Reading multiplicity of cusps , singularity etc from initial polynomial. Here I have an example which I found. Can someone help me to understand what's happening here? 
The following are my concerns:
1) What do we need to do co-ordinate transformation? 
2) How does the initial polynomial help us to see whether the singularity is ordinary or not?
3) How can we read tangents directly from the initial polynomial?

 A: To answer your first question, if we want to analyze a point $(x_0,y_0)$ of a curve $f(x,y)=0$ (let's just assume it's affine), we want to first see $f$ as a polynomial in the variables $x-x_0$ and $y-y_0$. This is done by writing $x=x-x_0+x_0$ and $y=y-y_0+y_0$. So, for example, if we have the curve $x^2+y^2-1=0$ and we want to analyze the point $(1,0)$, we just write
$$x^2+y^2-1=(x-1+1)^2+y^2-1=(x-1)^2+2(x-1)+y^2.$$
Step 2: To analyze a singularity, it helps to look at the tangent cone of your curve. If we are centered at $(x_0,y_0)$ as above, then this is just the vanishing set of the homogeneous part of your polynomial of lowest degree. So in the case of $x^2+y^2-1$, if we are centered at $(1,0)$ then the tangent cone corresponds to $x-1=0$. Notice that if a point is non-singular, then the tangent cone is just the tangent line.
Let's look at the example $y^2-x^2-x^3$ at $(0,0)$. Here we see that the tangent cone is $y^2-x^2=0$, that is $(y-x)(y+x)=0$. This gives us the union of two "tangent" lines $y=x$ and $y=-x$. This shows that the singularity at $(0,0)$ is a node. This also answers your question 3).
