standard_description: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
problem_activity: Task
Given below is a table that gives the populations of foxes and rabbits in a national park over a 12 month period. Note that each value of $t$ corresponds to the beginning of the month and $t = 0$ corresponds to the beginning of January.
{"table": [{"tbody": [{"tr": [{"th": [{"_value": "$t$, month"}], "td": [{"_value": "0"}, {"_value": "1"}, {"_value": "2"}, {"_value": "3"}, {"_value": "4"}, {"_value": "5"}, {"_value": "6"}, {"_value": "7"}, {"_value": "8"}, {"_value": "9"}, {"_value": "10"}, {"_value": "11"}]}, {"th": [{"_value": "$r$, number of rabbits"}], "td": [{"_value": "1000"}, {"_value": "750"}, {"_value": "567"}, {"_value": "500"}, {"_value": "567"}, {"_value": "750"}, {"_value": "1000"}, {"_value": "1250"}, {"_value": "1433"}, {"_value": "1500"}, {"_value": "1433"}, {"_value": "1250"}]}, {"th": [{"_value": "$f$, number of foxes"}], "td": [{"_value": "150"}, {"_value": "143"}, {"_value": "125"}, {"_value": "100"}, {"_value": "75"}, {"_value": "57"}, {"_value": "50"}, {"_value": "57"}, {"_value": "75"}, {"_value": "100"}, {"_value": "125"}, {"_value": "143"}]}]}]}]}
Note that the number of rabbits and the number of foxes are both functions of time.
Explain why it is appropriate to model the number of rabbits and foxes as trigonometric functions of time.
Find an appropriate trigonometric function that models the number of rabbits, $r(t)$, as a function of time, $t$, in months.
Find an appropriate trigonometric function that models the number of foxes, $f(t)$, as a function of time, $t$, in months.
Graph both functions and give one possible explanation why one function seems to “chase” the other function.


aligns: no