Quadratic Regression
Quadratic regression is a method for finding the best-fit quadratic function (a parabola) that matches a set of data points. It produces an equation in the form that most closely follows the pattern in the data.
Quadratic regression is a form of polynomial regression that models the relationship between an independent variable and a dependent variable using a second-degree polynomial. The method of least squares is used to determine the coefficients , , and in the equation such that the sum of the squared differences between the observed and predicted -values is minimized. This model is appropriate when data follows a curved, parabolic pattern rather than a straight line.
Key Formula
Where:
- = the coefficient of the squared term, which controls the direction and width of the parabola
- = the coefficient of the linear term
- = the constant term (the y-intercept when x = 0)
- = the independent variable
- = the predicted value of the dependent variable
Worked Example
Problem: A ball is launched upward, and its height (in meters) is recorded at different times (in seconds): (0, 2), (1, 18), (2, 26), (3, 26), (4, 18), (5, 2). Find the quadratic regression equation.
Step 1: Enter the data into a graphing calculator or statistics software. Input the x-values (time) into one list and the y-values (height) into another.
Step 2: Run the quadratic regression function (often labeled QuadReg). The calculator uses the least squares method to find the values of a, b, and c.
Step 3: Read the output. For this data, the calculator returns:
Step 4: Write the regression equation by substituting the coefficients.
Answer: The quadratic regression equation is . You can verify: at , the model gives , which matches the data exactly.
Visualization
Why It Matters
Many real-world situations produce curved data that a straight line can't capture — projectile motion in physics, profit optimization in business, and the trajectory of a kicked soccer ball are all parabolic. Quadratic regression gives you a precise equation to model these patterns, make predictions, and find key values like the maximum height or break-even point.
Common Mistakes
Mistake: Using linear regression when the data is clearly curved
Correction: Always plot your data first. If the scatterplot shows a U-shape or an arch, a quadratic model will fit much better than a line. Check the residuals or the value to compare fits.
Mistake: Extrapolating far beyond the range of the data
Correction: A quadratic model may fit well within your data range but give unrealistic predictions outside it. For example, a height model might predict negative heights for large time values. Use caution when predicting beyond your data.