Find the vertex of the graph of the equation $x - y^2 + 8y = 13$.

Rearrange as

x =   y^2  - 8y   +  13                complete the square on y

x = y^2  - 8y  + 16  + 13  - 16

x = ( y - 4)^2  - 3

(x + 3)  = (y - 4)^2

The  vertex is   ( -3, 4)

Here's the graph  : https://www.desmos.com/calculator/tksqqrj5dc

Completing the square, we get x = (y - 8)^2 - 51, so the vertex is (8,-51).