find a quadratic function that fits the set of data points (1,4) (-1,6) (-2,16)
To find a quadratic function that fits the given set of data points, we need to set up a system of equations using the general form of a quadratic function, which is:
y = ax^2 + bx + c
We have three data points, so we will set up three equations using these points:
Equation 1: (1,4)
4 = a(1)^2 + b(1) + c
Equation 2: (-1,6)
6 = a(-1)^2 + b(-1) + c
Equation 3: (-2,16)
16 = a(-2)^2 + b(-2) + c
Now, we can solve this system of equations to find the values of a, b, and c.
From Equation 1, we have:
4 = a + b + c ----(4a)
From Equation 2, we have:
6 = a - b + c ----(4b)
From Equation 3, we have:
16 = 4a - 2b + c ----(4c)
Let's solve this system of equations:
Subtracting Equation (4b) from Equation (4a) eliminates c:
4a - a = a = 4 - 6
3a = -2
a = -2/3
Substituting the value of a in Equation (4a):
4 = (-2/3) + b + c
12/3 + 2/3 = b + c
b + c = 14/3
Substituting the value of a in Equation (4c):
16 = 4(-2/3) - 2b + c
32/3 - 2b + c = 16
-2b + c = 16 - 32/3
-2b + c = 16/3
Now, we have two equations remaining:
b + c = 14/3 ----(4d)
-2b + c = 16/3 ----(4e)
Subtracting Equation (4e) from Equation (4d) eliminates c:
3b = 14/3 - 16/3
3b = -2/3
b = -2/9
Substituting the value of b in Equation (4d):
-2/9 + c = 14/3
c = 14/3 + 2/9
c = 46/9
Therefore, the quadratic function that fits the set of data points is:
y = (-2/3)x^2 - (2/9)x + 46/9
ax^2 + bx + c = y
plug in your points:
a + b + c = 4
a - b + c = 6
4a - 2b + c = 16
a = 3
b = -1
c = 2
y = 3x^2 - x + 2