I am working something where a picture like this one appeared : 
Say the curve is written in the form $$ y^2 = x^3 + ax^2 + bx + c $$ (if this is the wrong form of coefficients, feel free to correct me, I am guessing here) what are the conditions on $a,b,c$ so that the elliptic curve "looks like the one in red in the picture"? Essentially I'm asking this because I had an elliptic course that I can't recall (because my teacher barely gave course notes at all... so I have no reference), but I remember the pictures and I was doing something in graph theory where this precise picture showed up, so I want to try to fit an elliptic curve to the curve I have and see if my conjecture that the curve is elliptic is morally valid.
Thanks in advance!
If applying to your original equation for the curve
$$ y^2=x^3+ax^2+bx+c $$
then the curves similar to the red curve appear when $b$ and $c$ are $0$. The coefficient on $x^2$ can be anything within $(0,\infty)$. For an equation where the curve can be translated, consider
$$ (y-a)^2=(x-b)(x-c)^2 $$
with conditions that $b-c\gt 0$. The variable $a$ is the vertical translation of the curve, $b$ is the $x$-coordinate of the intersection, and $c$ is the $x$-coordinate of the loop. This is also why Gerry's answer worked, as $$ y^2=x(x-1)^2 $$ is of this form as well.