1. y = -1/4x^2 + 4x - 19
The x coordinate of the vertex is given by -4 / (2 * -1/4) = -4 / (-1/2) = -4 * -2 = 8
And the y coordinate is given by :
-1/4 (8)^2 + 4(8) - 19 =
-16 + 32 - 19
-3
So....the vertex is (8, -3)
Here's a graph : https://www.desmos.com/calculator/z76eahtb61
2. The center of the circle is ( 1, 4) and the radius is 8
So we have
(x - 1) ^2 + ( y - 4)^2 = 8^2 expand and simplify
x^2 - 2x + 1 + y^2 - 8y + 16 = 64
x^2 + y^2 - 2x - 8y + 17 = 64 subtract 64 from both sides
x^2 + y^2 -2x - 8y - 47 = 0
3. We have y = 1/8 x^2 + 4x + 20
We want to complete the square on x
y = (1/8) (x^2 + 32x + 160)
Take 1/2 of 32 = 16....square it = 256....add and subtract it within the parentheses
y = (1/8) (x^2 + 32x + 256 + 160 - 256) factor the first three terms, simplify the last two
y = (1/8) [ (x + 16)^2 - 96 ] apply the 1/8 over both terms in the parentheses
y = (1/8) (x + 16)^2 - 12
(y + 12) = (1/8)(x + 16)^2 multiply both sides by 8
8( y + 12) = (x + 16)^2
The vertex is at (-16, -12)
In the form
4p ( y - k) = (x - h)^2
4p = 8
So....p = 2
And the focus is given by
(-16 , -12 + p) = (-16, -12 + 2) = (-16, -10)
Here's the graph : https://www.desmos.com/calculator/cgsvcjy0ny