# Polynomial Regression

- Author:
- Walter M. Stroup

[1] Adjust slope and intercept to pass through A. Note, different combinations of slope and intercept will do this.
[2] You can add more points by increasing value of number of points slider. With d=2 adjust slope and intercept to pass through the two points shown. Are there still different combinations that will work? With two points the values of slope (1) and intercept (2) are determined.
[3] Click the regression box. Compare what is shown for f(x) to the values from your sliders. Notice values for R^2, Corr(elation), and Corr(elation)^2.
[4] Now turn off the regression box. Add more points. Now adjust the slope and intercept to estimate a best fit line.
[5] Click on regression box. Again, compare what is shown for f(x) to your slider values. Notice what happens to R^2, Corr, and Corr^2.
[6] What happens if you now you increase the "a" (related to degree of polynomial). Try adding (or subtracting points). What happens to R^2, Corr, and Corr^2. What does it mean when they are 1. When does undefined occur?
[7] With the number of points, n, greater than a, try moving points to increase or decrease R^2. What does R^2 seem to represent in relation to the locations of the points relative to the graph?
[8] With a=2 and n>2 what relationship is there between R^2, Corr, and Corr^2? Does this relationship hold when d is greater than 2 (you may need more points)?
[9] What relationship is there between the number of points and a when R^2=1. If you move the points when R^2 = 1 what happens?
[10] Things to try: Corr > R^2 ?, Corr^2 = R^2 ?, Corr^2 < R^2 ?, Corr negative and R^2 positive ?, R^2 < Corr^2 ?, Corr near zero and R^2 greater than 0.7 ?
[11] Move points to get close to a cubic (a=3 and d=6). Now decrease a to 2. Record R^2. Increase a to 3, record R^2 and so until a = 4. When does R^2 increase the most? Least?
Other links to try out:
Fit 1 Point https://www.desmos.com/calculator/q1nr73sg9n
Fit 2 Points https://www.desmos.com/calculator/77delynney
Fit 3 Point https://www.desmos.com/calculator/sz5bwpwiat
regression intro https://www.desmos.com/calculator/0yhiftu00g
poly regression 01 https://www.desmos.com/calculator/alkezr3jri
poly regression all no 2nd https://www.desmos.com/calculator/rl9m3boryn
poly regression all https://www.desmos.com/calculator/xisxwf2qql