A travel agency made the table below to represent the profits it make on a certain vacation package.

Number of People vsProfit
Number of People Who Buy the Package, x
Profit per Person (in dollars), f(x)
10
500
20
800
30
900
40
800
50
500

Which of the following explains the best model for the data in the table?

D. A quadratic function since the points are symmetric about the line  .

Step-by-step explanation:

Please consider the complete question.

A travel agency made the table below to represent the profits it make on a certain vacation package.

x                  Profit per Person (in dollars), f(x)

10                     500

20                     800

30                     900

40                     800

50                     500

Which of the following explains the best model for the data in the table?

A) An absolute value function since the values increase and then decrease

B) An exponential function since the ratio of the y-values are about the same.

C) A linear function since the function changes at a rate of 30 dollars per person

D) A quadratic function since the points are symmetric about the line x = 30.

First of all, we will graph all the given points on coordinate plane and then join the points to see the graph.

We can see that point (10,500) is symmetric to point (50,500). Point (20,800) is symmetric to point (40,800).

Upon looking at attached image, we can see that the data forms the shape of a downward opening parabola, which is symmetric to line .

Therefore, a quadratic function best represents the given data because the points are symmetric about the line .

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

26?

step-by-step explanation: