Question

1
select the correct answer.
as a result of changing sales figures, a company increased the size of its workforce very quickly at first and then more slowly, as shown in the table

years after opening	1	2	3	4	5	6	7

which type of function best models the data?
a. a quadratic function with a positive value of a
b. a quadratic function with a negative value of a
c. a square root function
d. a linear function with a positive slope

Explanation:

Step1: Calculate first differences

Differences: 348-100=248, 405-348=57, 575-405=170, 654-575=79, 704-654=50, 746-704=42

Step2: Calculate second differences

Second differences: 57-248=-191, 170-57=113, 79-170=-91, 50-79=-29, 42-50=-8

Step3: Analyze trend

The first differences decrease over time, indicating a concave down curve. A quadratic function with negative \(a\) (\(ax^2+bx+c, a<0\)) is concave down, matching the data's slowdown in growth. Linear functions have constant differences, square roots grow slower initially (opposite here), and positive \(a\) quadratics are concave up (growth speeds up). Thus, the best model is a quadratic with negative \(a\).

Answer:

B. a quadratic function with a negative value of a