How many points do I need for quadratic regression?

At least 3 with different x values. With exactly 3, the parabola passes through every point and R² is 1, so more points give a more meaningful fit.

Is this the same as TI-84 QuadReg?

Yes. QuadReg fits y = ax² + bx + c by least squares, and the coefficients and R² match.

What does R² mean for a quadratic fit?

The proportion of the variation in y that the parabola explains. R² = 0.95 means 95% of the variation is accounted for.

How do I find the vertex of the fitted parabola?

x = −b ÷ (2a), then substitute into the equation. The calculator shows it and says whether it is a maximum or minimum.

Quadratic Regression Calculator

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Quadratic Regression Calculator

This quadratic regression calculator finds the parabola y = ax² + bx + c that best fits a set of points, in the least-squares sense. Enter the x and y values and it returns a, b and c, the coefficient of determination R², each fitted value and residual, the vertex of the parabola and an optional prediction.

Use a quadratic model when a scatter plot shows one bend: projectile height against time, yield against fertiliser, cost against output with diminishing returns. The coefficients are found by regressing y on x and x² together, which is what the TI-84’s QuadReg, Excel’s LINEST with x and x² columns, and NumPy’s polyfit(x, y, 2) do; the answers agree.

R² measures the share of variation in y explained by the curve. Compare it with the straight-line fit from the linear regression calculator, and look at the residuals: if they still show a pattern, another model may suit better. For growth that speeds up in proportion to its size, try the exponential regression calculator.

x values

Separate values with commas, spaces or new lines.

y values

Separate values with commas, spaces or new lines.

Predict y at x = (optional)

Result and step-by-step solution

Best-fit parabola

y = 0.955952x² + 0.291667x + 1.710714

R² = 0.999962 (100% of the variation in y explained).

Model
y = ax² + bx + c, fitted by least squares (minimise Σ(y − ŷ)²)
Coefficients
a = 0.955952, b = 0.291667, c = 1.710714
Fitted values and residuals
x y ŷ residual
1 3.1 2.9583 0.1417
2 5.9 6.1179 −0.2179
3 11.2 11.1893 0.0107
4 18.1 18.1726 −0.0726
5 27.3 27.0679 0.2321
6 37.8 37.875 −0.075
7 50.6 50.594 0.006
8 65.2 65.225 −0.025
Goodness of fit
R² = 1 − SSE ÷ SST = 1 − 0.1331 ÷ 3476.92 = 0.999962
Vertex
x = −b ÷ 2a = −0.1526, y = 1.6885 (minimum)
Prediction
At x = 9: ŷ = 81.7679

x	y	ŷ	residual
1	3.1	2.9583	0.1417
2	5.9	6.1179	−0.2179
3	11.2	11.1893	0.0107
4	18.1	18.1726	−0.0726
5	27.3	27.0679	0.2321
6	37.8	37.875	−0.075
7	50.6	50.594	0.006
8	65.2	65.225	−0.025

Frequently asked questions

How many points do I need for quadratic regression?: At least 3 with different x values. With exactly 3, the parabola passes through every point and R² is 1, so more points give a more meaningful fit.
Is this the same as TI-84 QuadReg?: Yes. QuadReg fits y = ax² + bx + c by least squares, and the coefficients and R² match.
What does R² mean for a quadratic fit?: The proportion of the variation in y that the parabola explains. R² = 0.95 means 95% of the variation is accounted for.
How do I find the vertex of the fitted parabola?: x = −b ÷ (2a), then substitute into the equation. The calculator shows it and says whether it is a maximum or minimum.

Last reviewed: October 7, 2026. Calculations run in your browser and were validated against SciPy.

Quadratic Regression Calculator

Result and step-by-step solution

Frequently asked questions

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