Find the axis, vertex, focus, equation of directrix, latus rectum and length of the latus rectum for the parabola $\left(x - 4\right)^2 = 4 \left(y + 2\right)$. Also sketch its graph.
Given equation of parabola is: $\;\;$ $\left(x - 4\right)^2 = 4 \left(y + 2\right)$ $\;\;\; \cdots \; (1a)$
Let $\;$ $X = x - 4$ $\;$ and $\;$ $Y = y + 2$
Then equation $(1a)$ can be written as: $\;\;$ $X^2 = 4 Y$
i.e. $\left(X - 0\right)^2 = 4 \times 1 \times \left(Y - 0\right)$ $\;\;\; \cdots \; (1b)$
Comparing equation $(1b)$ with the standard equation: $\;\;$ $\left(X - h\right)^2 = 4 a \left(Y - k\right)$ $\;$ gives
$\left(h, k\right) = \left(0, 0\right)$; $\;\;$ $a = 1$
$\therefore$ $\;$ For the given parabola,
