# How to graph path of parametric equations

I am having a lot of trouble answering the following question as it is required to consider the value of t also. If t did not have to be considered, it would have been an easy ellipse.

Let an object move such that $$\left\{ \begin{array}{cr} x =& \cos{2t} \\ y =& -\sin{t} \end{array} \right.\qquad \text{where} \quad 0 \leq t \leq \frac{5\pi}{4}$$ Sketch a graph of the path followed by the object.

I greatly appreciate your contributions. Thank you in advance

## 1 Answer

Let $$u=\sin t$$, then since $$\cos2t=1-2\sin^2t$$, the curve can be reparametrised to $$(1-u^2,-u)$$ for $$-\frac{\sqrt2}2\le u\le1$$ (the range of $$\sin t$$ restricted to $$0\le t\le\frac{5\pi}4$$). This can then be implicitised as $$x=1-2y^2$$, so the curve is a parabola lying between the points obtained by substituting $$t=-\frac{\sqrt2}2$$ and $$t=1$$ into the parametric form (i.e. $$(0,\sqrt2/2)$$ and $$(-1,-1)$$. The rest of the sketch should be easy from here.

• Thanks, @ParclyTaxel ! – wendy Aug 12 at 9:24
• @wendy Would you please accept my answer by clicking the tick left of it? – Parcly Taxel Aug 12 at 9:39
• Hi, @ParclyTaxel , I didn't realise I hadn't accepted your answer! Sorry for not doing it sooner. – wendy Aug 13 at 11:38