Question

Given three points (-1,-7) (1, -5) (2, -1) how do you write a quadratic function in standard form with the points?

Answer 1

Use the 3 points and the standard form

`y = ax^2+bx+c" [1]"`

to write 3 simultaneous equations with the variables, a, b , and c and then solve the equations.

Explanation:

Substitute the point `(-1,-7)` into equation [1]:

`a(-1)^2+b(-1)+c = -7`

`a - b+ c = -7" [2]"`

Write equation [2] as the first row of an Augmented Matrix :

#[ (1,-1,1,|,-7) ]#

Substitute the point `(1,-5)` into equation [1]:

`a(1)^2+b(1)+c = -5`

`a + b+ c = -5" [3]"`

Add a row corresponding to equation [3] to the augmented matrix:

#[ (1,-1,1,|,-7), (1,1,1,|,-5) ]#

Substitute the point `(2,-1)` into equation [1]:

`a(2)^2+b(2)+c = -1`

`4a + 2b+ c = -1" [4]"`

Add a row corresponding to equation [4] to the augmented matrix:

#[ (1,-1,1,|,-7), (1,1,1,|,-5), (4,2,1,|,-1) ]#

Perform Elementary Row Operations on the matrix, until the left side becomes an Identity Matrix :

`R_2-R_1toR_2`

#[ (1,-1,1,|,-7), (0,2,0,|,2), (4,2,1,|,-1) ]#

`R_3-4R_1toR_3`

#[ (1,-1,1,|,-7), (0,2,0,|,2), (0,6,-3,|,27) ]#

`1/2R_2toR_2`

#[ (1,-1,1,|,-7), (0,1,0,|,1), (0,6,-3,|,27) ]#

`R_1+R_2toR_1`

#[ (1,0,1,|,-6), (0,1,0,|,1), (0,6,-3,|,27) ]#

`R_3-6R_2toR_3`

#[ (1,0,1,|,-6), (0,1,0,|,1), (0,0,-3,|,21) ]#

`-1/3R_3 to R_3`

#[ (1,0,1,|,-6), (0,1,0,|,1), (0,0,1,|,-7) ]#

`R_1-R_3toR_1`

#[ (1,0,0,|,1), (0,1,0,|,1), (0,0,1,|,-7) ]#

We have an identity matrix on the left, therefore, the solution is the column vector on the right:

`a = 1, b = 1 and c = -7`

check

`y = x^2+x-7" [5]"`

by substituting in all 3 points.

Substitute the point `(-1,-7)` into equation [5]:

`-7=(-1)^2+ (-1)-7`
`-7=-7`

Substitute the point `(1,-5)` into equation [5]:

`-5 = (1)^2+ 1 - 7`
`-5=-5`

Substitute the point `(2,-1)` into equation [5]:

`-1= (2)^2+(2)-7`
`-1= -1`

This checks, therefore, equation [5] is the correct answer.