+128475 1. 16(x - 3) = (y - 1)^2
The vertex of this parabola is at ( 3, 1)
This parabola opens to the right
In the form 4p(x - 3) = (y - k)^2......the focus is ( 3 + p, 1)
We can solve for p as 4p =16 ⇒ p = 4
So....the focus is ( 3 + 4, 1) = (7,1)
The dirctrix lies 4 units to the left of the vertex and is given by x = ( 3 - p ) = (3 - 4 ) = -1
Here's a graph : https://www.desmos.com/calculator/iwubncnnan
2. y = -1/6 x^2 + 7x - 80 we need to complete the square on x
y = (-1/6) [ x^2- 42x + 480 ]
Take 1/2 of 42 = 21....square it = 441....add and subtract it within the brackets
y = (-1/6) [x^2 - 42x + 441 + 480 - 441 ] factor the frist three terms....simplify the rest
y = (-1/6) [ (x - 21)^2 + 39 ]
y = (-1/6)(x - 21)^2 - 39/6
y = (-1/6)(x - 21)^2 - 13/2 add 13/2 to both sides
(y + 13/2) = (-1/6)(x - 21)^2 multiply through by -6
-6(y + 13/2) = (x - 21)^2
Because of the presence of a "-" in front of the "6"......this parabola turns downward
The vertex is ( 21, - 13/2)
The focus is given by ( 21 , -13/2 + p )
And we can find p as ..... 4p = -6 ⇒ p = -6/4 = -3/2
So....the focus is (21, -13/2 - 3/2) = ( 21 , -16/2) = (21, -8)
The directerix is given as y = (-13/2 - p) = ( -13/2 - (-3/2) ) = ( -13/2 + 3/2) = -10/2 = -5
Here's a graph : https://www.desmos.com/calculator/g9zfw92yos
