Question

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)?

Answer 1

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`