Determine whether a quadratic model exists for the set of values below. If so, write the model.
f(0)=-6 f(3)=3 f(-1)=-1
To determine whether a quadratic model exists for the given set of values, we can use the standard quadratic equation: f(x) = ax^2 + bx + c.
We are given the following values:
f(0) = -6
f(3) = 3
f(-1) = -1
Let's substitute these values into the equation:
1) When x = 0, f(0) = a(0)^2 + b(0) + c = c = -6
2) When x = 3, f(3) = a(3)^2 + b(3) + c = 9a + 3b + c = 3
3) When x = -1, f(-1) = a(-1)^2 + b(-1) + c = a + (-b) + c = -1
We can form a system of equations with these three equations:
Eq1: c = -6
Eq2: 9a + 3b + c = 3
Eq3: a - b + c = -1
Substituting Eq1 into Eq2 and Eq3, we get:
Eq2': 9a + 3b - 6 = 3
Eq3': a - b - 6 = -1
Simplifying Eq2' and Eq3', we get:
Eq2'': 9a + 3b = 9
Eq3'': a - b = 5
Next, we can rewrite Eq3'' as a = b + 5 and substitute it into Eq2'':
9(b + 5) + 3b = 9
12b + 45 = 9
12b = -36
b = -3
Substituting b = -3 into Eq3'', we get:
a - (-3) = 5
a + 3 = 5
a = 2
Finally, substituting a = 2 and b = -3 into Eq1 (c = -6), we get:
c = -6
Therefore, the quadratic model that fits the given set of values is:
f(x) = 2x^2 - 3x - 6