Math calculators

Quadratic Regression Calculator

Updated Sep 24, 2026 By Infinity Calculator
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Data Input

Input Mode
Separate values with commas, spaces or new lines.
Must contain the same number of values as X.
(Forces the parabola to pass through the origin.)

Introduction

This Quadratic Regression Calculator finds the best-fit parabola for your data. You type in your x and y values, and it gives you the equation y = ax² + bx + c. Use it when your data points curve instead of forming a straight line.

Every step is shown along with the answer: the sums table, the normal equations, the row work, and the back-substitution. You also get R², the correlation coefficient r, standard errors, the vertex, the axis of symmetry, and a scatter plot with the fitted curve drawn on top.

You can paste values as a list or type them into a table. Pick how many decimal places you want. Check the box to force the curve through the origin (c = 0) if your model needs that. Then use the prediction tool to find ŷ for any x, and download your full results as a file.

How to use our Quadratic Regression Calculator

Enter your x and y data points, and the calculator finds the best-fit parabola y = ax² + bx + c. You get the equation, R², the vertex, step-by-step math, and a scatter plot with the fitted curve.

Input Mode: Pick Text Mode to paste lists of numbers, or Table Mode to type each point in its own row.

X Values: Type your x numbers, split by commas, spaces, or new lines. You need at least 3 points and at least 3 different x values.

Y Values: Type your y numbers in the same order as the x values. The count must match the x list.

Add Row (Table Mode): Click it to add one more data point. Use the red trash button to delete a row you do not want.

Force c = 0: Check this box if the curve must pass through the origin. The model then becomes y = ax² + bx.

Decimal Places: Choose how many decimals to show in the results, from 2 up to 8.

Calculate: Click to run the regression and show your input, the full solution steps, the final equation, the stats, and the graph.

Clear / Reset: Click to bring back the sample data and default settings.

Download Results: Click to save your equation, stats, and tables as an HTML file you can keep or print.

Predict ŷ for x: Type any x value and click Predict to get the y value from your fitted curve.

Reset Zoom: Click to return the chart to its full view after you zoom in.

What Is Quadratic Regression?

Quadratic regression is a way to find the curved line that best fits a set of data points. The curve is a parabola, and its equation looks like this:

y = ax² + bx + c

You use it when your data does not follow a straight line. Instead, it rises and then falls, or falls and then rises. Things like the path of a thrown ball, profit that peaks and drops, or plant growth that slows down often fit a parabola better than a line.

What the Letters Mean

  • a sets how wide or narrow the curve is. If a is positive, the parabola opens up (it has a low point). If a is negative, it opens down (it has a high point).
  • b shifts the curve left or right and helps set the turning point.
  • c is the y-intercept. It is the y value when x = 0.

How the Best Curve Is Found

The method is called least squares. For each data point, you find the gap between the real y value and the y value the curve predicts. That gap is called a residual. The best curve is the one that makes the total of all the squared residuals as small as possible.

To get there, you add up columns of numbers from your data: Σx, Σx², Σx³, Σx⁴, Σy, Σxy, and Σx²y. Those sums go into three equations called the normal equations. Solving them gives you a, b, and c.

Key Terms to Know

  • R² (R-squared) shows how well the curve matches the data. It runs from 0 to 1. An R² of 0.98 means the curve explains 98% of the change in y. Higher is better.
  • Vertex is the turning point of the parabola. Its x value is −b ÷ (2a). This is the highest or lowest point of the curve.
  • Axis of symmetry is the vertical line through the vertex. The parabola looks the same on both sides of it.
  • Discriminant (Δ = b² − 4ac) tells how many times the curve crosses the x-axis: twice if Δ is positive, once if Δ is zero, never if Δ is negative.
  • Standard error shows how much each coefficient might be off. Smaller numbers mean more trust.

Things to Remember

  • You need at least 3 data points, and at least 3 different x values, to fit a parabola.
  • With exactly 3 points, the curve passes through all of them perfectly, so R² = 1. That does not prove the model is good.
  • Compare the quadratic R² with the linear R². If they are close, a straight line may be the simpler, better choice.
  • Predictions are safest inside your range of x values. Going far outside that range can give strange results, since a parabola always turns around at some point.
  • Force c = 0 only when you know the curve must pass through the origin, like distance that starts at zero.

Formulas used

Quadratic regression model
\hat{y} = ax^2 + bx + c
Normal equations for least-squares quadratic fit
\begin{aligned} a\sum x^4 + b\sum x^3 + c\sum x^2 &= \sum x^2 y \\ a\sum x^3 + b\sum x^2 + c\sum x &= \sum xy \\ a\sum x^2 + b\sum x + cn &= \sum y \end{aligned}
Coefficients by Cramer's rule
a = \frac{D_a}{D}, \quad b = \frac{D_b}{D}, \quad c = \frac{D_c}{D}, \quad D = \begin{vmatrix} \sum x^4 & \sum x^3 & \sum x^2 \\ \sum x^3 & \sum x^2 & \sum x \\ \sum x^2 & \sum x & n \end{vmatrix}
Coefficient of determination
R^2 = 1 - \frac{SS_{res}}{SS_{tot}} = 1 - \frac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2}
Vertex of the fitted parabola
x_v = \frac{-b}{2a}, \quad y_v = a x_v^2 + b x_v + c
Discriminant
\Delta = b^2 - 4ac
Standard errors of the coefficients
s^2 = \frac{SS_{res}}{n-3}, \quad SE(a) = \sqrt{s^2\,\frac{C_{11}}{D}}, \quad SE(b) = \sqrt{s^2\,\frac{C_{22}}{D}}, \quad SE(c) = \sqrt{s^2\,\frac{C_{33}}{D}}
Centered sums of squares and cross-products
S_{xx} = \sum x^2 - \frac{(\sum x)^2}{n}, \quad S_{xy} = \sum xy - \frac{(\sum x)(\sum y)}{n}, \quad S_{x^2x^2} = \sum x^4 - \frac{(\sum x^2)^2}{n}, \quad S_{x^2y} = \sum x^2 y - \frac{(\sum x^2)(\sum y)}{n}

Frequently asked questions

How do you calculate quadratic regression by hand?

