How do you find the quadratic function y=ax^2+ bx+ c whose graph passes through the given points. (1, -4), (-1, 12), (-3,- 12)?

1 Answer
Dec 12, 2016

Use the 3 points to write 3 equations and then solve them using an augmented matrix.

Explanation:

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