, or designing parabolic reflectors, this tool provides instant and accurate results for any quadratic function.
Frequently Asked Questions
What is a parabola?
A parabola is a U-shaped symmetrical curve where every point is equidistant from a fixed point called the focus and a fixed line called the directrix. It is the graph of a quadratic equation y = ax^2 + bx + c.
How do you find the vertex of a parabola?
The vertex of a parabola y = ax^2 + bx + c is found using h = -b/(2a) for the x-coordinate and k = c - b^2/(4a) for the y-coordinate. The vertex represents the minimum point (if a > 0) or maximum point (if a < 0) of the parabola.
How do you find the focus of a parabola?
The focus of a parabola y = ax^2 + bx + c has the same x-coordinate as the vertex: x = -b/(2a). The y-coordinate is given by y = c - (b^2 - 1)/(4a). The focus is a fixed point inside the parabola such that every point on the curve is equidistant from the focus and the directrix.
What is the directrix of a parabola?
The directrix of a parabola y = ax^2 + bx + c is a horizontal line given by y = c - (b^2 + 1)/(4a). For a parabola opening upward (a > 0), the directrix lies below the vertex; for a downward-opening parabola (a < 0), it lies above the vertex.
What is the axis of symmetry of a parabola?
The axis of symmetry of a parabola is the vertical line x = -b/(2a) that passes through the vertex and focus, dividing the parabola into two mirror-image halves. Every parabola is symmetric about this line.
What does the coefficient a tell you about a parabola?
The coefficient a in y = ax^2 + bx + c determines the direction and width of the parabola. If a > 0, the parabola opens upward and has a minimum vertex. If a < 0, it opens downward and has a maximum vertex. Larger |a| values make the parabola narrower, while smaller |a| values make it wider.
How is a parabola different from a catenary curve?
A parabola is the graph of a quadratic equation y = ax^2 + bx + c, while a catenary curve follows y = a cosh(x/a). A hanging cable under its own weight forms a catenary, not a parabola, though the two curves look similar near the vertex.
Can a parabola have a horizontal axis?
Yes, a parabola can open sideways with a horizontal axis of symmetry. Such parabolas are described by equations of the form x = ay^2 + by + c, where the roles of x and y are swapped. This calculator handles vertical parabolas of the form y = ax^2 + bx + c.