Modeling Real-Life Data Lake Pamela is the large lake that you see on Valencia West Campus as you enter the school from Kirkman Road. From about 1996 to 2000, vegetation was taken out around the lakeshore to make it look more cosmetically pleasing. The water quality quickly declined due to increased nutrient flow into the lake, making the lake unhealthy. Around 2000, the vegetation removal and cutting of the grass down to the shoreline stopped, and more trees were planted. Lake Pamela's TSI (trophic state index) data is given in the table below (the maximum value for each year is used). The TSI of a lake measures the biological condition of the lake, or how healthy it is. Generally speaking, the lower the TSI, the healthier a lake is. (http://www.hillsborough.wateratlas.usf.edu/shared/learnmore.asp?toolsection=lm_tsi) TSI (trophic state index) of Lake Pamela Plot the data on the graph below 1) Is TSI (y) a function of years (x)? How do you know this? 2) While the data does not form a single straight line, the shape of the data suggests a linear pattern. Pick any two data points so that when a straight line is drawn through those two points, it is a line that best fits the data. Draw your line of best-fit through your two chosen data points on the graph. Then, state your two chosen data points, as ordered pairs, below. (Hint: Your two points may be found through trial and error. Draw a straight line connecting any two data points and see if the line is a good approximation for the data; if not, try two other data points. They do NOT have to be consecutive data points.) Years since 2002, x TSI, y=f(x) 0 45.2 2 47.1 4 41 6 44.6 8 42.5 10 37.6 0 10 20 30 40 50 60 70 80 0 2 4 6 8 10 12 TSI Years since 2002 TSI of Lake Pamela 3) What is the slope of the line containing the two points you chose in #2? Round to the nearest hundredth if necessary. 4) The TSI of Lake Pamela is decreasing over this time period. Does your slope reflect this? Why? 5) Find a linear equation for the line that passes through the two points you chose in #2, using the slope you found in #3. Round to the nearest hundredth if necessary. Express the answer as a linear function, f(x). 6) The function you found in #5 is the mathematical model you found for the set of data. Using this function, find the value of f(12). Write your answer as a sentence, including what calendar year x = 12 represents. Round your answer to the nearest tenth. 7) Plot the data point you found in #6 on the graph on the first page of this project. Does this point fall on the line you drew as the line of best fit? If your point does not fall on the line you drew, figure out why not. 8) Using your mathematical model (the function found in #5), in what year would you expect the TSI of Lake Pamela to be 34? Round your answer to the nearest year. 9) You can use a calculator to find the (precise and accurate) equation for the line of best-fit or mathematical model of a set of data. The actual mathematical model for this set of data is: f(x) = -0.69x + 46.44 Refer to the graph below for how this function approximates the data points given. Rewrite the model (function) you found in #4 below. Now, compare (or contrast) the slope and the yintercept for the model above with the model you found