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Part 1
Determine whether a quadratic model exists for the set of values below. If so, write the model.
f(0)=5, f(2)=1 f(-1)=-2
To determine if a quadratic model exists for the given set of values, we need to check if it satisfies the general equation of a quadratic function: y = ax^2 + bx + c.
Given the points f(0) = 5, f(2) = 1, and f(-1) = -2, we can substitute the values into the equation to create a system of equations:
1) When x = 0, y = 5:
5 = a(0)^2 + b(0) + c
5 = c
2) When x = 2, y = 1:
1 = a(2)^2 + b(2) + c
1 = 4a + 2b + c
3) When x = -1, y = -2:
-2 = a(-1)^2 + b(-1) + c
-2 = a - b + c
From equation 1, we know that c = 5. Substituting c = 5 into equations 2 and 3, we get:
1 = 4a + 2b + 5 (equation 2)
-2 = a - b + 5 (equation 3)
Simplifying equation 2 and 3, we have:
4a + 2b = -4 (equation 4)
a - b = -7 (equation 5)
We can solve this system of equations (4 and 5) to find the values of a and b. First, multiply equation 5 by 2:
2a - 2b = -14 (equation 6)
Next, add equations 4 and 6:
4a + 2b + 2a - 2b = -4 + (-14)
6a = -18
a = -3
Substitute the value of a into equation 5:
-3 - b = -7
b = -4
Therefore, the quadratic model that represents the given set of values is:
f(x) = -3x^2 - 4x + 5