Parabola Calculator
Vertex, focus, directrix, latus rectum and roots of y = ax² + bx + c.
Results
Vertex, x
2.000000
Vertex, y
-1.000000
Distance from vertex to focus
0.250000
Focus, x
2.000000
Focus, y
-0.750000
Directrix, the line y =
-1.250000
Latus rectum, the width at the focus
1.000000
Discriminant
4.000000
Root x₁
3.000000
Root x₂
1.000000
Y-intercept (b)
3.000000
Which way the parabola opens
Upwards — the vertex is the lowest point
How many times it crosses the x axis
2
What this tool does
A parabola is every point that is equally far from one point and one line — the focus and the directrix. That definition is where all the rest comes from: the vertex sits halfway between them, and the focal distance is 1 ÷ 4a. Set a to zero and there is no parabola at all, only a straight line, and the page says so instead of dividing by nothing.
Formula
y = ax² + bx + c · vertex at x = −b ÷ 2a · focal distance p = 1 ÷ 4a · directrix y = yᵥ − p
Variables
| Symbol | Meaning | Unit |
|---|---|---|
a | a | — |
b | b | — |
c | c | — |
VX | Vertex, x | — |
VY | Vertex, y | — |
FD | Distance from vertex to focus | — |
FX | Focus, x | — |
FY | Focus, y | — |
DI | Directrix, the line y = | — |
LR | Latus rectum, the width at the focus | — |
DS | Discriminant | — |
R1 | Root x₁ | — |
R2 | Root x₂ | — |
YI | Y-intercept (b) | — |
OP | Which way the parabola opens | — |
NC | How many times it crosses the x axis | — |
Worked example
- a1
- b-4
- c3
- Vertex, x2.000000
- Vertex, y-1.000000
- Distance from vertex to focus0.250000
- Focus, x2.000000
- Focus, y-0.750000
- Directrix, the line y =-1.250000
- Latus rectum, the width at the focus1.000000
- Discriminant4.000000
- Root x₁3.000000
- Root x₂1.000000
- Y-intercept (b)3.000000
- Which way the parabola opensUpwards — the vertex is the lowest point
- How many times it crosses the x axis2
Limitations
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.