ES
Home › Calculators › Quadratic Regression Calculator

Quadratic Regression Calculator

Paste your x and y data to find the least-squares parabola y = ax² + bx + c, or switch to a cubic, quartic, quintic or degree-6 polynomial. You get the equation, R², residuals, a prediction for any x and a scatter plot with the fitted curve.

Last updated:

Data format

One point per line, x and y separated by a comma, space or tab. Paste straight from a spreadsheet.

Enter your values to see the result.
Advertisement
Advertisement

How to use the Quadratic Regression Calculator

  1. Enter your data as x, y pairs (one per line) or as two separate lists.
  2. Pick the degree: 2 for a quadratic (parabola), up to 6 for higher-order curves.
  3. Optionally enter an x value to predict its y.
  4. Read the equation and R², then check the graph and residuals.
  5. Compare degrees in the table to see whether a higher degree really improves the fit, and download the CSV.

Quadratic regression formula

For n points (xi, yi) the least-squares parabola y = ax² + bx + c minimises Σ(yi − axi² − bxi − c)². Setting the partial derivatives to zero gives three normal equations:

Equationa termb termc term= right side
1Σx⁴ · aΣx³ · bΣx² · cΣx²y
2Σx³ · aΣx² · bΣx · cΣxy
3Σx² · aΣx · bn · cΣy

Higher degrees work the same way with more powers of x (in matrix form, (XᵀX)β = Xᵀy). Fit quality is measured with R² = 1 − SSres / SStot, where SSres = Σ(y − ŷ)² and SStot = Σ(y − ȳ)², and the standard error of the estimate is √(SSres / (n − k − 1)) for a degree-k polynomial.

Choosing the polynomial degree

DegreeNameShapeMinimum pointsTypical use
1LinearStraight line2Steady trends
2QuadraticParabola, one turning point3Projectile paths, profit vs price, acceleration
3CubicUp to two turning points, S-shape4Growth that speeds up then slows
4QuarticUp to three turning points5More complex calibration curves
5–6Quintic, sexticUp to 4–5 turning points6–7Smooth interpolation of many points; risk of overfitting

Tip: if moving from one degree to the next barely changes the adjusted R², keep the simpler model. Always look at the residuals: a curved pattern in the residuals means the model is missing part of the shape.

Frequently asked questions

What is quadratic regression?

Quadratic regression finds the parabola y = ax² + bx + c that best fits a set of points by minimising the sum of the squared vertical distances (residuals) between the points and the curve. It is used when data rises and then falls, or curves upward, instead of following a straight line.

How many data points do I need?

At least three points with different x values for a quadratic, and in general degree + 1 points for a polynomial of that degree. With exactly that many points the curve passes through all of them (R² = 1), so use more points to get a meaningful fit.

What does R² mean?

R² (the coefficient of determination) is the share of the variation in y that the model explains: R² = 1 − SSres/SStot. A value of 1 is a perfect fit and 0 means the curve explains nothing beyond the average of y. Adjusted R² penalises extra coefficients, which helps when comparing degrees.

Should I always use a higher degree?

No. A higher degree always raises R² on the data you have, but it can overfit and swing wildly between and beyond your points. Choose the lowest degree that captures the shape; the degree comparison table and adjusted R² help with that.

How is the quadratic fit calculated?

The coefficients solve the normal equations: Σx⁴a + Σx³b + Σx²c = Σx²y, Σx³a + Σx²b + Σxc = Σxy and Σx²a + Σxb + nc = Σy. This tool shows those sums with your numbers and solves the system with a numerically stable QR decomposition.

Can I trust predictions outside my data range?

Be careful. Predicting beyond the smallest or largest x (extrapolation) assumes the curve keeps the same shape, which polynomials often do not. The calculator flags predictions that are outside your data range.

Is my data uploaded anywhere?

No. All calculations run in your browser and the data never leaves your device.

Are all the tools on FreeToolCabin tested for quality and accuracy?

Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.

