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Quadratic Regression — Parabola Fitting

Fit y = ax² + bx + c. Vertex, R², linear comparison, residuals, F-test.

Concept Fundamentals
y = ax² + bx + c
Quadratic Model
Polynomial regression
Minimize Σ(yᵢ−ŷᵢ)²
Least Squares
Best fit criterion
1 − SS_res/SS_tot
R² Value
Goodness of fit
Curved relationships
Application
Non-linear trends
Fit Quadraticy = ax² + bx + c

Why This Statistical Analysis Matters

Why: Quadratic models capture curvature. Used for projectiles, revenue, dose-response.

How: Enter (x,y) pairs. Get coefficients, vertex, R², F-test vs linear.

  • ●x_v = -b/(2a)
  • ●R² = 1 - SS_res/SS_tot
  • ●F-test: quad vs linear
Sources:NIST Regression HandbookWolfram MathWorld
y=x²
REGRESSION2026

Quadratic Regression — Parabola Fitting

Fit y = ax² + bx + c to your data. Get vertex, axis of symmetry, R², comparison with linear, residuals, and F-test.

Linear Regression →Cubic Regression →

Real-World Scenarios — Click to Load

xy
quadratic_regression_results.sh
CALCULATED
$ quadratic_regression --least-squares --gauss
a (x²)
1.0000
b (x)
0.0000
c (intercept)
1.0000
R²
1.0000
Vertex
(0.000, 1.000)
Axis of symmetry
x = 0.000
Adjusted R²
1.0000
F (quad vs linear)
0.0000
Share:
Quadratic Regression Result
y = 1.000x² + 0.000x + 1.000
R² = 1.0000Vertex: (0.00, 1.00)Axis: x = 0.00
numbervibe.com/calculators/statistics/quadratic-regression-calculator

Calculation Breakdown

GAUSS ELIMINATION
Normal equation matrix A
[Σ1, Σx, Σx²; Σx, Σx², Σx³; Σx², Σx³, Σx⁴]
Right-hand side b
[Σy, Σxy, Σx²y]
n=6 points
Solve via Gaussian elimination
β = A⁻¹b
ext{Partial} ext{pivoting}
Coefficient a (x²)
1.000000
y = ext{ax}^{2} + ext{bx} + c
Coefficient b (x)
0.000000
y = ext{ax}^{2} + ext{bx} + c
Coefficient c (intercept)
1.000000
y = ext{ax}^{2} + ext{bx} + c
GOODNESS OF FIT
R² (quadratic)
1.0000
1 - SS_res/SS_tot
Adjusted R²
1.0000
1 - (1-R^{2}) rac{n-1}{n-3}
Standard Error
0.0000
√(SS_res/(n-3))
Vertex & Axis
Vertex x
0.0000
-b/(2a)
Vertex y
1.0000
c - b^{2}/(4a)
Axis of symmetry
x = 0.0000
F-TEST
F (quad vs linear)
0.0000
(SS_res_lin - SS_res_quad)/(SS_res_quad/(n-3))

Scatter with Parabola & Linear Fit

Residuals Plot

For educational and informational purposes only. Verify with a qualified professional.

📈 Statistical Insights

y=ax²

Parabola

— Model

x_v

-b/(2a)

— Vertex

R²

Variance

— Fit

Key Takeaways

  • • Model: y = ax² + bx + c. Vandermonde X: [1, x, x²]. Normal equations: (X'X)β = X'y. Solved via Gaussian elimination.
  • • Vertex: x_v = −b/(2a), y_v = c − b²/(4a). Axis of symmetry: x = x_v.
  • • Opens: Up if a > 0, down if a < 0. Discriminant Δ = b² − 4ac.
  • • Roots: x = (−b ± √Δ)/(2a). Real roots if Δ ≥ 0.
  • • R²: 1 − SS_res/SS_tot. Adjusted R² = 1 − (1−R²)(n−1)/(n−3) for 3 parameters.
  • • F-test: F = ((SS_res_linear − SS_res_quad)/1) / (SS_res_quad/(n−3)). Tests if quadratic improves fit.

