Draw the graph of $y=e^x \sin x$ (without using a graph calculator)

Question

I am having a whole lot of problem with these graphs.

If I try to sketch the graph taking the function $y=e^x \sin x$ itself,it becomes a bit complex.

I can seperately plot the graphs of $y=\sin x$ & $y=e^x$ by studying their function.(On a common X & Y axis).

But i cannot figure out how to transform these 2 to the given one.

Answer 1

Evaluate some points of $f$ for intuition. Note that $f(x\pm \pi k) = 0$. Now look at the first and second derivative for increasing/decreasing and for concavity. The general graph will be of an increasing, unbounded oscillation as $e^x$ is increasing without bound and $\sin x$ oscillates.

Answer 2

For graphs of the form $y=f(x)sinx$, draw the graphs of $y=f(x)$ and $y=-f(x)$. Then draw the graph of $y=sinx$ with the usual roots but with the amplitude increasing or decreasing according to the values of $f$. The graphs should touch whenever $sinx$ reaches its maximum and minimum.