Quadratic Regression Calculator: Curve Fit & Formula

A quadratic regression calculator fits a second-degree polynomial (parabola) to a set of (x, y) data points, returning the coefficients a, b, and c in the equation y = ax² + bx + c. You enter paired x and y values, and the calculator gives you the best-fit curve along with a measure of how well it fits, such as R². This is useful when your data shows a curved trend that a straight line cannot capture.

Formula

The quadratic regression model is:

y = a·x² + b·x + c

where:

• x is the independent variable (input),
• y is the dependent variable (output),
• a, b, and c are the coefficients that define the parabola.

The coefficients are found by solving the normal equations derived from the least-squares method. For n data points (xᵢ, yᵢ), the system is:

a·Σx⁴ + b·Σx³ + c·Σx² = Σx²y
a·Σx³ + b·Σx² + c·Σx = Σxy
a·Σx² + b·Σx + c·n = Σy

Here Σ denotes summation over all data points. The solution gives the parabola that minimizes the sum of squared vertical distances between the data points and the curve.

Worked Example

Using the default data: x = 1, 2, 3, 4, 5 and y = 3, 5, 8, 12, 17.

Step 1: Compute the sums needed for the normal equations.

• n = 5
• Σx = 1+2+3+4+5 = 15
• Σx² = 1+4+9+16+25 = 55
• Σx³ = 1+8+27+64+125 = 225
• Σx⁴ = 1+16+81+256+625 = 979
• Σy = 3+5+8+12+17 = 45
• Σxy = 1·3 + 2·5 + 3·8 + 4·12 + 5·17 = 3+10+24+48+85 = 170
• Σx²y = 1·3 + 4·5 + 9·8 + 16·12 + 25·17 = 3+20+72+192+425 = 712

Step 2: Set up the normal equations:

979a + 225b + 55c = 712
225a + 55b + 15c = 170
55a + 15b + 5c = 45

Step 3: Solve the system (using elimination or a matrix solver). The solution is:

a = 0.5, b = 1.1, c = 1.4

Step 4: Write the fitted equation: y = 0.5x² + 1.1x + 1.4

Step 5: Compute R² to see how well the curve fits. The predicted y values are: for x=1, y=3.0; x=2, y=5.6; x=3, y=9.2; x=4, y=13.8; x=5, y=19.4. The R² is approximately 0.998, meaning the curve explains 99.8% of the variation in y.

When to Use Quadratic Regression

Quadratic regression is appropriate when your data shows a single bend (a maximum or minimum) and the relationship is not linear. Common examples include:

• Projectile motion: height vs. time follows a parabola.
• Cost vs. production quantity: often U-shaped due to economies and diseconomies of scale.
• Temperature vs. time of day: a simple parabolic approximation for daily temperature cycles.
• Yield vs. fertilizer amount: often increases then decreases, forming an inverted U.

If your data has two bends, you would need a cubic or higher-degree polynomial. If it has no bend, linear regression is simpler and more appropriate.

How to Read the Results

The coefficient a tells you the direction and curvature of the parabola:

• If a > 0, the parabola opens upward (minimum at the vertex).
• If a < 0, it opens downward (maximum at the vertex).
• The larger |a|, the steeper the curve.

The vertex (the peak or trough) is at x = -b / (2a). The value of c is the y-intercept, the predicted y when x = 0.

R² (coefficient of determination) ranges from 0 to 1. A value close to 1 means the model fits the data well. However, R² alone is not enough; always check a scatter plot to ensure the parabola is a sensible shape for your data.

Common Mistakes and Limitations

• Unit mix-ups: If x and y have units, the coefficients will have fractional units (e.g., if y is in meters and x is in seconds, then a has units m/s², b has m/s, and c has m). Keep units consistent.
• Overfitting: A quadratic curve will always fit better than a straight line, but that does not mean the relationship is truly quadratic. Use domain knowledge.
• Extrapolation danger: The parabola continues indefinitely; predictions far outside your data range can be wildly wrong, especially if the curve turns sharply.
• Rounding too early: Keep many decimal places during calculation; round only the final coefficients.
• Not checking residuals: A systematic pattern in residuals (the differences between observed and predicted y) suggests a better model exists.

Quadratic vs. Linear vs. Polynomial Regression

Linear regression fits a straight line y = mx + b. Quadratic regression adds an x² term to capture curvature. Higher-degree polynomial regression (cubic, quartic, etc.) can fit even more complex curves, but with more flexibility comes the risk of overfitting—following random noise rather than the true trend.

As a rule of thumb, start with linear, then try quadratic if the scatter plot shows a clear bend. Only go to cubic or higher if you have a strong theoretical reason or if the data clearly has multiple bends.

Frequently Asked Questions

How is quadratic regression different from linear regression?

Linear regression fits a straight line and has one slope; quadratic regression fits a parabola and includes a squared term, allowing the rate of change to vary. Use quadratic when the data shows a curve that goes up then down (or vice versa).

What does R² mean in quadratic regression?

R² is the proportion of variance in y explained by the model. For example, an R² of 0.95 means 95% of the variation in y is accounted for by the quadratic relationship with x. It is a goodness-of-fit measure, but a high R² does not guarantee the model is appropriate—always inspect the plot.

Can I use quadratic regression for any data set?

Technically yes, the math will always produce a parabola, but it may be a poor fit. If the data is linear, the a coefficient will be near zero, and you might as well use linear regression. If the data has more than one bend, a higher-degree polynomial may be needed.

What is the vertex of the parabola?

The vertex is the point where the curve reaches its maximum or minimum. Its x-coordinate is -b / (2a). Plug that x into the equation to get the y-coordinate. This is often the most useful result, e.g., the time of maximum height or the optimal production level.

How do I choose between quadratic and cubic regression?

Look at the scatter plot. If the data has exactly one turn, quadratic is enough. If it has two turns (like an S-shape), you need a cubic. More turns require even higher degrees, but beware of overfitting—use the simplest model that adequately describes the trend.

Why does my calculator give a different answer than Excel?

Differences can arise from rounding, the algorithm used, or how missing data is handled. Most calculators and spreadsheets use the same least-squares method, so results should match if you enter the same data and use enough precision. Check that you have not accidentally swapped x and y.

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in x | x values (comma-separated) | - | 1, 2, 3, 4, 5 | 0
in y | y values (comma-separated) | - | 3, 5, 8, 12, 17 | 0
out a | Coefficient a | - | 0.5 | 4
out b | Coefficient b | - | 1.1 | 4
out c | Coefficient c | - | 1.4 | 4
out r2 | R² (goodness of fit) | - | 0.998 | 4