Question

Find the equation of the parabola that passes through the points

1. 0,5 2,-3 -1,12

2. 2,0 3,-2 1,-2

Answer 1

1. The equation of parabola is
`y=x^2-6x+5`.

2. The equation of parabola is
`y=-2x^2+8x-8`.

Explanation:

1.

Let the equation of parabola be

`y=ax^2+bx+c`

It is given that the parabola passing through points (0,5), (2,-3) and (-1,12).

`5=a(0)^2+b(0)+c`

`5=c`

The value of c is 5.

`y=ax^2+bx+5`

The equation must be satisfied by the points (2,-3) and (-1,12).

`-3=a(2)^2+b(2)+5`

`-8=4a+2b`

Divide both sides by 2.

`-4=2a+b` .... (1)

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

`7=a-b` .... (2)

From (1) and (2), we get

`a=1,b=-6`

Therefore equation of parabola is
`y=x^2-6x+5`.

2.

Let the equation of parabola be

`y=ax^2+bx+c`

It is given that the parabola passing through points (2,0), (3,-2) and (1,-2).

`0=a(2)^2+b(2)+c`

`0=4a+2b+c` .... (3)

The equation must be satisfied by the points (3,-2) and (1,-2).

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

`-2=9a+3b+c` .... (4)

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

`-2=a+b+c` .... (5)

On solving (1), (2) and (3), we get

`a=-2,b=8,c=-8`

Therefore equation of parabola is
`y=-2x^2+8x-8`.