Write the equation of the parabola in vertex form that passes through (-6, -7), (-11, -2) and (-8, 1).

Answer 1

y = Ax^2 + Bx + C

(-6,-7)

-7 = 36A - 6B + C

(-11,-2)

-2 = 121A - 11B + C

(-8,1)

1 = 64A - 8B + C

Using the last two equations:

1 = 64A - 8B + C

-2 = 121A - 11B + C

subtracts them:

3 = -57A + 3B <--- equation ALPHA

Using the first two equations:

-2 = 121A - 11B + C

-7 = 36A - 6B + C

subtracts them:

5 = 85A - 5B <--- equation BETA

So the 2x2 system is:

3 = -57A + 3B <--- equation ALPHA

5 = 85A - 5B <--- equation BETA

Solves by elimination: 5 times ALPHA and 3 times BETA:

15 = -285A + 15B

15 = 255A - 15B

Adds them:

30 = -30A

A = -1

Plugs into ALPHA:

3 = -57(-1) + 3B

3 = 57 + 3B

-54 = 3B

B = -18

Plugs into BETA:

5 = 85(-1) - 5B

5 = -85 - 5B

90 = - 5B

B = -18

Plugs into original 1st equation:

1 = 64(-1) - 8(-18) + C

1 = -64 + 144 + c

1 = 80 + c

c = =-79

Plugs into original 2nd equation:

-2 = 121 (-1) - 11(-18) + c

-2 = -121 + 198 + c

-2 = 77 + c

c = -79

Plugs into original 3rd equation:

-7 = 36(-1) - 6(-18) + c

-7 = -36 + 108 + c

-7 = 72 + c

c = -79

the parabola is: -x^2 - 18x - 79 = - ( x^2 + 18x + 79)

Answer 2

This is essentially a quadratic regression done by hand. If you simply enter your x-values into L1 and y-values into L2 (using the STAT button on your TI-84), then go to STAT --> CALC --> QuadReg (making sure that L1 is x-values and L2 is y-values), hit enter a bunch of times and it will tell you your a, b and c coefficients for a standard form quadratic equation. That can be done in 30 seconds. But when you have to do it by hand... watch this video :)