Substitute the 3 points, (1, -4), (-1, 12), and (-3, 12) into and make 3 linear equations where the variables are a, b, and c:
Point (1, -4): -4 = a(1)^2 + b(1) + c" [1]"
Point (-1, 12): 12 = a(-1)^2 + b(-1) + c" [2]"
Point (-3, 12): 12 = a(-3)^2 + b(-3) + c" [3]"
You have 3 equations with 3 unknown values, a, b, and c.
Here is what they look like in standard linear form:
a + b + c = -4" [1]"
a - b + c = 12" [2]"
9a - 3b + c = 12" [3]"
Here is their Augmented Matrix :
[
(1,1,1,|,-4),
(1,-1,1,|,12),
(9,-3,1,|,12)
]
Perform Elementary Row Operations
R_2 - R_1 to R_2:
[
(1,1,1,|,-4),
(0,-2,0,|,16),
(9,-3,1,|,12)
]
R_3 - 9R_1 to R_3:
[
(1,1,1,|,-4),
(0,-2,0,|,16),
(0,-12,-8,|,48)
]
-1/2R_2:
[
(1,1,1,|,-4),
(0,1,0,|,-8),
(0,-12,-8,|,48)
]
12R_2 + R_3 to R_3:
[
(1,1,1,|,-4),
(0,1,0,|,-8),
(0,0,-8,|,-48)
]
-1/8R_3:
[
(1,1,1,|,-4),
(0,1,0,|,-8),
(0,0,1,|,6)
]
R_1 - R_3 to R_1:
[
(1,1,0,|,-10),
(0,1,0,|,-8),
(0,0,1,|,6)
]
R_1 - R_2 to R_1:
[
(1,0,0,|,-2),
(0,1,0,|,-8),
(0,0,1,|,6)
]
a = -2, b = -8 and c = 6
The equation is:
y = -2x^2 - 8x + 6