Question

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.

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

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.)

Answer 1

(1) Mathematically speaking, for every value of x, there is a single value of y. No value of x has two different values of y. Thus y is a function of x. Practically it is obvious that time will deteriorate the lake, which means y is a function of x.

(2)

x	y
0	45.2
2	47.1
4	41
6	44.6
8	42.5
10	37.6

The two points that I used are (0, 45.2) and (10, 37.6)