Saved calculation history
No saved calculations yet.
Formula used
For a vertical parabola, the polynomial model is y = ax² + bx + c. Its vertex occurs at h = -b/(2a) and k = f(h).
The focal parameter is p = 1/(4a). Therefore, the focus is (h, k + p), and the directrix is y = k - p.
For a horizontal parabola, exchange the roles of x and y. The equation becomes x = ay² + by + c.
The discriminant is Δ = b² - 4ac. Its sign determines whether the roots are distinct, repeated, or complex.
How to use this calculator
Select the equation form or builder matching your known information. Choose vertical or horizontal orientation before entering values.
Enter coefficients, points, roots, focus data, or regression coordinates. Fractions such as 3/4 are accepted in numeric fields.
Use the optional calculus fields for tangents, normals, integrals, areas, arc lengths, and intersections. Press Calculate to generate complete results.
Explore the graph using zooming and panning controls. Download reports, images, CSV data, or JSON when finished.
Understanding advanced parabola analysis
Equation forms reveal different properties
Standard form displays the polynomial coefficients directly. Vertex form makes the turning point immediately visible. Factored form highlights real roots and axis crossings.
Conic form connects the curve with focus geometry. Parametric form describes every point using one parameter. Conversion tools make these views easy to compare.
The leading coefficient controls shape
The sign of a controls the opening direction. Its magnitude controls the apparent width. Larger magnitudes create narrower curves near the vertex.
Small magnitudes create wider curves. A zero value removes the quadratic term. That equation becomes linear and is not a parabola.
Focus and directrix define the curve
Every parabola point is equally distant from its focus and directrix. This geometric rule explains reflection behavior. It also supports antenna and reflector design.
The focal distance is represented by p. The latus rectum passes through the focus. Its total length equals four times absolute p.
Calculus describes local behavior
The first derivative gives the tangent slope. The second derivative describes constant concavity. The vertex appears where the first derivative becomes zero.
Definite integration measures signed accumulation. Absolute integration estimates geometric area. Arc length measures distance along the curved path.
Regression fits measured data
Quadratic regression estimates a curve from many points. Residuals measure differences between observations and predictions. Smaller residuals generally indicate closer agreement.
R² summarizes explained variation within the dataset. RMSE reports typical prediction error. Both should be interpreted with practical context.
Frequently asked questions
Can the calculator handle horizontal parabolas?
Yes. Select horizontal orientation to analyze x as a quadratic function of y.
Can I enter fractions?
Yes. Numeric fields accept fractions such as 1/2, -3/4, and 7/5.
What happens when a equals zero?
The quadratic term disappears, so the equation no longer represents a parabola.
How are complex roots displayed?
The calculator reports the real and imaginary components when the discriminant is negative.
Can I create a parabola from points?
Yes. Use three points for an exact model or regression for many measurements.
Does the graph show the focus and directrix?
Yes. The graph marks the vertex, focus, directrix, axis, roots, and selected point.
How is the area calculated?
Signed area uses the polynomial antiderivative. Absolute area uses numerical Simpson integration.
Can I compare several parabolas?
Yes. Enter additional coefficient triples and update the comparison graph and table.
How do I save a PDF?
Choose Print or save PDF, then select your browser's PDF destination.