A new hospital tracked the number of births during its first 6 months of operation, as shown in the table. Find a quadratic function that models the data. Round numerical values to the nearest whole number. f(x) = x2 + x + Use the function to predict the number of births for month 8. There will be about births..

Let    x refers to the Month numbers
And  y refers to Number of births

There is 5 steps to find the quadratic function that models the data.

First step           ⇒ Plot the data pairs (x, y) in a coordinate plane.
                              which are the blue circles in the attached figure

Second step     ⇒ Draw the parabola you think best fits the data.
                             Which is the red function in the attached figure.

Third step         ⇒ Estimate the coordinates of three points on the
                             parabola, which are (2, 108), (4, 68), and (6, 75).
                             which are the black squares in the attached figure

Fourth step       ⇒ Substitute the coordinates of the points into the model
                            y = ax2 + bx + c
                            to obtain a system of three linear equations.

                            4a + 2 b + c = 108   ---------(1)
                            16a + 4 b + c = 68   ---------(2)
                            36a + 6 b + c = 75   ---------(3)   

Fifth step   ⇒ Solve the linear system.
                      By using the calculator
                      a = 5.875 , b = -55.25 , c = 195

So, the quadratic model for the data is ⇒ y = 5.875 x²- 55.25 x + 195
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To predict the number of births for month 8
substitute with x = 8 ⇒  y = 5.875 * 8²- 55.25 * 8 + 195 = 129