What quadratic regression does
Quadratic regression finds the parabola y = ax² + bx + c that best fits a set of (x, y) data points. "Best fit" has a precise meaning: the calculator chooses the three coefficients a, b, and c so that the sum of the squared vertical distances between each data point and the curve — the residuals — is as small as possible. This is the ordinary least-squares criterion, the same principle behind linear regression, extended to a curve that can bend once.
You reach for a quadratic model instead of a straight line when the data rises then falls (or falls then rises), or when its rate of change speeds up or slows down across the x-range. Projectile height over time, stopping distance versus speed, and revenue versus price are all situations where a single bend describes the relationship far better than a line.
The formula and how it is solved
To minimize the squared error, the least-squares method sets up three "normal equations" — one for each unknown — using sums computed from your data. Writing n for the number of points, they are:
- a·Σx⁴ + b·Σx³ + c·Σx² = Σx²y
- a·Σx³ + b·Σx² + c·Σx = Σxy
- a·Σx² + b·Σx + c·n = Σy
This calculator computes those seven sums (Σx, Σx², Σx³, Σx⁴, Σy, Σxy, Σx²y) from your inputs, then solves the resulting 3×3 system by Gaussian elimination with partial pivoting to return exact a, b, and c.
Reading the coefficients
- a controls the curvature. If a > 0 the parabola opens upward (a U shape with a minimum); if a < 0 it opens downward (an ∩ shape with a maximum). The larger |a| is, the tighter and steeper the curve.
- b is the linear term; together with a it locates the turning point.
- c is the y-intercept — the model's predicted value when x = 0.
- The vertex (turning point) sits at x = −b / (2a), the input where the fitted quantity is minimized or maximized.
What R² tells you
R² (the coefficient of determination) measures how much of the variation in y the parabola explains, on a scale from 0 to 1. It is defined as R² = 1 − (SSres / SStot), where SSres is the sum of squared residuals and SStot is the total squared variation of y about its mean. An R² of 1.0 means the curve passes exactly through every point; 0 means the parabola does no better than a flat line at the mean. Values above about 0.9 usually indicate a strong quadratic relationship, but always plot the residuals too — a high R² can still hide a systematic pattern the model misses.
Frequently Asked Questions
What is quadratic regression?
When should I use a linear model instead?
How many data points do I need?
When should I use a linear model instead?
Can I use the fitted curve to predict outside my data range?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |