Calculator
Example Data Table
| Method | Input | Standard form | Vertex |
|---|---|---|---|
| Vertical vertex form | y = 2(x - 3)2 - 4 | (x - 3)2 = 0.5(y + 4) | (3, -4) |
| Focus and directrix | Focus (2, 5), directrix y = -1 | (x - 2)2 = 12(y - 2) | (2, 2) |
| Quadratic function | y = x2 - 6x + 5 | (x - 3)2 = 1(y + 4) | (3, -4) |
Formula Used
Vertical standard form is (x - h)2 = 4p(y - k).
Horizontal standard form is (y - k)2 = 4p(x - h).
For y = ax2 + bx + c, use h = -b / (2a), k = c - b2 / (4a), and p = 1 / (4a).
For a vertical focus-directrix pair, use h = focus x, k = (focus y + d) / 2, and p = (focus y - d) / 2.
For a horizontal focus-directrix pair, use h = (focus x + d) / 2, k = focus y, and p = (focus x - d) / 2.
How to Use This Calculator
- Select the input method that matches your problem.
- Enter all known numbers without units.
- Press Submit to view the standard form above the form.
- Check the vertex, focus, directrix, and axis.
- Use CSV or PDF download for saving the report.
Understanding Parabola Standard Form
A parabola can be written in several ways. Standard form is useful because it shows the vertex, direction, focus, directrix, and focal distance. This calculator converts common input styles into a clean standard equation. It also keeps the working steps visible, so the answer is easier to check.
Why Standard Form Matters
For a vertical parabola, the standard form is (x - h)2 = 4p(y - k). For a horizontal parabola, it is (y - k)2 = 4p(x - h). The point (h, k) is the vertex. The value p gives the signed distance from the vertex to the focus. A positive p opens upward or rightward. A negative p opens downward or leftward.
Input Options
You may start with vertex form, focus and directrix data, three points, or a quadratic function. Each option answers a different classroom problem. Three points create a fitted quadratic when the x-values are different. Focus and directrix data creates the equation by locating the vertex halfway between them. Quadratic coefficients are completed by the square to reveal the vertex.
Checking the Result
The calculator lists the standard equation, vertex, focus, directrix, axis, opening direction, and latus rectum length. These values give several ways to verify the same parabola. If the focus and directrix are equal distances from the vertex, the standard form is consistent.
Study Use
Use the step list before copying the final equation. It explains which formula was chosen and how important values were found. The downloadable CSV file is helpful for spreadsheets. The PDF report is helpful for homework notes, teaching handouts, and saved examples.
Common Mistakes
Many errors come from signs. The expression x - h means the vertex x-value is h, not negative h. The same rule applies to y - k. Another common mistake is forgetting that 4p is the coefficient beside the shifted variable. When 4p is large, the curve is wider. When its absolute value is small, the curve is narrower. If a three point input gives a value of a near zero, the points may describe a line instead of a true parabola. Recheck repeated x-values, rounded numbers, and copied signs before using the final equation. These checks make each result safer for exam practice.
FAQs
What is standard form of a parabola?
Standard form is (x - h)2 = 4p(y - k) for vertical parabolas, or (y - k)2 = 4p(x - h) for horizontal parabolas.
What do h and k mean?
The values h and k form the vertex. The vertex is the turning point or starting point of the curve.
What does p mean?
The value p is the signed distance from the vertex to the focus. It also sets the distance from the vertex to the directrix.
Can this calculator use three points?
Yes. It fits three points to y = ax2 + bx + c, then converts that equation into standard vertical form.
Why must the x-values be different?
For three point vertical fitting, repeated x-values can make the system unsolvable or invalid. Different x-values define the quadratic more reliably.
How do I know the opening direction?
For vertical form, positive p opens upward and negative p opens downward. For horizontal form, positive p opens rightward and negative p opens leftward.
Does it show work?
Yes. The result includes calculation steps, computed vertex, p value, focus, directrix, axis, and latus rectum length.
What can I download?
You can download a CSV file for spreadsheet use. You can also download a PDF report with the main result and steps.