Question

Please answer this question

the curve y=ax^2+bx+c passes through (2, 5), (3, 12) and (-1, -4). Find the equation of the curve

Answer 1

y =
`x^(2)` - 2x - 3

Explanation:

We know: y=ax^2+bx+c

Given: (2, 5), (3, 12), and (-1, -4)

so...

We plug in these numbers:

5 = a*(2^2) + b*2 + c

12 = a*(3^2) + b*3 + c

-4 = a*((-1)^2) + b*(-1) + c

simplify:

5 = 4a + 2b + c

12 = 9a + 3b + c

-4 = a - b + c

Solve this 3 variable system

Combine 5 = 4a + 2b + c and -4 = a - b + c

-8 = 2a - 2b + 2c from multiplying the equation -4 = a - b + c by 2

5 - 8 = 4a + 2a + 2b - 2b + c + 2c

-3 = 6a + 3c

-1 = 2a + c

next multiply -4 = a - b + c by 3

-12 = 3a - 3b + 3c add to 12 = 9a + 3b + c

-12 + 12 = 3a + 9a + -3b + 3b + 3c + c

0 = 12a + 4c

0 = 3a + c

combine -1 = 2a + c and 0 = 3a + c

1 = -2a - c

0 + 1 = 3a - 2a + c - c

1 = a

so... -1 = 2*1 + c

-1 = 2 + c

c = -1 -2 = -3

use equation -4 = a - b + c , solve for b

-4 = 1 - b + (-3)

-4 = -2 - b

b = -2 + 4 = -2

so a = 1, b = -2, c = -3

y = x^2 - 2x - 3

y =
`x^(2)` - 2x - 3