Example 1 — Perfect quadratic data
Data from \(y=2x^2-3x+1\): \((0,1),(1,0),(2,3),(3,10)\). The fit returns \(a=2,\ b=-3,\ c=1\) (exact), \(R^2=1\). Vertex: \(x_v=\tfrac{3}{4}\), \(y_v=\tfrac{-7}{8}\).
Quadratic Regression Calculator
Fit \(y \approx ax^2 + bx + c\) to your data. Enter X and Y lists and press Calculate.
Comma, space, or newline separated numbers.
Same count and order as X values.
Equation Preview
Your fitted equation will render here…
Helping notes
\( \text{Model: } y \approx ax^2 + bx + c \text{ with design matrix } X=\begin{bmatrix}x_1^2&x_1&1\\ \cdots&\cdots&\cdots\\ x_n^2&x_n&1\end{bmatrix}. \)
\( \text{Normal equations: } (X^{\mathsf T}X)\,\beta = X^{\mathsf T}y,\; \beta=[a,b,c]^{\mathsf T}. \)
\( R^2 = 1-\dfrac{\text{SSE}}{\text{SST}},\; \text{SSE}=\sum(y_i-\hat y_i)^2,\; \hat y_i=ax_i^2+bx_i+c. \)
Results
Coefficients
Model Fit
Predicted Values (ŷ)
What is a Quadratic Regression Calculator?
A Quadratic Regression Calculator estimates a best-fit parabola for paired data \((x_i,y_i)\) using least squares. Instead of a straight line, the model allows curvature, capturing accelerating or decelerating trends common in physics (projectiles), business (diminishing returns), biology (growth curves), and engineering. The tool returns the coefficients \(a,b,c\) in \(y=ax^2+bx+c\), reports goodness-of-fit (e.g., \(R^2\)), and highlights the vertex—the point where the fitted curve changes direction. Beyond raw numbers, it computes residuals, summarizes error (SSE, MSE), and can evaluate or forecast \(y\) at any \(x\). All steps appear clearly, with algebra shown so students can follow the derivation and practitioners can audit results.
About the Quadratic Regression Calculator
The calculator builds the design matrix with columns \([x_i^2,\;x_i,\;1]\) and solves the normal equations for the coefficient vector. Centering/scaling options can stabilize the fit when \(x\) is large, since \(x\) and \(x^2\) are often correlated. Diagnostics include \(R^2\) for explained variance, residual summaries to spot heteroscedasticity or outliers, and the vertex \((x_v,y_v)\) for interpretation (maximum or minimum depending on the sign of \(a\)). While quadratic models are flexible, they can overfit extremes; the tool flags extrapolation beyond the observed range and encourages inspection of residual patterns to validate assumptions.
How to Use this Quadratic Regression Calculator
- Paste or type paired values \((x_i,y_i)\) (comma/space/newline separated). The tool cleans and sorts as needed.
- Click Fit to compute \(a,b,c\), the vertex, \(R^2\), SSE, and MSE. Optional: enable centering of \(x\).
- Enter any \(x_0\) to get the predicted value \(\hat y(x_0)\) and its residual if an observed \(y_0\) is supplied.
- Review diagnostics and, if residuals look patterned, consider transforming variables or trying higher-order or different models.
- Export coefficients and steps for reports, labs, or code integrations.
Core Formulas (LaTeX)
Model: \[ y = a x^2 + b x + c. \]
Matrix form (least squares): \[ \hat{\boldsymbol\beta}=\begin{bmatrix}\hat a\\ \hat b\\ \hat c\end{bmatrix} =(X^\top X)^{-1}X^\top \mathbf{y},\quad X=\begin{bmatrix} x_1^2 & x_1 & 1\\ \vdots & \vdots & \vdots\\ x_n^2 & x_n & 1 \end{bmatrix}. \]
Normal equations (summation form): \[ \begin{bmatrix} \sum x_i^4 & \sum x_i^3 & \sum x_i^2\\ \sum x_i^3 & \sum x_i^2 & \sum x_i\\ \sum x_i^2 & \sum x_i & n \end{bmatrix} \begin{bmatrix} a\\ b\\ c\end{bmatrix} = \begin{bmatrix} \sum x_i^2 y_i\\ \sum x_i y_i\\ \sum y_i \end{bmatrix}. \]
Predictions & residuals: \[ \hat y_i = a x_i^2 + b x_i + c,\qquad r_i = y_i-\hat y_i. \]
SSE, MSE, and \(R^2\): \[ \mathrm{SSE}=\sum r_i^2,\quad \mathrm{MSE}=\frac{\mathrm{SSE}}{n-3},\quad R^2=1-\frac{\sum r_i^2}{\sum (y_i-\bar y)^2}. \]
Vertex of the parabola: \[ x_v=-\frac{b}{2a},\qquad y_v = c - \frac{b^2}{4a}. \]
Examples (Illustrative)
Example 2 — Noisy arc
\((0,1.1),(1,2.9),(2,9.2),(3,18.8),(4,32.1)\). The model yields coefficients near \(a\approx2,\ b\approx-1,\ c\approx1\) with \(R^2\) high; residuals look random—fit is adequate.
Example 3 — Forecasting
With \(a,b,c\) from your dataset, predict \(\hat y(5)=a\cdot25+b\cdot5+c\). Use the vertex to reason about maxima/minima when planning operating points.
FAQs
When should I prefer quadratic over linear regression?
When residuals from a line show curvature or theory suggests accelerating/decelerating change with \(x\).
What does the vertex tell me?
It’s the fitted maximum/minimum. If \(a<0\) it’s a peak; if \(a>0\) it’s a trough.
Is a high \(R^2\) always good?
It indicates explained variance, but check residuals and beware of overfitting or extrapolation beyond observed \(x\).
How many points do I need?
At least three noncollinear points; more data improves stability and allows error estimation (\(\mathrm{MSE}\)).
Why do my coefficients look unstable?
Large \(x\) values cause collinearity between \(x\) and \(x^2\). Try centering/scaling \(x\).
Can I constrain the vertex or force \(c=0\)?
Yes—use transformed variables or constrained fitting; the calculator can expose these options if needed.
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