How to draw contour plot of a function $f(x)=y^2 - x^2 y -2x^4 + \epsilon x$

Question

$f(x) = y^2 - x^2 y -2x^4 + \epsilon x$ , where $\epsilon>0$ and $\epsilon$ is small.

Draw a contour plot in $\mathbb{R^2}$of f(x).

How do I do this? I have no idea, it is not a simple circle.. Please, help

Edit: I forgot to tell, we CANNOT use computer programs. That is why I dont know how to start, I don't know methods for such task.

Answer 1

For $\epsilon = 0$, $f(x)$ factors as $(y - 2 x^2)(y + x^2)$. So the contour $f = 0$ would consist of two parabolas $y = 2 x^2$ and $y = -x^2$. Above the top parabola and below the bottom one $f$ would be positive, between them it would be negative. In particular, on the $x$ axis it would be $-2x^4$. Now how would adding a term $\epsilon x$ affect this?

Answer 2

In Mathematica:

Plot3D[y^2 - x^2 y - 2 x^4 + 1 x, {x, -4, 4}, {y, -4, 4}]