# Is it possible to convert the parametric curve defined by $x = t^3 - 3t$ and $y = t^2 - 4$ to an implicit function?

Is it possible to convert the parametric curve defined by $$x = t^3 - 3t \text{ and } y = t^2 - 4$$ to an implicit function?

• It is a plane curve. Are you asking if it's possible to express it as $y=f(x)$? – rogerl Feb 23 '19 at 21:49
• @rogerl No, it does not necesserily have to be expressible as an explicit function. An implicit function would be just fine. I just want to eliminate the $t$ and express the curve only in the varibles $x$ and $y$. – K. Claesson Feb 23 '19 at 21:52

If you want it in a more user friendly cartesian form, you can start with $$\frac xt=t^2-3=y+1$$ So $$t=\frac{x}{y+1}$$ $$\implies y=\frac{x^2}{(y+1)^2}-4$$ $$\implies x^2=(y+4)(y+1)^2$$
• Should get $t = \dfrac{x}{y + 1}$ – DanielV Feb 23 '19 at 22:42
• $t^2-3=t^2-4+1=y \,\mathbf\, + 1$ – J. W. Tanner Feb 24 '19 at 1:29
$$x/t = y + 1$$, hence $$x^2 / t^2 = (y + 1)^2$$, hence $$\frac{x^2}{y + 4} = (y + 1)^2$$ However! This is not completely equivalent form ...
$$x^2 = t^6 - 6t^4+9t^2$$, and $$t^2 = y+4$$; substituting gives $$x^2 = (y+4)^3 - 6(y+4)^2 + 9(y+4) = (y+4)(y+1)^2.$$
If you mean can $$y$$ be expressed explicitly in term of $$x$$ or vice versa no, but this is an attempt: $$y=t^2-4$$ $$t=\pm\sqrt{y+4}$$ $$x=t^3-3t=\pm(y+4)^{3/2}\mp3\sqrt{y+4}$$