Calculator
Choose an equation form, enter the needed values, and calculate the focus.
Formula Used
Vertical Vertex Form
y = a(x - h)² + k
p = 1 / 4a
Focus = (h, k + p)
Directrix = y = k - p
Horizontal Vertex Form
x = a(y - k)² + h
p = 1 / 4a
Focus = (h + p, k)
Directrix = x = h - p
General Form
For y = Ax² + Bx + C, the calculator finds:
h = -B / 2A
k = C - B² / 4A
For x = Ay² + By + C, the same square-completion idea is used with x and y reversed.
How to Use This Calculator
- Select vertex form or general quadratic form.
- Choose vertical or horizontal orientation.
- Enter the coefficient values from your equation.
- Choose the number of decimal places.
- Press the calculate button.
- Review the focus, vertex, directrix, axis, and steps.
- Use CSV or PDF buttons to save the result.
Example Data Table
| Equation Type | Orientation | Values | Expected Focus |
|---|---|---|---|
| Vertex Form | Vertical | a = 0.25, h = 0, k = 0 | (0, 1) |
| Vertex Form | Horizontal | a = 0.5, h = 2, k = 1 | (2.5, 1) |
| General Form | Vertical | A = 1, B = -4, C = 3 | (2, -0.75) |
| General Form | Horizontal | A = -0.25, B = 2, C = 1 | (5, 4) |
About This Equation to Focus Calculator
This calculator helps you find the focus of a parabola from common equation forms. It supports vertex form and expanded quadratic form. You can use vertical or horizontal orientation. The tool also returns the vertex, directrix, axis, focal length, latus rectum length, and opening direction. These details help students check graph work and prepare clean solution notes.
Why Focus Matters
The focus is a fixed point inside a parabola. Every point on the curve has equal distance from the focus and the directrix. This property explains the shape. It also supports real uses in reflectors, antennas, mirrors, headlights, and satellite dishes. A correct focus makes the graph easier to understand.
Supported Equation Forms
For vertex form, enter values for a, h, and k. A vertical parabola uses y = a(x - h)^2 + k. A horizontal parabola uses x = a(y - k)^2 + h. For general form, enter A, B, and C. The calculator completes the square internally. Then it extracts the vertex and focus.
Better Study Workflow
Manual conversion can create sign mistakes. This calculator shows each important value in one place. You can compare the result with your textbook answer. You can also export the work as CSV or PDF. This makes it useful for assignments, tutorials, and quick lesson examples.
Accuracy Tips
Use the correct orientation before calculating. Keep negative signs in coefficient fields. Do not enter zero for the quadratic coefficient. Select enough decimal places for your class needs. Rounded answers are useful for reports. Exact fraction work may still be needed for proofs.
Interpreting Results
If the focal length is positive on a vertical equation, the parabola opens upward. If it is negative, it opens downward. For horizontal equations, positive means right. Negative means left. The directrix sits the same distance from the vertex, but on the opposite side.
When to Use It
Use this converter when a problem asks for the focus from an equation. It is also helpful when creating graph labels. You can test several coefficients quickly. The example table gives starting values. Change them to match your own homework or project equation with reliable clear output.
FAQs
What does this calculator find?
It finds the focus of a parabola from vertex form or general quadratic form. It also gives the vertex, directrix, axis, opening direction, focal length, and latus rectum details.
What is the focus of a parabola?
The focus is a fixed point used to define the parabola. Every point on the parabola is the same distance from the focus and the directrix.
What is p in the formula?
The value p is the focal length. It is the signed distance from the vertex to the focus. This calculator uses p = 1 / 4a.
Can I use general quadratic equations?
Yes. You can enter A, B, and C for general form. The calculator completes the square and converts the equation into vertex-based focus information.
What happens if a is negative?
A negative coefficient changes the opening direction. A vertical parabola opens downward. A horizontal parabola opens left. The focus moves in that same signed direction.
Why can a not equal zero?
If a equals zero, the equation is not quadratic. It will not create a parabola. So the focus formula cannot be applied correctly.
Can I download the result?
Yes. After calculation, you can download the result as a CSV file or a PDF file. Both options include the main output values.
Is this useful for graphing?
Yes. The focus, vertex, directrix, and axis help you draw a more accurate parabola. They also help verify graphing work.