Problem of Harsthorne page 35 problem 5.1. Q Which is which in Figure?
a)$x^2=x^4+y^4$
b)$xy=x^6+y^6$
c)$x^3=y^2+x^4+y^4$
d)$x^2y+xy^2=x^4+y^4$

a)$x^2=x^4+y^4$
This is invariant under the transformation $x\mapsto -x$ and  $y\mapsto -y$. Thus it is Tacnode.
b)$xy=x^6+y^6$
It is invariant under the map  $(x,y) \mapsto (y,x)$, thus Node or Triple point. 
Because triple points meat the small circle of origin in six times and six is the degree of this polynomial, thus I guess this curve is triple point but I cannot prove it in some precise manner.
c)$x^3=y^2+x^4+y^4$
This curve is invariant under the map $y \mapsto -y$ and it is not invariant under the map $x \mapsto -x$, thus Cusp.
d)$x^2y+xy^2=x^4+y^4$
It is invariant under the map  $(x,y) \mapsto (y,x)$, thus Node or Triple point. 
Because triple points meat the small circle of origin in 4 times and 4 is the degree of this polynomial, thus I guess this curve is Node but I cannot prove it in some precise manner.
I  cannot answer for cuves b) and d).

Hint: The homogeneous term of lowest degree tells you the tangent directions at the origin. For instance, the lowest order term of $x^2 - x^4 - y^4 = 0$ is $x^2 = x \cdot x$, so the line $x=0$ is a double tangent line at the origin. Only one of your graphs has this property--can you see which one?
You can match the other graphs and equations similarly. Just zoom in on the portion of the graph near the origin and see what the tangent lines are.
