Quadratic curve and straight-line alternative over the same observationsUpdates with every input
Math & Statistics
Regression Graphing Calculator
Fit and graph a quadratic least-squares model through six entered points, display the linear alternative, residual stems, turning point, fit improvement, and model behavior across the observed x domain.
Quadratic equation -
Quadratic coefficient -
Linear coefficient -
Intercept -
Quadratic R-squared -
Linear R-squared -
Turning point -
Evaluated fitted y -
CURVATURE COMPARISON
Quadratic curve and straight-line alternative over the same observations
The chart draws both models, observation residuals to the quadratic curve, and the turning point only when it lies in the observed domain.
MODEL COMPARISON LEDGER
Point-level quadratic and linear residuals
The same observations are scored against both models so improvement is attributable to curvature rather than to a changed sample.
| Point | x | Observed y | Quadratic fit | Quad residual | Linear fit | Linear residual |
|---|
CURVE SETUP
Use curvature because the pattern supports it, not because R-squared must rise
- Enter six distinct or sufficiently varied x values.
- Plot observations before comparing models.
- Keep the linear model as a visible benchmark.
- Inspect whether the turning point lies within observed support.
- Treat extrapolated quadratic growth or decline with exceptional caution.
MODEL COMPLEXITY
Adding x squared cannot increase training SSE, but it can overfit
The quadratic model contains the linear model as a special case, so in-sample fit can only stay equal or improve. That fact alone is not evidence that curvature is stable.
Residual shape, domain knowledge, repeated data, and out-of-sample performance matter more than a small R-squared increase.
QUADRATIC OLS
Solve three normal equations for a, b, and c
The model treats x squared as a second predictor and solves the normal-equation system. Linear fit is computed separately as a benchmark.
Detailed calculation process and general formulas
yhat = a x^2 + b x + c[sum x4, sum x3, sum x2; sum x3, sum x2, sum x; sum x2, sum x, n][a,b,c]' = [sum x2y, sum xy, sum y]'SSE_q = sum(y-yhat_q)^2R2_q = 1-SSE_q/SSTx_vertex = -b/(2a)Symbols, meanings, and units
- a
- quadratic curvature coefficienty per x squared
- b
- linear coefficienty per x
- c
- intercepty units
- SSE_q
- quadratic residual sum of squaresy units squared
- x_vertex
- quadratic turning-point x coordinatex units
CURVATURE EVIDENCE
Four questions for a quadratic graph
A bend must be interpretable, supported, and located within the data range.
Fit gain
-Compare quadratic and linear R-squared on the same points.
Turning point
-A vertex outside the observed range is not an observed reversal.
Residual pattern
Curvature should remove a systematic bend rather than chase one point.
Boundary behavior
Quadratic extrapolation grows rapidly and should be constrained by subject knowledge.
Decision takeaway: Prefer the simpler line unless curvature is visible, explainable, and stable across the observed domain.
Applied decisions
Quadratic graph decisions
Process optimum
Yield improves and then declines across a controlled setting range.
What the result clarifies: An in-range turning point can summarize the observed optimum.
Learning curve fragment
Six early observations bend but later behavior is unknown.
What the result clarifies: The quadratic graph describes the window and should not be extended indefinitely.
Worked current scenario
Substitution, intermediate values, and reconciliation
Method references
Sources for this calculator's specific method
Scope and limitations
This is an unweighted six-point quadratic least-squares model. Polynomial fits can be unstable with clustered x values, outliers, extrapolation, or high collinearity between x and x squared. A higher in-sample R-squared does not guarantee better prediction.
Regression Graphing Calculator | Quadratic Least-Squares Curve FAQ
Why must quadratic R-squared be at least linear R-squared?
The quadratic model includes all terms in the linear model plus one additional term.
Is the vertex always meaningful?
No. It is most defensible when inside the observed x range and supported by the point pattern.
Can I infer causation from the curve?
No. The graph summarizes association in the entered observations.