Question

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 linear function with a positive slope
b. a quadratic function with a positive value of a
c. a square root function
d. a quadratic function with a negative value of a

Explanation:

Step1: Analyze Linear Function

A linear function has a constant rate of change (slope). Calculate the differences between consecutive number of employees:
- From year 1 to 2: \( 348 - 100 = 248 \)
- From year 2 to 3: \( 405 - 348 = 57 \)
- From year 3 to 4: \( 575 - 405 = 170 \)
- From year 4 to 5: \( 654 - 575 = 79 \)
- From year 5 to 6: \( 704 - 654 = 50 \)
- From year 6 to 7: \( 746 - 704 = 42 \)
The differences are not constant, so not a linear function (eliminate A).

Step2: Analyze Square Root Function

A square root function (\( y = \sqrt{x} \)) increases very slowly and the rate of increase decreases, but the initial jump (from year 1 to 2: 248) is too large for a square root function. Also, square root of 1 is 1, 2 is ~1.41, etc., which doesn't match the data (eliminate C).

Step3: Analyze Quadratic Function

A quadratic function \( y = ax^2 + bx + c \) has a parabola shape. If \( a > 0 \), it opens upwards (concave up), meaning the rate of increase first increases then decreases? Wait, no—wait, the data here: the increase is fast at first (year 1-2: 248), then slower (year 2-3: 57, then 170? Wait, no, year 3-4 is 170, which is a big jump. Wait, maybe I miscalculated. Wait, year 1:100, year 2:348 (diff 248), year3:405 (diff 57), year4:575 (diff 170), year5:654 (diff 79), year6:704 (diff 50), year7:746 (diff 42). Wait, maybe the data is increasing, and the second differences? Wait, no, maybe the key is the shape. Wait, the problem says "increased very quickly at first and then more slowly"—so the rate of increase (slope) is decreasing. Wait, but a quadratic with \( a < 0 \) opens downward (concave down), meaning the slope (rate of change) decreases after a peak. Wait, no—wait, let's think about the graph of quadratic functions. If \( a > 0 \), the parabola opens up, so the function is concave up, meaning the rate of increase (slope) is increasing. If \( a < 0 \), concave down, slope decreasing. Wait, but the data here: the company increased workforce quickly at first, then more slowly—so the slope (rate of change) is decreasing. So that would be a concave down parabola, which is a quadratic with \( a < 0 \)? Wait, no, wait—wait, maybe I got it reversed. Wait, let's take an example: \( y = -x^2 + 10x \). The derivative (slope) is \( -2x + 10 \), which decreases as x increases (since coefficient of x is -2). So the slope decreases, meaning the function increases quickly at first (when x is small, slope is large) then more slowly (as x increases, slope decreases). That matches the problem statement: "increased very quickly at first and then more slowly". So that's a quadratic with \( a < 0 \)? Wait, but wait, the data: year 1 to 2: 248, year 2 to 3: 57 (slope decreased), year 3 to 4: 170 (slope increased? Wait, that's a problem. Wait, maybe the data has a typo, or I misread. Wait, year 3:405, year4:575: 575-405=170. So from year2-3:57, year3-4:170—slope increased. Then year4-5:79 (slope decreased), year5-6:50, year6-7:42. Hmm, maybe the initial jump is big, then a small jump, then a big jump, then smaller. But the problem says "very quickly at first and then more slowly"—maybe the overall trend is that after some point, the increase slows. Wait, maybe the key is between the options. Option B: quadratic with positive a (opens up, slope increasing), option D: quadratic with negative a (opens down, slope decreasing). Wait, the problem says "increased very quickly at first and then more slowly"—so the rate of increase (slope) is decreasing. So that's a concave down function, which is a quadratic with \( a < 0 \) (option D). Wait, but let's check the options again. Wait, maybe I made a mistake in the first differences. Wait, year1:100, year2:348 (diff 248), year3:405 (diff 57), year4:575 (diff 170), year5:654 (diff 79), year6:704 (diff 50), year7:746 (diff 42). So the differences are 248, 57, 170, 79, 50, 42. The first difference (slope) is not constant (so not linear). The second difference for a quadratic should be constant. Let's calculate second differences:

First differences: 248, 57, 170, 79, 50, 42

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

These are not constant, so maybe it's not a perfect quadratic, but the question is which type best models. The key is the shape: "increased very quickly at first and then more slowly"—so the slope is decreasing, which is a concave down curve, which is a quadratic with \( a < 0 \) (option D). Wait, but wait, maybe I messed up. Wait, the problem says "increased very quickly at first and then more slowly"—so the rate of increase (slope) is decreasing. A quadratic with \( a < 0 \) has a maximum point, and after that, the slope decreases. So the data here: starts with a big increase (year1-2), then a small one (year2-3), then a big one (year3-4), then smaller ones. Wait, maybe the problem's description is simplified: "very quickly at first and then more slowly"—so overall, the trend is that the increase slows down, which is a concave down curve, hence quadratic with \( a < 0 \) (option D). Wait, but let's check the options again. Option B: quadratic with positive a (opens up, slope increasing), which would mean the increase gets faster and faster. Option D: opens down, slope decreasing (increase slows down). So the problem's description matches option D.

Wait, but maybe I made a mistake in the first differences. Let's recalculate the number of employees:

Year 1: 100

Year 2: 348 (diff: 348 - 100 = 248)

Year 3: 405 (diff: 405 - 348 = 57)

Year 4: 575 (diff: 575 - 405 = 170)

Year 5: 654 (diff: 654 - 575 = 79)

Year 6: 704 (diff: 704 - 654 = 50)

Year 7: 746 (diff: 746 - 704 = 42)

So the differences are 248, 57, 170, 79, 50, 42. The first big jump, then a small one, then a big one, then smaller. But the problem says "increased very quickly at first and then more slowly"—maybe the "at first" is year1-2, then after that, even with a big jump in year3-4, overall the trend is slowing down (year4-5:79, year5-6:50, year6-7:42). So the later increases are smaller. So the slope is decreasing over time (after year2), which fits a quadratic with \( a < 0 \) (concave down), because the slope (rate of change) decreases. So the correct answer should be D: a quadratic function with a negative value of a.

Answer:

D. a quadratic function with a negative value of a