Parabola Calculator
Find the vertex, focus, directrix and axis of symmetry of a parabola y=ax²+bx+c.
Inputs
Vertex X
2.0000
Vertex Y
-1.0000
Focus Y
-0.7500
Directrix (y = )
-1.2500
Focal Distance
0.2500
Step by step
Vertex x-coordinate
x = -b/(2a) = 4/2
= 2.0000
Vertex y-coordinate
y = a×(2.0000)² + b×(2.0000) + c
= -1.0000
Focal distance
f = 1/(4|a|) = 1/4
= 0.2500
How it works
A parabola y = ax² + bx + c has its vertex at x = −b/(2a). The focus lies 1/(4|a|) units from the vertex along the axis of symmetry, and the directrix is the same distance on the opposite side.
Formulas
Vertex
x_v = −b/(2a), y_v = f(x_v)
- a
- Leading coefficient
- b
- Linear coefficient
Focal Distance
f = 1/(4|a|)
- a
- Leading coefficient
Frequently Asked Questions
What does the sign of 'a' determine?
If a > 0 the parabola opens upward; if a < 0 it opens downward.
How do I find the axis of symmetry?
It is the vertical line x = −b/(2a), passing through the vertex.