Equation Focus and Directrix Calculator

Calculator

Choose the equation type, enter values, and submit the form.

Equation Type

Decimal Places

Graph Range Start

Used as x-range for vertical curves and y-range for horizontal curves.

Graph Range End

a

h

k

a

h

k

A

B

C

A

B

C

Focus x

Focus y

Directrix y = d

Focus x

Focus y

Directrix x = d

Formula Used

Vertical parabola: (x - h)^2 = 4p(y - k)

Horizontal parabola: (y - k)^2 = 4p(x - h)

Coefficient relation: p = 1 / 4a

Vertical focus: (h, k + p). Vertical directrix: y = k - p.

Horizontal focus: (h + p, k). Horizontal directrix: x = h - p.

General vertical form: for y = Ax^2 + Bx + C, use h = -B / 2A and k = C - B^2 / 4A.

General horizontal form: for x = Ay^2 + By + C, use k = -B / 2A and h = C - B^2 / 4A.

How to Use This Calculator

Select the equation type that matches your problem.
Enter the required coefficient, vertex, focus, or directrix values.
Choose decimal precision for the final answer.
Set the graph range for a wider or closer visual view.
Press the calculate button.
Review the result panel above the form.
Use the CSV button for spreadsheet export.
Use the PDF button for a printable report.

Example Data Table

Input Type	Input Values	Vertex	Focus	Directrix	Opening
Vertical vertex form	y = 0.25(x - 2)^2 + 1	(2, 1)	(2, 2)	y = 0	Upward
Horizontal vertex form	x = -0.5(y + 1)^2 + 3	(3, -1)	(2.5, -1)	x = 3.5	Left
General vertical form	y = 2x^2 - 8x + 5	(2, -3)	(2, -2.875)	y = -3.125	Upward
Focus and directrix	Focus (0, 3), directrix y = -1	(0, 1)	(0, 3)	y = -1	Upward

Article

Understanding Focus and Directrix

A parabola is more than a curved graph. It is a set of points with one shared rule. Every point on the curve is the same distance from a fixed point and a fixed line. The point is the focus. The line is the directrix. This calculator turns that rule into usable numbers.

Why These Values Matter

Focus and directrix values explain the shape, opening, and position of a parabola. They also connect algebra with geometry. When the focus moves farther from the vertex, the curve becomes wider. When it moves closer, the curve becomes narrower. This relationship is controlled by p, the directed distance from the vertex to the focus.

Supported Parabola Forms

The tool supports vertical, horizontal, and general forms. You can enter vertex form, such as y equals a times x minus h squared plus k. You can also enter general quadratic forms. For focus and directrix inputs, the calculator builds the equation from geometry. This helps when a problem gives the focus and line instead of coefficients.

Reading the Results

The result panel shows the vertex, focus, directrix, axis of symmetry, opening direction, standard equation, and coefficient form. It also shows the value of p and focal width. These details make it easier to check homework, prepare graphs, or compare different parabola equations.

Using the Graph

The Plotly graph gives a visual check of the answer. The curve, focus, vertex, directrix, and axis are shown together. This makes mistakes easier to catch. For example, if the focus appears below the vertex, the directrix should be above it for a downward opening curve.

Exporting Your Work

CSV export helps with spreadsheets and classroom records. PDF export creates a cleaner report for study notes or project files. Keep units consistent if your values represent real measurements. The calculator does not assume inches, meters, or feet. It only follows the numbers you enter. Always review signs carefully, because a negative p changes the opening direction. For best accuracy, use decimals instead of rounded fractions when needed. After calculating, compare the graph with your expected opening direction before downloading the final report for later careful review.

FAQs

1. What is the focus of a parabola?

The focus is a fixed point inside the parabola. Every point on the curve has the same distance from the focus and the directrix.

2. What is the directrix?

The directrix is a fixed line outside the parabola. It works with the focus to define every point on the curve.

3. What does p mean?

The value p is the directed distance from the vertex to the focus. It also controls the directrix location and focal width.

4. Can p be negative?

Yes. A negative p means the parabola opens downward or left. A positive p means it opens upward or right.

5. Which forms does this calculator support?

It supports vertical vertex form, horizontal vertex form, two general quadratic forms, and focus-directrix input forms.

6. What is focal width?

Focal width is the length of the latus rectum. Its value is the absolute value of 4p.

7. Why is coefficient a important?

The coefficient a controls the curve width and opening direction. It connects to p through the formula p equals 1 divided by 4a.

8. Can I export the answer?

Yes. Use CSV for spreadsheet data. Use PDF for a clean report with results and a graph image.