Make an equation for a parabola in the form is y=ax^2+bx+c.

Question

The parabola must pass through the points (1,4) (-1,6) (2,6)?

Answer 1

The parabola form is y = ax² + bx + c.

The points (1, 4), ( - 1, 6) and ( 2, 6) passes through the above function.

Substitute for (x , y) = (1, 4) in y = ax² + bx + c.

4 = a(1)² + b(1) + c

a + b + c = 4 ---> (1)

Substitute for (x , y) = (- 1, 6) in y = ax² + bx + c.

6 = a(- 1)² + b(- 1) + c

 a - b + c = 6 ---> (2)

Substitute for (x , y) = (2, 6) in y = ax² + bx + c.

6 = a(2)² + b(2) + c

 4a + 2b + c = 6 ---> (3)

 

Now solve above three equations by elimination method.

To eliminate the b variable, add the equations (1) and (2).

(a + b + c) + (a - b + c) = 4 + 6

2a + 2c = 10  ---> (4)

 Multiple to each side of equation (2) by 2.

2a - 2b + 2c = 12 ---> (5)

To eliminate the b variable, add the equations (3) and (5).

(4a + 2b + c) + (2a - 2b + 2c) = 6 + 12

6a + 3c = 18

2a + c = 6 ---> (6)

To eliminate the a variable, Subtract equation (6) from equation (4).

(2a + 2c) - (2a + c) = 10 - 6

c = 4

Substitute c value in equation (6).

2a + 4 = 6

2a = 2

a = 1

Substitute a, c values in equation (1).

1 + b + 4 = 4

b = - 1

Substitute a, b, c values in y = ax² + bx + c.

Required equation is y = x² - x + 4.