Parametric curve and its orientation

Question

I am having trouble understanding how to find the orientation of the graph of the curve. I plotted the two equations on Wolfram Alpha (see picture of graph above). Where should I be drawing the arrows on the graph to show the orientation? Thank you!

Answer 1

The choice of parametrization for a curve can change its orientation. There are common ones, say for the circle $\langle \cos t, \sin t \rangle$, which you can remember is oriented counterclockwise starting at $(1,0)$ with this choice. But for a generic curve, you will have to choose a particular $t$ value and see what happens as you vary $t$, i.e. does the $x,y$ coordinate increase/decrease for a 'small' change in $t$, using derivatives or 'derivative' estimates if the derivative does not exist. If the curve is simple, plotting might be easier. Plot for a few $t$ values with $t$ consistently increasing/decreasing to see what happens.

For your example, notice that as $t \to \infty$, both $x,y \to \infty$ so the curve most 'move upwards' and 'to the right'.

Answer 2

Note that $\lim_{t\to \pm\infty} \frac {y(t)}{x(t)}=1$, which indicates the slope of the orientation, i.e. at 45 degrees with the $x$-axis.