1. When finding a line through two points, we typically find the slope and then use the point-slope form through one of the points. In this problem, we show that we can also use the center of the the points (defined by the means) as the point as well. (a) Find the line through the point (2, 1) and (5,3). (b) Find the average (mean) of the coordinates and y-coordinates of the points in #1. Write the means a the point (8,9). Use the point-slope form of a line to find the equation of the line through (4,0) and same slope in part (a). Show algebraically that this is the same lines in part (a). 2. This problem goes through a simple example of finding the best-fit line through the following set of points (1,2), (5,3), (6,5) and (8,6). Also, in this problem, keep all numbers as fractions, do not we decimals (via a calculator) (a) Plot the points in Desmos, plot the line y-me+b and give yourself a slider for mand. Use the slider to try to find a good fit to the data. Print out the data, line and make sure you show the sliders or clearly show what your values of m and b are. (b) Find the mean of both the r- and y-coordinates of the points above. (c) Let S = (1 -m(41 - 2) - 2 + (ya - mr, - ) - 92 + (8's - m(x - 1) - 2 + (y - m(x4-1)-9) Plug in the values of wand and simplify. You should have a function that only depends on m. (d) Plot the function Sim) in Desmos and estimate the minimum value from the graph. What value of m results in that minimum? (e) Solve S'(m) - 0 for m. Keep mas a fraction, not a decimal. (1) Find S" (m) and explain using the second derivative test why the value found in (f) is a minimum. (g) Write down the line y = m( - ) + using the slope in part (e) and values of Pandy found in part (b). Write the function in slope intercept form. Plot your result and the data. Make sure that your screen includes both the plot as well as the calculations that you used to find the slope and y-intercept. (h) Compare your results in (g) with that in part (a). Comment on the values of m and 6 for each. Also comment on your results in part (d) for the slope. Overall, explain which m results in the best-fit-line