Sketch the graph $x=e^{-t}\sin t$,$ t\ge 0$

Question

My graph is always negative though, and that doesn't make sense cause $t$ is supposed to be positive. I substituted $x$ as $y$ and $t$ as $x$.

Answer 1

Here are some portions of the graphs:

Answer 2

To draw this graph without the use of a graphics calculator or graphing software, first sketch the graphs of $x=e^{-t}$ and the reflection of this in the $t$ axis, $x=-e^{-t}$

Now sketch the graph of $x=\sin t$ so that it has the same roots as usual, but the amplitude decreases so that it fits between the two exponential graphs, only touching them when $\sin t=\pm1$.

Note that the stationary points are not the same points as where the diminishing sine wave and its exponential "envelopes" touch.