Question

Given three points (1, -2) (-2, 1) (3, 6) 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` to write 3 equations as rows in an augmented matrix. Use elementary row operations to solve for a, b, and c.

Explanation:

Substitute the point `(1,-2)` into the standard form:

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

The first row of the augmented matrix is:

`[(1,1,1,|,-2)]`

Substitute the point `(-2,1)` into the standard form:

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

Write the above equation as a second row into the matrix:

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

Substitute the point `(3,6)` into the standard form:

`6 = a(3)^2+ b(3)+c`

Write the above equation as a third row into the matrix:

#[ (1,1,1,|,-2), (4,-2,1,|,1), (9,3,1,|,6) ]#

Perform elementary row operations.

We want the coefficient is position `1,1` to be 1 and it is, therefore, no operation is required.

We want the other two coefficients in column 1 to be zero, therefore, we perform the following two row operations:

`R_2-4R_1toR_2`

#[ (1,1,1,|,-2), (0,-6,-3,|,9), (9,3,1,|,6) ]#

`R_3-9R_1toR_3`

#[ (1,1,1,|,-2), (0,-6,-3,|,9), (0,-6,-8,|,24) ]#

We want the coefficient in position `2,2` to be 1 but we shall save this for later.

We can make the coefficient in position `3,2` become zero by performing the following row operation:

`R_3-R_2toR_3`

#[ (1,1,1,|,-2), (0,-6,-3,|,9), (0,0,-5,|,15) ]#

Make the coefficient in position `3,3` become 1 by performing the follow row operation:

#R_3/-5 to R_2

#[ (1,1,1,|,-2), (0,-6,-3,|,9), (0,0,1,|,-3) ]#

Make the other coefficients of column 3 become 0 by performing the following two row operations:

`R_2+3R_3toR_2`

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

`R_1-R_3toR_1`

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

Because the second row is obviously zero we can make the following adjustments:

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

`a = 1, b= 0 and c = -3`

The standard form is:

`y = x^2-3`

Check:

`(1,-2)`
`-2 = 1^2-3`
`-2=-2`

`(-2,1)`
`1 = (-2)^2-3`
`1=1`

`(3,6)`
`6 = 3^2-9`
`6=6`

All 3 points check.