Did You Know?

📈Projectile motion follows a parabola: height = −½gt² + v₀t + h₀. Quadratic regression fits this perfectly.Source: Physics
💰Revenue vs price often has an inverted U: low price = low revenue, high price = fewer sales. Optimal price at the vertex.Source: Economics
🌡️Daily temperature over 24 hours often follows a quadratic curve: cool at night, warm at midday.Source: Meteorology
🧪Dose-response curves can be inverted U: low dose = low effect, optimal dose = peak, high dose = toxicity.Source: Pharmacology
📊R² always increases with more parameters. Use adjusted R² and F-test to compare quadratic vs linear.Source: Model Selection
⛽Fuel efficiency vs speed is typically quadratic: optimal at moderate speed; too slow or too fast reduces mpg.Source: Automotive

How Quadratic Regression Works

1. Normal equations

3×3 system: [Σ1, Σx, Σx²; Σx, Σx², Σx³; Σx², Σx³, Σx⁴] β = [Σy, Σxy, Σx²y]. Solve via Gaussian elimination.

2. Vertex and axis

Vertex at x = −b/(2a). Parabola opens up if a &gt; 0, down if a &lt; 0. Axis of symmetry through vertex.

3. Roots

Quadratic formula: x = (−b ± √(b²−4ac))/(2a). Real roots when discriminant ≥ 0.

4. R² and SE

R² = 1 − SS_res/SS_tot. SE = √(SS_res/(n−3)). Compare with linear R² to see improvement.

5. F-test

F = (SS_res_linear − SS_res_quad) / (SS_res_quad/(n−3)). Tests H₀: quadratic term adds nothing. Large F = quadratic helps.

Expert Tips

Minimum sample size

Use at least 4 points for quadratic. More points reduce overfitting.

Check vertex

Vertex marks min or max. For revenue curves, vertex gives optimal price.

Compare with linear

If R²_quad only slightly higher than R²_linear, linear may suffice.

Extrapolation

Parabolas diverge outside data range. Avoid extrapolation.

Formulas Reference

y = ax² + bx + c

Vertex: x_v = −b/(2a), y_v = c − b²/(4a)

Discriminant: Δ = b² − 4ac

Roots: x = (−b ± √Δ)/(2a)

R² = 1 − SS_res/SS_tot

Adjusted R² = 1 − (1−R²)(n−1)/(n−3)

SE = √(SS_res/(n−3))

F = ((SS_res_lin − SS_res_quad)/1) / (SS_res_quad/(n−3))

Frequently Asked Questions

When should I use quadratic vs linear regression?

Use quadratic when the relationship shows curvature (parabola, one hump or valley). Linear for roughly straight relationships.

What does a negative a (x² coefficient) mean?

Parabola opens downward. Vertex is a maximum. Common in revenue vs price, efficiency vs speed.

How do I interpret the vertex?

Vertex is the min (if opens up) or max (if opens down). For revenue curves, vertex gives optimal price.

What is the F-test for?

Tests whether the quadratic term significantly improves fit over linear. Large F and small p-value mean quadratic is justified.

Can I use this for extrapolation?

Cautiously. Parabolas can diverge quickly outside the data range. Prefer interpolation.

Quadratic Regression by the Numbers

3
Parameters
n≥4
Min points
1
Vertex
0–2
Real roots

Applications

ApplicationInterpretation
Projectile motionHeight vs time: parabola. Vertex = max height.
Revenue vs priceInverted U. Vertex = optimal price.
Temperature vs yieldOptimal temp at vertex.
Dose-responseInverted U. Vertex = optimal dose.
Age vs performancePeak performance at vertex.
Altitude vs pressurePressure decay with altitude.
Speed vs fuelOptimal speed at vertex.

Disclaimer: Quadratic regression can overfit with few points. Use adjusted R² and F-test to assess whether quadratic improves over linear. Avoid extrapolation.

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