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Quadratic regression is a method to model a relationship between the two sets of the independent variables. It is referred as the process of determining the equation of the parabola, which best fits the data set. This data set is a given set of the graph points, which make up a parabola shape. The equation of parabola is y= ax2+bx+c.

y=ax2+bx+cy=ax2+bx+c where a≠0a≠0.

The quadratic function’s graph has a non-linear U shape with the exponential curves on either side of the single interception y value.

** Figure 1**: Quadratic Regression

The best way of determining a parabola equation without a calculator of quadratic regression is using the least-square method. By using the given data set, one requires to find out the values of a, b, and c, so that there is a minimal squared vertical distance between each of the given points i.e., x and y, and the parabola equation that is also referred as the quadratic curve. The distance should be minimal to make sure that one has a precisely determined equation of the parabola. Hence, for this process, one needs to follow the below steps:

The first step involves making a table with all the values of x and y. After these values are plugged into a graphing calculator, they must form a parabola.

The second step involves creating five additional columns for the calculation of the quadratic regression- x, XY, and y values. For this step, a calculator or Microsoft excel needs to be used.

The sums must be calculated at the bottom of each column.

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The below matrix equation is used to find out the parabolic curve. ∑ signifies the summation that implies plugging the relevant sum into the equation. For instance, ∑xi^4 would be the total sum of the column x^4, 9,66. All the sums need to be filled by using the matrix equation.

The a, b, and c need to be solved by isolating each of the variables by using the online calculator. The outcome must be the following:

a= -0.3660714

b= 3.015714

c- 30.42179

These values should be inserted into the quadratic equation, and rounded off to the third decimal point.

There are some situations where data does not follow a straight line, but rather forms a parabola that seems to be just right. Hence, a quadratic equation is a kind of multiple linear equation through which a parabola equation of best fit is determined for a data set. It is quite similar to the linear equation, where one looks for a straight line to the cubic regression, where one deals with the curves of degree three, or to the exponential regression, where one fits exponential curves to the data.

In the quadratic regression, R-Squared is referred to as the coefficient of the determination and it shows the extent to which variation in y can be described by the x variables. Hence, the r-squared value allows assessing the way the differences in one variable can be described by a difference in the second variable. The R-squared can take the values between 0- 1, which shows 0 percent variation and 1 shows a 100 percent variation.

Quadratic regression model is the way of modeling a relationship between the two sets of variables.

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**Figure 2**: Quadratic Regression

The man aim of the quadratic regression is finding the equation in the form- y = a + bx + cx², which best fits the data points (x1, y1), ..., (xn, yn). In the case of c = 0, the quadratic regression model boils down to the simpler linear regression. For computing the coefficients of the quadratic regression equation, the least squares methods will be used, in which the a, b, and c values need to be determined, so that the squared distance between each point of data (xi, yi), and the point (xi,a + bxi + cxi²) is determined by the equation of parabola- y = a + bx + cx² is minimal.

It is somewhat tougher to calculate the quadratic regression by hand than finding out the linear model by oneself. It is the reason why the quadratic regression calculator is made to perform all the tedious work in just a second. However, still, if one requires to do this by hand and are not happy with the formula; there is an alternative way. The coefficients a, b and c satisfy the system of 3 linear equations, which are as follows:

**or in the matrix form:**

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Hence, quadratic regression can be calculated by computing the sum of x1, ..., xn, and then making the sums of the cubes, squares, and fourth powers; and then solve the systems of 3 linear equations.

Ans. The steps involved in performing quadratic regression on a calculator include pressing STAT and pressing ENTER button; entering x-variables; scrolling across to L2 by using scroll keys; entering y-variables; pressing the STAT button; and arrowing right to the calculator and then pressing down arrow to QuadReg and press enter; type the parameters L1, L2, Y1; and press enter to calculate regression.

Ans. The coefficients of the quadratic regression equation can be computed by using the least-squares method. The values of a, b. and c need to be determined so that the squared distance between each point of data and the point determined by the equation of a parabola is minimal.

Ans. The regression equation is determined by the equation of a parabola that best fits the data set. The equation of a parabola is y= ax2+bx+c, where a can never be equal to zero.

**Q4. **How do you find a quadratic equation from data?

Ans. Firstly, understand the values in the table, find the slope, determine the point where the line crosses the vertical axis, and check the work.

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