Question

The table below shows the number of visitors to a museum over a seven day period.

Museum Visitors
Day of Week, x 1 2 3 4 5 6 7
Number of Visitors, N(x) 71 84 91 92 87 76 59

Which function best models this set of data?
A.
N(x) = -3x2 + 22x + 52
B.
N(x) = 22x + 49
C.
N(x) = 3x + 68
D.
N(x) = -3x2 + 74

Answer 1

Answer:

$N(x)=-3x^2+22x+52$

Step-by-step explanation:

Day of Week, x             :  1       2     3      4      5    6     7

Number of Visitors, N(x): 71    84   91    92   87   76   59

To check whether it is linear

First difference

84-71 = 13

91-84=7

92-91=1

87-92=-5

To check whether it is quadratic

Second difference

7-13=- 6

1-7=-6

-5-1 =-6

So, It is quadratic

Now to find the equation :

General quadratic equation : $y=ax^2+bx+c$

Day of Week, x             :  1       2     3      4      5    6     7

Number of Visitors, N(x): 71    84   91    92   87   76   59

for x =1, $71 = a (1)^2 + b (1) + c=a+b+c$

for x =2, $84 = a (2)^2 + b (2) + c=4a+2b+c$

for x =3, 91 = $a (3)^2 + b (3) + c=9a+3b+c$

So, :a+b+c=71 ---1

4a+2b+c=84 ---2

9a+3b+c=91 ---3

Substitute the value of c from 1 in 2 and 3

4a+2b+71-a-b=84

3a+b=13 ---4

9a+3b+71-a-b=91

8a+2b=20 ---5

Substitute the value of b from 4 in 5

8a+2(13-3a)=20

8a+26-6a=20

2a=-6

a=-3

Substitute the value of a in 5

8(-3)+2b=20

-24+2b=20

2b=44

b=22

Substitute the values in 1

-3+22+c=71

19+c=71

c=52

Substitute the values in the general equation :

$N(x)=-3x^2+22x+52$

So, Option A is true