Related tools

Linear Regression CalculatorLeast-squares line of best fit: slope, intercept, r, r², predictions and a scatter plot.Quadratic Equation SolverSolve ax² + bx + c = 0 with exact roots, steps, vertex and a parabola graph.Parabola CalculatorVertex, focus, directrix, axis, intercepts and graph of a parabola from its equation.Standard Deviation CalculatorStandard deviation, variance, mean, median, mode and quartiles.Graphing CalculatorPlot multiple functions with zoom, pan, intersections and a value table.Exponent CalculatorRaise any number to a power, including negative and fractional exponents.Logarithm CalculatorCalculate log in any base, ln, log10, log2 and antilog with change-of-base steps.Matrix CalculatorAdd, multiply, invert and row-reduce matrices up to 5×5 with exact fractions and steps.Prime Number CheckerCheck if a number is prime, factor it with a factor tree and list primes in a range.GCF and LCM CalculatorFind the GCF and LCM of two or more numbers with Euclid steps and prime factorization.Cubic Yard CalculatorConvert length, width and depth in any unit to cubic yards, cubic feet, m³ and material weight.Factorial CalculatorExact n! up to 5000! with digit count, trailing zeros and double factorial.Prime Factorization CalculatorFind prime factors in exponent form with a factor tree, divisor count and divisors list.Modulo CalculatorCalculate a mod n and the quotient, with floored, truncated and Euclidean rules.Exponential Growth CalculatorExponential growth and decay: final amount, rate or time, discrete or continuous.System of Equations SolverSolve 2×2, 3×3 and 4×4 linear systems with exact fractions and step-by-step working.Cubic Equation SolverSolve ax³ + bx² + cx + d = 0 for real and complex roots with steps and a graph.Summation CalculatorEvaluate sigma notation: sum any expression in n from a to b with partial sums and a chart.Arithmetic Sequence CalculatorFind the n-th term, sum and full list of an arithmetic sequence, or d from two terms.Geometric Sequence CalculatorFind the n-th term, sum and infinite sum of a geometric sequence, or r from two terms.Fibonacci CalculatorExact n-th Fibonacci number (BigInt), sequence lists, Fibonacci check and golden ratio.Factors CalculatorAll factors and factor pairs of a number, prime factors, divisor count and perfect check.Depth of Field CalculatorNear and far focus limits, depth of field and hyperfocal distance for any camera or lens.Crop Factor CalculatorCrop factor, full-frame equivalent focal length and aperture, and field of view for any sensor.Completing the Square CalculatorRewrite ax² + bx + c in vertex form a(x − h)² + k step by step and solve it.Derivative CalculatorSymbolic derivative of any f(x) up to 5th order, with steps, slope at a point and a graph.Exposure Value CalculatorEV from aperture, shutter and ISO, or solve any missing setting with sunny 16 presets.Factoring Trinomials CalculatorFactor trinomials and polynomials: GCF, ac method, difference of squares, cubes, with steps.Integral CalculatorDefinite integral of f(x) from a to b with adaptive Simpson, area graph and antiderivative.Least Common Denominator CalculatorFind the LCD of two or more fractions and rewrite them as equivalent fractions, with steps.Limiting Reactant CalculatorFind the limiting reactant, theoretical yield and leftover excess from an equation.Polynomial Root FinderFind all real and complex roots of a polynomial up to degree 10, with the factored form.Prime Number List GeneratorList all primes in a range or the first N primes with a fast sieve; copy or download the list.Scatter Plot MakerPaste X,Y pairs for a scatter plot with trendline, equation, r and R². Download PNG.Synthetic Division CalculatorDivide polynomials by x − c (synthetic table) or any polynomial (long division) with steps.3D Graphing CalculatorPlot z = f(x, y) surfaces and 3D parametric curves; rotate, zoom, add contours, save PNG.Algebra CalculatorSimplify, expand and factor algebraic expressions with exact answers and step-by-step working.Determinant CalculatorDeterminant of a 2×2 to 8×8 matrix with exact fractions and cofactor or row-reduction steps.Antilog CalculatorInverse logarithm (antilog) in base 10, e, 2 or any base, with steps and scientific notation.Asymptote CalculatorFind vertical, horizontal and slant asymptotes and holes of a function, with graph and reasoning.Cramer's Rule CalculatorSolve 2x2 and 3x3 linear systems by Cramer's rule with every determinant shown step by step.Dot Plot MakerMake a dot plot or line plot from data or a frequency table; download PNG or SVG.Exponential Regression CalculatorFit y = a·bˣ or y = a·e^(kx) to data with r², predictions, residuals and a plot.Extended Euclidean Algorithm CalculatorFind gcd(a, b) and Bézout coefficients x, y with the full step-by-step division table.Factor Tree GeneratorDraw the prime factor tree of any number as a printable diagram, with exponent form and steps.Function Composition CalculatorFind f(g(x)), g(f(x)), f + g, f · g and f / g with values at a point and domains.X and Y Intercept CalculatorFind the x- and y-intercepts of lines, curves and functions with steps and a graph.Line Intersection CalculatorFind where two lines cross from equations, points or slopes, with parallel checks and a graph.Linear Programming SolverSolve max/min linear programs with the simplex method: tableau steps and a 2-variable graph.Modular Exponentiation CalculatorCompute a^b mod m for huge numbers with square-and-multiply steps (BigInt).Modular Inverse CalculatorFind a⁻¹ mod m with the extended Euclidean algorithm, step by step, for huge numbers too.Nth Term CalculatorFind the nth term formula of a sequence: arithmetic, geometric, quadratic or cubic.Partial Fraction Decomposition CalculatorSplit a rational function into partial fractions with the setup, equations and solution.Perpendicular Line CalculatorEquation of the perpendicular and parallel line through a point, with distance and a graph.Polar Graphing CalculatorGraph polar equations r = f(θ): roses, cardioids, spirals, with a θ range slider and PNG export.Polynomial CalculatorAdd, subtract, multiply, divide and evaluate polynomials; see roots, degree and a graph.Polynomial Long Division CalculatorDivide polynomials with the full long division layout, quotient, remainder and steps.Scale Factor CalculatorScale factor between similar figures, plus scaled lengths, areas (k²) and volumes (k³).Set CalculatorUnion, intersection, difference and complement of up to 4 sets, with a Venn diagram and counts.Table of Values CalculatorMake an x/y table for any function with a start, end and step, then plot it or copy CSV.Differential Equation SolverSolve first- and second-order ODEs step by step, with initial conditions, an RK4 check and a graph.Domain and Range CalculatorFind the domain and range of a function in interval notation, with steps and a graph.Eigenvalue and Eigenvector CalculatorEigenvalues (real or complex), eigenvectors and the characteristic polynomial of a square matrix.Equation SolverSolve any equation for x: exact roots, complex roots, numeric answers, steps and a graph.Inequality CalculatorSolve linear, quadratic, rational and absolute value inequalities with steps and a number line.Inverse Function CalculatorFind the inverse f⁻¹(x) of a function with steps, domain restrictions and a graph.Laplace Transform CalculatorLaplace transform of f(t) with the table rule used for each term and the region of convergence.Limit CalculatorEvaluate limits at a point or at infinity, one-sided or two-sided, with steps, table and graph.
See all 74 Algebra & Calculus Calculators →