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

Nov 8, 2016

$y = 3 {x}^{2} - 2 x + 5$

#### Explanation:

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

1. The type that opens up or down:

$y = a {x}^{2} + b x + c$

1. The type that opens left or right:

$x = a {y}^{2} + b x + c$

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

$y = a {x}^{2} + b x + c$

Write 3 different equations using the 3 points:

$6 = a {\left(1\right)}^{2} + b \left(1\right) + c$$\text{ [1]}$
$26 = a {\left(3\right)}^{2} + b \left(3\right) + c$$\text{ [2]}$
$21 = a {\left(- 2\right)}^{2} + b \left(- 2\right) + c$$\text{ [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) ]

$a = 3 , b = - 2 \mathmr{and} c = 5$

Check:

$6 = 3 {\left(1\right)}^{2} - 2 \left(1\right) + 5$$\text{ [1]}$
$26 = 3 {\left(3\right)}^{2} - 2 \left(3\right) + 5$$\text{ [2]}$
$21 = 3 {\left(- 2\right)}^{2} - 2 \left(- 2\right) + 5$$\text{ [3]}$

$6 = 6$
$26 = 26$
$21 = 21$

This checks

The equation is:

$y = 3 {x}^{2} - 2 x + 5$