(2.1)Examples of parametrised cubics.Some plane cubic curves can be parametrised, just as the conics:

Nodal cubic$\displaystyle C: (y^2=x^3+x^2) \subset \mathbb{R}^2$ is the image of the map $\displaystyle \varphi: \mathbb{R}^1 \to \mathbb{R}^2$ given by $\displaystyle t\mapsto (t^2-1, t^3-t)$

Cuspidal cubic. $\displaystyle C: (y^2 = x^3)\subset \mathbb{R}^2$ is the image of $\displaystyle \varphi:\mathbb{R}^1 \to \mathbb{R}^2$ given by $\displaystyle t\mapsto (t^2,t^3)$

(1) Let $\displaystyle C: (y^2 = x^3 + x^2) \subset \mathbb{R}^2.$ Show that a varible line though $\displaystyle (0,0)$ meets $\displaystyle C$ at one further point, and hence deduce the parametrisation of $\displaystyle C$ given in (2.1). Do the same for $\displaystyle (y^2 = x^3)$ and $\displaystyle (x^3 = y^3-y^4).$

If anyone could show me how to do any of those in (1), I would really appreciate it.. I have no idea how to tackle this problem. Thanks!