given the points graphed in the following figure, use quadratic regression to find the quadratic function of…

1. General form of a quadratic function:
   • The general form of a quadratic function is (y = ax^{2}+bx + c). For quadratic - regression, if we have (n) data points ((x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)), we want to minimize the sum of the squared residuals (S=\sum_{i = 1}^{n}(y_i-(ax_i^{2}+bx_i + c))^{2}).
   • To find the coefficients (a), (b), and (c), we can use matrices or a statistical software/calculator. Here, we'll assume we are using a graphing - calculator (like TI - 84 Plus) or a software like Excel or Python.
   • First, we need to identify the data points from the graph. Let's assume the data points are ((x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)).
   • For a TI - 84 Plus calculator:
     • Step 1: Enter the data
       • Press STAT and then 1:Edit.... Enter the (x) - values in the (L_1) list and the corresponding (y) - values in the (L_2) list.
     • Step 2: Perform quadratic regression
       • Press STAT, then arrow over to CALC, and select 5:QuadReg. Press ENTER. Then type L_1,L_2,Y_1 (where (Y_1) is a function variable in the calculator). Press ENTER.
       • The calculator will output the values of (a), (b), and (c) in the form (y = ax^{2}+bx + c).
   • In Python, we can use the numpy and scipy.stats libraries.
     • Step 1: Import the necessary libraries

       import numpy as np
       from scipy.optimize import curve_fit
       

     • Step 2: Define the quadratic function and the data points

       def quadratic(x, a, b, c):
           return a * x ** 2 + b * x + c
       # Assume x and y are numpy arrays of the x - and y - values of the data points
       x = np.array([x1, x2, ..., xn])
       y = np.array([y1, y2, ..., yn])
       

     • Step 3: Perform the regression

       popt, pcov = curve_fit(quadratic, x, y)
       a, b, c = popt
       

2. Let's assume we got the following values from the regression (example values):
   • Suppose the calculator or software gives us (a = 0.234), (b=-1.234), and (c = 3.456).
   • The quadratic function of best - fit is (y=0.234x^{2}-1.234x + 3.456).

Since we don't have the actual data points from the graph, we can't give the exact coefficients. But the general process is as above. If we assume we have performed the regression and got coefficients (a), (b), and (c):

Answer:

(y = ax^{2}+bx + c) (where (a), (b), and (c) are the values obtained from the regression process)