Question

Write the quadratic function in three forms using the three points it goes through (-1,24),(5,0), and (-4,18)

Answer 1

The Standard form becomes - y =
`-(2)/(3)` x²
`-(4)/(3)` x +
`(70)/(3)`

Explanation:

To find - Write the quadratic function in three forms using the three points it goes through (-1,24),(5,0), and (-4,18)

Proof -

There are there forms of quadratic equation :

1) Standard form: y = ax²+ bx + c

where the a,b, and c are constants

2) Factored form: y = (ax + c)(bx + d)

where a,b,c, and d are constants

3) Vertex form: y = a(x + b)² + c

where a, b, and c are constants

Now,

For Standard form -

Given that the quadratic equation passes through three points

(-1, 24), (5, 0), (-4, 18)

24 = a(-1)²+ b(-1) + c

⇒24 = a - b + c ............(1)

0 = a(5)²+ b(5) + c

⇒0 = 25a + 5b + c ............(2)

18 = a(-4)²+ b(-4) + c

⇒18 = 16a - 4b + c ............(3)

By solving equation (1), (2), (3), we get

a =
`-(2)/(3)`, b =
`-(4)/(3)`, c =
`(70)/(3)`

So,

The Standard form becomes - y =
`-(2)/(3)` x²
`-(4)/(3)` x +
`(70)/(3)`