Similar to functions, quadratic regression is a way to model a relationship between two sets of independent variables. Quadratic regression is the process of determining the equation of a parabola that best fits a set of data. This set of data is a given set of graph points that make up the shape of a parabola. The equation of the parabola is y = ax2 + bx + c, where a can never equal zero.

The graphs of quadratic functions have a nonlinear “U”-shape with exponential curves on either side of a single intercepting y-value. We’ll show you how to use this equation.

## Applying the Quadratic Regression Equation

The best way to determine the equation of a parabola without a quadratic regression calculator is to use the least-squares method. Using a given set of data, you need to determine the values of a, b, and c so that the squared vertical distance between each given (x, y) point and the equation of the parabola, otherwise known as the quadratic curve, is minimal. This distance must be minimal to assure that you’ve most accurately determined the parabola’s equation.

For this process, you must follow the following steps:

### Step 1

Make a table with all your x and y values. When you plug these values into a graphing calculator they should form a parabola:

### Step 2

Create 5 additional columns for and calculate. You’ll want to use Microsoft Excel or a calculator for this step:

### Step 3

At the bottom of each column, calculate the sums:

### Step 4

Below is the matrix equation for determining the parabolic curve. ∑ represents the summation, meaning that you will plug the relevant sum into the equation. For example, ∑xi^4 would be the sum of column x^4, 9,669. Using the matrix equation, fill in all the sums:

### Step 5

Solve for *a, b,* and *c* by isolating each of these variables using an online calculator. Your result should be the following:

*a* = -0.3660714

*b* = 3.015714

*c* = 30.42179

### Step 6

Insert these values (rounding to the 3rd decimal point) into our quadratic equation:

## Quadratic Regression Tools

Quadratic Regression is a tough method to tackle by hand. Luckily there are plenty of websites that provide online calculators that make solving the quadratic regression model much easier. However, if that option is not available, follow the steps above.

While the tables and equations above may seem intimidating, with a little practice, you'll be a pro at finding quadratic regression in no time.