Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression quadratic for the given points. Then plot the points and graph the least squares regression quadratic. $$ (-4,5),(-2,6),(2,6),(4,2) $$

Step-by-Step Solution

Answer

The answer is a set of coefficients a, b, and c, generated by the least squares regression quadratic. These will form the equation of the parabolic curve that best fits the given points.

Step 1: Prepare the Data

Formulate the given points into an array, where each point is a pair of (x, y). The prepared data should look like this: [(-4,5),(-2,6),(2,6),(4,2)].

Step 2: Implement Least Squares Regression Quadratic

Through a spreadsheet program (like MS Excel) or a graphing utility (like Google Colab or Python program), perform the least squares regression quadratic to this set of data. In case of using python, you can use numpy's polyfit function with degree 2, like 'np.polyfit(x_coordinates, y_coordinates, 2)'. After doing this, the tool will yield a set of 3 coefficients a, b, and c, in the quadratic equation \(ax^2 + bx + c\)

Step 3: Plot the Points

Using the same graphing tool or spreadsheet, do a scatter plot of the points using their x and y coordinates. This will provide the raw data on which the regression model is based.

Step 4: Graph The Quadratic Regression Model

Still within the plotting tool, draw the graph of the quadratic function obtained in Step 2 across the range of your x-coordinates. This will graph the quadratic regression model (curve) that best fits your data points.