Standard & Vertex & Focus-Directrix
Works for vertical y = a x² + b x + c
and horizontal x = a y² + b y + c parabolas.
Inputs
Pick how you’ll define the parabola.
Note: a ≠ 0. For vertical parabolas, positive a opens upward; negative opens downward. For horizontal parabolas, positive a opens rightward; negative opens leftward.
Results
Orientation:
Given:
Canonical Forms
Standard (general):
Vertex form:
Focus-Directrix:
Geometry
- Vertex (h, k):
- Focus:
- Directrix:
- Axis of symmetry:
Latus Rectum & Intercepts
- p (focus distance):
- Latus rectum length:
- Latus rectum endpoints:
- Intercepts:
Hint: You can export all fields as CSV or a tidy PDF.
Example Data
| # | Mode | Orientation | a | b | c | h | k | p | Action |
|---|---|---|---|---|---|---|---|---|---|
| 1 | Coefficients | vertical | 1 | 0 | 0 | ||||
| 2 | Coefficients | vertical | 0.5 | -3 | 2 | ||||
| 3 | Coefficients | horizontal | 0.25 | 0 | 0 | ||||
| 4 | Coefficients | horizontal | -0.5 | 2 | 1 | ||||
| 5 | Vertex | vertical | 1 | -2 | |||||
| 6 | Focus-Directrix | vertical | 0 | 0 | 0.25 |
Formulas Used
Vertical parabola y = a x² + b x + c
- Vertex: \( h = -\frac{b}{2a}, \; k = a h^2 + b h + c \)
- Focus distance: \( p = \frac{1}{4a} \)
- Focus: \( (h,\, k + p) \) Directrix: \( y = k - p \)
- Axis of symmetry: \( x = h \)
- Vertex form: \( y = a(x - h)^2 + k \)
- Focus-Directrix: \( (x - h)^2 = 4p (y - k) \)
- Latus rectum length: \( |4p| = \left|\frac{1}{a}\right| \)
- Latus rectum endpoints: \( (h \pm 2p,\; k + p) \)
- x-intercepts: solve \( a x^2 + b x + c = 0 \)
- y-intercept: \( (0, c) \)
Horizontal parabola x = a y² + b y + c
- Vertex: \( k = -\frac{b}{2a}, \; h = a k^2 + b k + c \)
- Focus distance: \( p = \frac{1}{4a} \)
- Focus: \( (h + p,\, k) \) Directrix: \( x = h - p \)
- Axis of symmetry: \( y = k \)
- Vertex form: \( x = a(y - k)^2 + h \)
- Focus-Directrix: \( (y - k)^2 = 4p (x - h) \)
- Latus rectum length: \( |4p| = \left|\frac{1}{a}\right|
- Latus rectum endpoints: \( (h + p,\; k \pm 2p) \)
- y-axis intercepts: solve \( a y^2 + b y + c = 0 \) for \(y\), at \(x=0\)
- x-intercept: \( (c, 0) \) when \( y = 0 \)
How to Use This Calculator
- Choose Input mode: coefficients, vertex form, or focus-directrix.
- Select the parabola Orientation: vertical or horizontal.
- Enter the required values. Ensure
a ≠ 0when using coefficients or vertex mode. - Click Calculate to view canonical forms, focus, directrix, and more.
- Use Download CSV to export all results as a spreadsheet row.
- Use Download PDF to capture the results panel as a tidy PDF.
- Try the Example Data table and press Use on any row.
FAQs
For vertical axis parabolas: y = a x² + b x + c. For horizontal axis parabolas: x = a y² + b y + c. We also provide vertex and focus-directrix forms.
For vertical parabolas, \( h = -\frac{b}{2a} \) and \( k = a h^2 + b h + c \). For horizontal parabolas, swap roles of \(x\) and \(y\): compute \( k \) first, then \( h \).
The value \( p \) is the directed distance from the vertex to the focus. It also determines the directrix: \( y = k - p \) for vertical, or \( x = h - p \) for horizontal parabolas.
The relationship is \( a = \frac{1}{4p} \) (assuming vertex form orientation). Positive \( a \) opens up/right; negative opens down/left respectively.
Yes. We compute x/y intercepts depending on orientation, latus rectum length \( |4p| \), and endpoints at the focus level.
Absolutely. Choose the horizontal orientation to work with x = a y² + b y + c. All forms and geometric elements are computed accordingly.
Use the Download CSV button to export a single-row CSV of all current outputs. Use Download PDF to generate a neat PDF snapshot of the results panel.
Attribution
This is an educational tool. Verify results for critical work. Not a substitute for professional advice.
Parabola Standard Form — Quick Reference Data
| Orientation | Standard form | Vertex form | Focus–Directrix form | Relation between a and p |
|---|---|---|---|---|
| Vertical | y = a x² + b x + c | y = a (x - h)² + k | (x - h)² = 4 p (y - k) | a = 1 / (4 p) |
| Horizontal | x = a y² + b y + c | x = a (y - k)² + h | (y - k)² = 4 p (x - h) | a = 1 / (4 p) |
Worked Examples — Reference Data
| # | Input equation | Orientation | Vertex (h, k) | Focus | Directrix | p | Latus rectum length |
|---|---|---|---|---|---|---|---|
| 1 | y = x² | Vertical | (0, 0) | (0, 0.25) | y = -0.25 | 0.25 | 1 |
| 2 | y = 0.5 x² − 3 x + 2 | Vertical | (3, -2.5) | (3, -2) | y = -3 | 0.5 | 2 |
| 3 | x = 0.25 y² | Horizontal | (0, 0) | (1, 0) | x = -1 | 1 | 4 |
Numbers are exact values; rounding may occur on-screen.