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Mathematics (09) Practice TestFIRST SAMPLE WEAK RESPONSE FOR OPEN-RESPONSEITEM ASSIGNMENT #1The formula for the annual profit is E(n) = 50(10) n . This works for all years thus far. This modelwould predict that in the fourth year there would be an annual profit of $500,000. This can beshown by the equation:E(n) = 50(10) 4E(n) = $500,000There are no other formulas that work for this data. By using the method of differences, one cansee that a quadratic formula would be impossible in these circumstances.1 5002 50003 500004 5000005 5000000450045000450000450000405004050004050000This process would go on in this manner for thousands of differences. It is not practical or reallypossible to find a quadratic formula given the behavior of the data, therefore it can only bedetermined through the exponential function, E n= 50(10) n .ANALYSIS FOR FIRST WEAK RESPONSE TO OPEN-RESPONSEITEM ASSIGNMENT #1This is an example of a weak response because it is characterized by the following:Purpose: The purpose of the assignment is partially achieved. The candidate derives one of the functions andincorrectly assumes that the second model is impossible. The candidate then correctly uses the first model topredict the fourth year. Because the candidate does not use the second model, it is impossible to calculate thepercent error as requested in the last part of the problem.Subject Matter Knowledge: The statement, "There are no other formulas that work for this data," shows limitedunderstanding of quadratics and mathematical modeling. The candidate is missing the fundamental concept thatthree distinct nonlinear points can always be uniquely fit with a quadratic function. The candidate's use of seconddifferences is not relevant to solving this problem because the problem only specifies three data points. Thecandidate extrapolates the data using the first model and then attempts to fit the results to the second model.This shows a lack of understanding of the distinction between observed data and data predicted by mathematicalmodeling.Support: Although the response lacks support for the derivation of the exponential function, it shows thecalculation of the 4th year's profit using that function. The candidate fails to explain that the percent error couldnot be calculated without the quadratic model. This response is not made stronger by the verbal explanationbecause it is clear that the candidate has a limited understanding of the content.Rationale: The candidate reaches the conclusion that a quadratic model is not possible by assuming anexponential pattern from the first three data points and inappropriately applying a difference method to determinethat a quadratic pattern is impossible. In this way, the candidate shows a limited understanding of themathematical modeling process in general, and of quadratic functions in particular. The candidate also fails todemonstrate the reasoning behind the solution to the first part of the problem and the decision not to act on the lastpart of the problem.83