To draw all details of a given equation
First thing first, when x = 0 find the values of y
When y = 0 find the values of x
That gives you all the intercepts
Differentiate, implicit or not
Sometimes you have to force the equation in terms of x
ie from your original equation
y = (a^.5 - x^.5)^2
Find all the x values for dy/dx = 0
That tells you stationary points, which you sub into the original equation to find the stationary points
Now you have where the graph cuts all the axis, and high above and below the x axis it goes (Assuming the graph is a function, and not a relation)
Now you also need to find out any asymptotes, or lines the graph does not touch.
There are multiple ways to find these, essentially finding the x and y values that do not exist (eg for the graph y = 1/x x cannot equal zero so x = 0 is an asymptote and so on and so forth)
Draw these as dotted lines
At the end, you need to figure out where the graph ends, both in the positive and negative sides of the graph.
This is achieved by using limits
y value as x approaches infinite
in the example y = x^2, as x approaches infinite, y too approaches infinite, so y =x becomes the asymptote in the positive and negative axis
Thats pretty much the basics of the graphing, however depending on the detail required, simply using the second derivative instead of finding all stationary points will suffice to give the graphs general shape
