Question

How do you find the equation of the quadratic function given (1,6), (3,26), (-2,21)?

Answer 1

Please see the explanation for steps leading to the equation:
`y = 3x^2 - 2x + 5`

Explanation:

There are two standard equations for a parabola for the two different types of parabolas.

1. The type that opens up or down:

`y = ax^2 + bx + c`

1. The type that opens left or right:

`x = ay^2 + bx + c`

The points look like they belong to a parabola that opens up, therefore, we shall try to fit them the first equation:

`y = ax^2 + bx + c`

Write 3 different equations using the 3 points:

`6 = a(1)^2 + b(1) + c` `" [1]"`
`26 = a(3)^2 + b(3) + c` `" [2]"`
`21 = a(-2)^2 + b(-2) + c` `" [3]"`

Write these equations into an augmented matrix:

`[ (1,1,1,|,6), (9,3,1,|,26), (4,-2,1,|,21) ]`

Multiply row 1 by -9 and add to row 2:

`[ (1,1,1,|,6), (0,-6,-8,|,-28), (4,-2,1,|,21) ]`

Multiply row 1 by -4 and add to row 3:

`[ (1,1,1,|,6), (0,-6,-8,|,-28), (0,-6,-3,|,-3) ]`

Subtract row 2 from row 3:

`[ (1,1,1,|,6), (0,-6,-8,|,-28), (0,0,5,|,25) ]`

Divide row 3 by 5:

`[ (1,1,1,|,6), (0,-6,-8,|,-28), (0,0,1,|,5) ]`

Multiply row 3 by 8 and add to row 2:

`[ (1,1,1,|,6), (0,-6,0,|,12), (0,0,1,|,5) ]`

Divide row 2 by -6:

`[ (1,1,1,|,6), (0,1,0,|,-2), (0,0,1,|,5) ]`

Subtract row 3 from row 1

`[ (1,1,0,|,1), (0,1,0,|,-2), (0,0,1,|,5) ]`

Subtract row 2 from row 1:

`[ (1,0,0,|,3), (0,1,0,|,-2), (0,0,1,|,5) ]`

`a = 3, b = -2 and c = 5`

Check:

`6 = 3(1)^2 - 2(1) + 5` `" [1]"`
`26 = 3(3)^2 - 2(3) + 5` `" [2]"`
`21 = 3(-2)^2 - 2(-2) + 5` `" [3]"`

`6 = 6`
`26 = 26`
`21 = 21`

This checks

The equation is:

`y = 3x^2 - 2x + 5`