Quadratic Regression Calculator - Fit y = ax² + bx + c to paired data

A quadratic regression calculator is a curve-fitting tool that takes paired (x, y) data and returns the coefficients a, b, and c of the parabola y = a·x² + b·x + c that best fits the points by least squares. It solves the 3×3 normal equations from sums of 1, x, and x², reports R² and the vertex of the fitted curve, and predicts y for any new x.

• • Classroom regression homework: Verify a hand-solved quadratic fit on small data sets from algebra or introductory statistics courses.
• • Fitting physics data: Fit parabolas to projectile position data, free-fall distance versus time, or pendulum-energy curves where theory predicts a quadratic form.
• • Calibrating sensor responses: Capture the parabolic shape of an instrument's response curve when the spec sheet predicts a quadratic calibration polynomial.
• • Modelling cost and dose-response: Estimate the minimum of a quadratic cost function, or fit a parabolic dose-response curve around the inflection region.

The fit is purely deterministic: paste paired observations and the calculator rebuilds the normal equations every time a value changes, reporting the equation, three coefficients, R², vertex, predicted y, SSE, and standard error.

Use this calculator when the data form a single symmetric curve. If the data look linear, fit a straight line instead; if they grow geometrically, fit an exponential curve.

When the same data grow by a roughly constant percentage per unit x, the Exponential Regression Calculator fits y = a·e^(b·x) instead.

Quadratic regression fits the model y = a·x² + b·x + c by least squares. The three coefficients solve a 3×3 linear system whose entries are sums of powers of x and cross-products with y. The calculator builds this system every time a value changes.

• xᵢ: Independent variable value for the i-th data point. Any real number.
• yᵢ: Observed dependent variable value for the i-th data point. Any real number.
• a: Quadratic coefficient. a > 0 opens the parabola upward, a < 0 opens it downward.
• b: Linear coefficient. Tilts the parabola left or right along with a.
• c: Constant term. Equal to the predicted y when x = 0.
• n: Number of paired observations used in the fit. Must be at least three, ideally five or more.

R² in the result panel is the squared correlation between observed y and the fitted curve. R² near 1 means the parabola hugs the points; R² close to 0 means a quadratic shape is not supported and a different model family should be tried.

The vertex (xᵥ, yᵥ) is reported separately so users can read the minimum or maximum of the fitted parabola directly. xᵥ = -b / (2a) when a ≠ 0; the calculator returns 0, 0 when a is essentially zero, which signals the data are effectively linear.

According to NIST/SEMATECH e-Handbook of Statistical Methods, polynomial regression including quadratic models is fit by solving the normal equations XᵀXβ = Xᵀy, where X contains powers of the predictor.

When the data look linear instead of curved, the Linear Regression Calculator runs the same least-squares idea without the x² term.

Four small ideas decide how the calculator behaves and how the fitted equation should be read.

These ideas recur in any polynomial fit. The same approach extends to cubic and higher-degree fits by adding more powers of x to the design matrix.

When the same data look reasonable under several models, compare R², residuals, and domain knowledge rather than chasing the smallest R².

For a quick read on the strength of the linear association before fitting a curve, the Correlation Calculator returns r and r² for paired data.

Paste paired observations into the two text areas, optionally set a prediction x, and read the fitted parabola on the right.

1. 1 Enter at least three (x, y) pairs: Use the same units for every x and every y. Paste comma, space, or line-separated values into each list.
2. 2 Match the list lengths: Each row of the X list pairs with the same row of the Y list. A status message replaces the result when the lengths differ.
3. 3 Vary the X values: If every x is the same number the design matrix is singular. Spread x across the prediction range.
4. 4 Set a prediction x: Type any x value to read the fitted curve at that point. Leave the field blank to skip the prediction.
5. 5 Read the equation, coefficients, and R²: The primary line shows the equation; the next three lines are a, b, and c; R² tells how much of the variance the parabola explains.
6. 6 Check the vertex for a minimum or maximum: The vertex (xᵥ, yᵥ) is the turning point of the fitted parabola. Use it directly when the goal is to locate the minimum or maximum.

An engineering student measures bicycle stopping distance at 20, 30, 40, 50, 60, 70 km/h and gets 6, 14, 26, 42, 62, 86 m. Paste the six pairs, set prediction x = 80, and the calculator returns a = 0.02, b = -0.2, c = 2.0 with R² = 100%.

If you also need the mean, standard deviation, and other descriptive statistics on the same data, the Statistics Calculator sits alongside this one.

The form replaces a normal-equation solve, an R² calculation, a vertex calculation, and a prediction with a single screen.

• • One form for the full quadratic fit: Enter paired observations once and the calculator shows the fitted equation, the three coefficients, R², the vertex, SSE, standard error, and the predicted y in one place.
• • Vertex and minimum point built in: The vertex (xᵥ, yᵥ) is reported automatically, so users looking for a parabola minimum or maximum can read it without doing the -b / (2a) calculation by hand.
• • Standard error and SSE for sanity checks: SSE and standard error sit next to R² so users can spot a fit that looks good on R² but has large residuals in absolute terms.
• • Live recalculation on every change: Each input change rebuilds the normal equations and refreshes the result panel, so editing a row in either list updates coefficients and R² immediately.
• • Honest status messages: Fewer than three pairs, mismatched list lengths, identical x values, and singular design matrices each produce a labelled status instead of a silently wrong number.

When you want to convert the SSE on the result panel into a comparable spread across datasets, the Standard Deviation Calculator returns σ and s for a single column.

Most surprises with quadratic regression come from the data shape rather than the math itself.

According to Wikipedia - Polynomial regression, higher-degree polynomial fits can drive training residuals toward zero without improving real-world fit, so the degree of the polynomial is itself a modelling decision rather than something to maximise blindly.

• • Predictions outside the fitted x range are extrapolations. The parabola grows without bound, so a small change in x can produce a large change in predicted y when x is far from the vertex.
• • This calculator reports coefficients, R², and the standard error of the residuals. Users who need standard errors or confidence intervals on a, b, c themselves should use a dedicated statistics package.
• • When the data are essentially linear the quadratic coefficient a collapses toward zero and the vertex becomes meaningless. Switch to a linear regression when a is smaller than the typical residual.

When the same data look reasonable under several models, compare R², residuals, and domain knowledge rather than chasing the smallest R².

According to Penn State STAT 200 - Linear Models, least-squares regression chooses the coefficients that minimize the sum of squared residuals, and the same approach extends directly to higher-degree polynomial models.

When the next step is comparing two fitted curves from different samples, the T-Test Calculator covers the standard two-sample t-test workflow.