Familiarization with math

Standard Parabola Equation

Find the standard equation:
We have S(a, 0); P(x,y); M(-a, y).
By the definition of conic, we have SP/PM = e
SP / PM =1.
Therefore, SP = PM.
Therefore, SP² = PM^2.
(x - a)² + (y - 0)² = (x + a)² + (y - y)²
x² - 2ax + a² + y² = x² + 2ax + a²
Simplifying, we will get y² = 4ax.
The standard equation the parabola is y² = 4ax.
If the origin is moved to the point V(h, k) and if the new axes X, Y are parallel to x, y axes then the equations for transformations of axes are
x = X + h
y= Y + k
or
X = x - h
Y = y - k

Suppose they think about a parabola with vertex V(h, k) & open to the right.transformations of functions

In terms of Y, X coordinates, the equation of the parabola is of the form
Y² = 4aX
" style="vertical-align: middle;" src="http://www.tutorvista.com/js/jsMath/wysiwyg_asciimath/mimetex/mimetex.cgi?%5Cdisplaystyle%5CRightarrow"> (y - k)² = 4a (x - h)