Question

Vertex for equation
Opens up or down, and passes
through (−6, −7), (−11, −2), and (−8, 1)

Answer 1

Brute force, check my arithmetic
y = a x^2 + b x + c
first two points:
-7 = 36 a - 6 b+ c
-2 = 121 a -11 b + c
---------------------------------subtract
-5 = -85 a + 5 b
second and third points:
-2 = 121 a - 11 b + c
1 = 64 a - 8 b + c
--------------------------- subtract again
-3 = +57 a - 3 b multiply by 5
-5 = -85 a + 5 b from above, multiply by 3
so
-15 = +285 a - 15 b
-15 = -540 a + 15 b
-------------------------------add
-30 = - 255 a
a = 30/255 = 0.118
-3 = +57 a - 3 b
-3 = 57 (0.118) - 3 b
3 b = 9.7
b = 3.24
-7 = 36 a - 6 b+ c
-7 = 36(0.118) - 6(3.24) + c
c = -7 - 4.25 + 19.44
c = 8.19
so
y = 0.118 x^2 + 3.24 x + 8.19 opens up, holds water
now the vertex, complete square
0.118 x^2 + 3.24 x = y - 8.19 divide by 0.118
x^2 + 27.5 x = 8.47 y - 69.4
add (b/2)^2
x^2 + 27.5 x + 189 = 8.47 y - 69.4 + 189
(x+13.75)^2 = 8.47 ( y + 14.1)
(-13.75 , -14.1)

Answer 2

using the three points,
36a-6b+c = -7
121a-11b+c = -2
64a-8b+c = 1
So, (a,b,c) = (-1,-18,-79)
y = -x^2-18x-79
= -(x^2+18x+81) + 2
= -(x+9)^2 + 2
opens down, vertex at (-9,2)