A biologist took a count of the number of migrating waterfowl at a particular lake, and recounted the lake’s population of waterfowl on each of the next six weeks.
Week
0
1
2
3
4
5
6
Population
635
644
719
860
1,067
1,340
1,679
Find a quadratic function that models the data as a function of x, the number of weeks. Use the model to estimate the number of waterfowl at the lake on week 8.
(1 point)
Responses
P(x) = 33x2 – 24x + 635; 2,555 waterfowl
P ( x ) = 33 x 2 – 24 x + 635; 2,555 waterfowl
P(x) = 38x2 + 24x + 585; 2,615 waterfowl
P ( x ) = 38 x 2 + 24 x + 585; 2,615 waterfowl
P(x) = 33x2 – 24x + 635; 2,084 waterfowl
P ( x ) = 33 x 2 – 24 x + 635; 2,084 waterfowl
P(x) = 38x2 + 24x + 585; 3,209 waterfowl
The correct answer is:
P(x) = 33x^2 - 24x + 635; 2,555 waterfowl
To find a quadratic function that models the data, we can use the given population values for weeks 0 to 6.
We can form a quadratic function of the form P(x) = ax^2 + bx + c, where P(x) represents the population at week x.
Using the population values, we have the following system of equations:
635 = a(0)^2 + b(0) + c
644 = a(1)^2 + b(1) + c
719 = a(2)^2 + b(2) + c
860 = a(3)^2 + b(3) + c
1,067 = a(4)^2 + b(4) + c
1,340 = a(5)^2 + b(5) + c
1,679 = a(6)^2 + b(6) + c
Simplifying these equations, we get:
c = 635
a + b + c = 644
4a + 2b + c = 719
9a + 3b + c = 860
16a + 4b + c = 1,067
25a + 5b + c = 1,340
36a + 6b + c = 1,679
Solving the system of equations, we find:
a = 33
b = -24
c = 635
Therefore, the quadratic function that models the data is:
P(x) = 33x^2 - 24x + 635.
To estimate the number of waterfowl at the lake on week 8, we can substitute x = 8 into the quadratic function:
P(8) = 33(8)^2 - 24(8) + 635
P(8) = 33(64) - 192 + 635
P(8) = 2,112 - 192 + 635
P(8) = 2,555
So the estimated number of waterfowl at the lake on week 8 is 2,555.
Therefore, the correct answer is:
P(x) = 33x^2 - 24x + 635; 2,555 waterfowl
To find a quadratic function that models the data, we can use the given data points and solve for the coefficients of the quadratic equation.
Let's start by assigning the data to variables:
x = number of weeks
P(x) = population of waterfowl
Using the given data:
Week (x) Population (P(x))
0 635
1 644
2 719
3 860
4 1,067
5 1,340
6 1,679
We can see that the population seems to be increasing in a quadratic pattern.
We can express the quadratic function as follows:
P(x) = ax^2 + bx + c
We need to find the values of a, b, and c.
To solve for the coefficients, we can use a system of equations. We can set up three equations using three data points:
Using (0, 635):
(0) = a(0)^2 + b(0) + c
0 = 0 + 0 + c
c = 635
Using (1, 644):
644 = a(1)^2 + b(1) + 635
644 = a + b + 635
Using (2, 719):
719 = a(2)^2 + b(2) + 635
719 = 4a + 2b + 635
Now, solve the system of equations:
From the second equation, we can rearrange it to get:
a + b = 9
Subtracting the first equation from the third equation, we get:
(4a + 2b + 635) - (a + b + 635) = 719 - 644
4a + 2b - a - b = 75
3a + b = 75
We now have a system of two equations:
a + b = 9
3a + b = 75
Solve this system of equations for a and b:
Multiply the first equation by 3:
3(a + b) = 3(9)
3a + 3b = 27
Subtract the second equation from this new equation:
(3a + 3b) - (3a + b) = 27 - 75
2b = -48
b = -24
Substitute the value of b into the first equation:
a + (-24) = 9
a - 24 = 9
a = 9 + 24
a = 33
So, we have found a = 33 and b = -24.
Now, we can plug these values into the quadratic function equation:
P(x) = 33x^2 - 24x + 635
Finally, to estimate the number of waterfowl at the lake in week 8 (x = 8), substitute the value of x into the equation:
P(8) = 33(8)^2 - 24(8) + 635
P(8) = 33(64) - 192 + 635
P(8) = 2112 - 192 + 635
P(8) = 3555
Therefore, the estimated number of waterfowl at the lake in week 8 is 3,555.
The correct answer is:
P(x) = 33x^2 - 24x + 635; 2,615 waterfowl