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Polynomial Regression: When to Use Degree 2 (Quadratic) vs. Degree 3 (Cubic) with Real-World Examples

Polynomial regression is a powerful machine learning technique used to model non-linear relationships between variables. Unlike linear regression, which assumes a straight-line relationship, polynomial regression fits a curved trend line to the data.

In this tutorial, we will explore when to use quadratic regression (degree 2) and cubic regression (degree 3) with real-world examples, datasets, and graphical representations.

Polynomial regression is useful when the relationship between variables is non-linear. The choice of degree depends on how complex the curve needs to be.

How do you choose the degree of a polynomial regression?

Use Degree 2 (Quadratic) When:

• The data follows a U-shape or inverted U-shape (parabolic trend).
• The rate of change increases or decreases but doesn’t have multiple turning points.
• The relationship is simple but non-linear.

Use Degree 3 (Cubic) When:

• The data has multiple turning points (S-shaped or wave-like pattern)
• The relationship changes direction more than once.
• The pattern is more complex and cannot be captured by a quadratic function.

Example 1: Car Speed vs. Braking Distance (Quadratic- Degree 2) 

Why Degree 2?
Braking distance increases at a non-constant rate as speed increases. The relationship follows a parabolic trend.

Speed (km/h)	Braking Distance (m)
10	2
20	6
30	12
40	20
50	30
60	42

Quadratic Regression Formula:

Distance=a⋅(Speed)²+b⋅(Speed)+c

There are no multiple turning points (humps). A quadratic curve is sufficient.

Example 2: Age vs. Happiness Level (Cubic- Degree 3)

Why Degree 3?
Happiness varies at different life stages:

1. Young Age: People are happy.
2. Middle Age: Happiness drops due to responsibilities.
3. Old Age: Happiness increases again after retirement.

Age (Years)	Happiness Score (0-100)
10	80
25	70
40	50
60	65
80	85

 

Cubic Regression Formula:

Happiness=a⋅(Age)³+b⋅(Age)²+c⋅(Age)+d

A quadratic equation would miss the second turning point (increase in happiness in old age). A cubic model is better.