You build sums from your data, then solve three equations.

  1. Make columns for x, y, x², x³, x⁴, xy, and x²y. Add each column.
  2. Put the sums into the three normal equations:
    a·Σx⁴ + b·Σx³ + c·Σx² = Σx²y
    a·Σx³ + b·Σx² + c·Σx = Σxy
    a·Σx² + b·Σx + c·n = Σy
  3. Solve the three equations for a, b, and c using elimination or determinants.
  4. Write the answer as y = ax² + bx + c.

The math is simple but long, so mistakes are easy with more than a few points.

Is quadratic regression a type of linear regression?

Yes. "Linear" here means the equation is linear in the unknowns a, b, and c, not in x. You treat x² as just another input variable. That is why the same least squares method and the same normal equations work for both. The fitted line looks curved, but the math behind it is the same family as straight-line regression.

How do you do quadratic regression on a TI-84 calculator?

Follow these steps:

  1. Press STAT, choose 1:Edit, and type your x values in L1 and y values in L2.
  2. Press STAT, move to CALC, and pick 5:QuadReg.
  3. Set Xlist to L1, Ylist to L2, then choose Calculate.

The screen shows a, b, and c. If R² does not show, press 2nd then 0 (CATALOG), scroll to DiagnosticOn, press ENTER twice, and run QuadReg again.

What is a good R² value for a quadratic model?

It depends on the subject. In physics or lab work, R² above 0.99 is normal because the data is clean. In business, sports, or social data, 0.70 can be strong. Two rules help:

  • Compare it to the straight-line R² for the same data. If the curve only adds a tiny bit, keep the line.
  • A high R² does not prove the model is right. Always look at the residuals to see if they scatter randomly.

Can R² be negative in a quadratic fit?

With a normal fit that includes the c term, no. R² stays between 0 and 1 because the curve can always do at least as well as the mean of y.

But if you force the curve through the origin (c = 0) and your data does not really pass near the origin, the fit can be worse than just using the average y. Then R² can come out negative. That is a sign the forced model is wrong for your data.

What is the difference between quadratic regression and polynomial regression?

Quadratic regression is one kind of polynomial regression. Polynomial regression fits any power of x:

  • Degree 1: y = bx + c (a straight line)
  • Degree 2: y = ax² + bx + c (quadratic, one bend)
  • Degree 3: cubic, with up to two bends

Higher degrees bend more and hug the data more closely, but they often follow noise instead of the real pattern. Degree 2 is usually the first curve to try.

How do you know if the x² term is really needed?

Test the coefficient a. Divide it by its standard error to get a t value:

t = a ÷ SE(a)

With a decent number of points, a t value larger than about 2 (ignoring the sign) means the curve is doing real work. If t is small, the x² term may just be fitting noise, and a straight line is the better, simpler model. Comparing the two R² values gives the same clue.

What does it mean if the coefficient a is almost zero?

It means your data barely curves. When a is near zero, the equation y = ax² + bx + c acts almost like the straight line y = bx + c. The parabola becomes so wide that the bend is not visible in your x range. In that case, use linear regression instead. It is easier to explain and less likely to give wild predictions.

What real-life data fits a quadratic curve?

Quadratic models fit patterns that rise then fall, or fall then rise. Common examples:

  • The height of a thrown or kicked ball over time
  • Profit as price goes up: it climbs, peaks, then drops
  • Stopping distance of a car versus speed
  • Crop yield versus amount of fertilizer, which peaks then declines
  • Area of a shape as one side length changes

How do you find the peak value, like maximum profit, from a quadratic regression equation?

Use the vertex. First find the x value:

x = −b ÷ (2a)

Then put that x back into y = ax² + bx + c to get the peak y value. If a is negative, the vertex is the maximum (the highest point). If a is positive, it is the minimum. For example, with y = −2x² + 40x + 100, x = −40 ÷ (2 × −2) = 10, and y = 300, so profit peaks at 300 when x = 10.

What does the correlation coefficient r mean for a quadratic model?

Plain r measures straight-line strength only, so it can be misleading for a curve. Data in a clean U shape can have r near 0 even though the parabola fits perfectly. For curved fits, use R² instead, since it measures how much of the change in y the model explains no matter the shape. The r reported for a quadratic is just the square root of R², so treat it as a rough guide, not proof of a straight-line link.

Why do large x values cause rounding trouble in quadratic regression?

Because the math needs Σx⁴. If x is a year like 2024, then x⁴ is over 16 trillion. Adding huge numbers and then subtracting almost equal huge numbers wipes out the small digits that matter, so the coefficients can come out wrong.

The fix is coding: subtract a base value so x becomes 0, 1, 2, 3 instead of 2020, 2021, 2022, 2023. The curve shape stays the same, and the numbers stay small and safe.

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