Calculator
Enter one known parabola form. The calculator expands it into Ax² + Bxy + Cy² + Dx + Ey + F = 0.
Formula Used
General conic form: Ax² + Bxy + Cy² + Dx + Ey + F = 0
Vertical vertex form: y = a(x - h)² + k
Vertical general result: ax² - 2ahx - y + ah² + k = 0
Horizontal vertex form: x = a(y - k)² + h
Horizontal general result: ay² - x - 2aky + ak² + h = 0
Parabola test: B² - 4AC = 0
The calculator expands the selected input form. It then moves all terms to the left side. Finally, it reports A, B, C, D, E, and F.
How to Use This Calculator
- Select the parabola form that matches your problem.
- Enter the values needed for that form.
- Use decimal places to control rounding.
- Use scale factor when you want an equivalent multiplied equation.
- Press submit to see the general form above the form.
- Download the result as CSV or PDF for records.
Example Data Table
| Input form | Values | Expanded equation | General form |
|---|---|---|---|
| y = a(x - h)² + k | a = 2, h = 3, k = -4 | y = 2x² - 12x + 14 | 2x² - 12x - y + 14 = 0 |
| y = ax² + bx + c | a = 1, b = -6, c = 8 | y = x² - 6x + 8 | x² - 6x - y + 8 = 0 |
| y = a(x - r₁)(x - r₂) | a = 3, r₁ = 1, r₂ = 5 | y = 3x² - 18x + 15 | 3x² - 18x - y + 15 = 0 |
| x = a(y - k)² + h | a = 4, h = -2, k = 1 | x = 4y² - 8y + 2 | 4y² - x - 8y + 2 = 0 |
Advanced Guide to Parabola General Form
Why general form matters
A parabola can be written in many useful ways. Vertex form shows the turning point. Standard polynomial form shows coefficients quickly. Intercept form shows roots. Focus-directrix form shows the geometric definition. General form brings these versions into one shared structure. That structure is Ax² + Bxy + Cy² + Dx + Ey + F = 0. It is useful for comparison, graphing, storage, and algebraic checking.
How expansion works
The conversion is mainly an expansion process. For a vertical parabola, start with y = a(x - h)² + k. Square the binomial first. Then multiply by a. Then move y to the left side. The result is ax² - 2ahx - y + ah² + k = 0. The coefficients can then be read directly.
Vertical and horizontal equations
Vertical parabolas use x as the squared variable. Their general form usually has an x² term and a y term. Horizontal parabolas use y as the squared variable. Their general form usually has a y² term and an x term. This calculator handles both directions. That makes it useful for analytic geometry, conic sections, engineering paths, and coordinate transformations.
Coefficient meaning
The coefficient A belongs to x². The coefficient B belongs to xy. The coefficient C belongs to y². A standard axis-aligned parabola normally has B equal to zero. A vertical parabola has A present and C absent. A horizontal parabola has C present and A absent. The D, E, and F values complete the linear and constant parts.
Discriminant check
The conic discriminant is B² - 4AC. A true parabola has a discriminant equal to zero. For axis-aligned parabolas, this often happens because either A or C is zero and B is also zero. If scaling is applied, the equation remains equivalent. The discriminant still confirms the conic type when the equation is valid.
Focus and directrix use
The focus-directrix option is helpful when a parabola is described geometrically. The vertex is halfway between the focus and directrix. The focal distance p measures the signed distance from the vertex to the focus. After p is found, the calculator uses a = 1 / 4p. Then it expands the matching vertex form into general form.
Using the scale factor
Some teachers and systems prefer integer-looking equations. Others prefer a direct decimal result. The scale factor lets you multiply every coefficient by the same value. This does not change the graph. It only changes the presentation. For example, multiplying every term by two gives an equivalent equation with larger coefficients.
Practical checking
You can test the result by moving the linear variable back to the other side. Then simplify the equation into the original form. You can also substitute a point from the parabola. If both sides satisfy the general equation, the conversion is consistent. Always keep enough decimal places when working with focus-directrix data or fractional coefficients.
FAQs
1. What is parabola general form?
Parabola general form is a conic equation written as Ax² + Bxy + Cy² + Dx + Ey + F = 0. For common axis-aligned parabolas, B is usually zero.
2. Can this calculator convert vertex form?
Yes. It converts y = a(x - h)² + k and x = a(y - k)² + h. It expands the square and moves all terms to one side.
3. Can it convert standard polynomial form?
Yes. It converts y = ax² + bx + c into ax² + bx - y + c = 0. It also supports horizontal polynomial form.
4. What does A mean in general form?
A is the coefficient of x². If A is present and C is zero, the parabola usually opens upward or downward.
5. What does C mean in general form?
C is the coefficient of y². If C is present and A is zero, the parabola usually opens left or right.
6. Why is B usually zero?
B is the coefficient of xy. Axis-aligned parabolas do not have rotation, so the xy term is usually absent.
7. What is the discriminant test?
The conic discriminant is B² - 4AC. For a parabola, this value should equal zero, allowing for small rounding differences.
8. What is the scale factor?
The scale factor multiplies every coefficient by the same number. The equation changes appearance, but the graph stays equivalent.
9. Can I use decimal values?
Yes. Decimal values are accepted for all numeric inputs. You can also set the number of decimal places used in the output.
10. Can it convert intercept form?
Yes. It converts y = a(x - r₁)(x - r₂). The calculator expands the product and moves y to the left side.
11. What is focal distance p?
Focal distance p is the signed distance from the vertex to the focus. It controls how wide or narrow the parabola is.
12. Why does a cannot equal zero?
If a equals zero, the squared term disappears. The equation becomes linear, not a parabola.
13. Does the CSV include coefficients?
Yes. The CSV download includes the result table values, including A, B, C, D, E, F, discriminant, vertex, and focal distance.
14. Does the PDF include the final equation?
Yes. The PDF download includes the general equation and the reported values from the result table.