Find a quadratic model for the sequence.
-4, -4, -3, -1, 2
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To find a quadratic model for the sequence, we need to determine the quadratic equation that best fits the given data points.
Let's form the general equation for a quadratic function in the form of ax^2 + bx + c, and substitute the x and corresponding y values from the sequence to form a system of equations.
For the given sequence: (-4, -4), (-3, -4), (-1, -3), (2, -1)
1. Substitute the x and y values into the quadratic equation:
a(-4)^2 + b(-4) + c = -4
16a - 4b + c = -4
2. Substitute the x and y values into the quadratic equation:
a(-3)^2 + b(-3) + c = -4
9a - 3b + c = -4
3. Substitute the x and y values into the quadratic equation:
a(-1)^2 + b(-1) + c = -3
a - b + c = -3
4. Substitute the x and y values into the quadratic equation:
a(2)^2 + b(2) + c = -1
4a + 2b + c = -1
Now, we have a system of equations with three unknowns (a, b, and c).
To solve the system of equations, we can use methods such as substitution, elimination, or matrices. Once we solve the system of equations, we will obtain the values of a, b, and c, which will allow us to form the quadratic model for the given sequence.
To find a quadratic model for the given sequence, we need to find a quadratic equation of the form y = ax^2 + bx + c that can represent the given data points.
Let's analyze the given sequence:
-4, -4, -3, -1, 2
To find the general form of the quadratic equation, we can use the differences between consecutive terms to find the pattern. Let's find the differences between each term:
2 - (-1) = 3
-1 - (-3) = 2
-3 - (-4) = 1
-4 - (-4) = 0
Since the second differences are constant (in this case, 0), we know that the sequence can be represented by a quadratic equation.
Now, let's create a table to find the values of a, b, and c:
x | y
-------
1 | -4
2 | -4
3 | -3
4 | -1
5 | 2
At this point, we have five data points, so we can set up a system of equations using these data points and solve for the coefficients a, b, and c.
Using the quadratic equation form y = ax^2 + bx + c, we can substitute the values of x and y from the table:
(1) a(1)^2 + b(1) + c = -4
(2) a(2)^2 + b(2) + c = -4
(3) a(3)^2 + b(3) + c = -3
(4) a(4)^2 + b(4) + c = -1
(5) a(5)^2 + b(5) + c = 2
Simplifying these equations, we get:
a + b + c = -4 (1)
4a + 2b + c = -4 (2)
9a + 3b + c = -3 (3)
16a + 4b + c = -1 (4)
25a + 5b + c = 2 (5)
Now, we have a system of equations that can be solved simultaneously to find the values of a, b, and c. We can use techniques like substitution or elimination to solve this system and find the quadratic model for the given sequence.