# How do you graph y = x sin(1/x)?

As $\sin \left(\theta\right) \in \left[- 1 , 1\right]$, the $x$ prior to $\sin \left(\frac{1}{x}\right)$ acts as a scaling factor. As $x$ grows large, the amplitude of the oscillations of the sine function also grow. Similarly, as $x$ approaches $0$, the amplitude shrinks.
Next, looking at $\sin \left(\frac{1}{x}\right)$ we note that $\frac{1}{x} \to \infty$ as $x \to 0$. This means that as $x \to 0$ the sine function cycles through periods of $2 \pi$ more and more rapidly. Similarly, $\frac{1}{x} \to 0$ as $x \to \pm \infty$, meaning it will take greater and greater changes in $x$ to go through a full period of $2 \pi$. After $| x | > \frac{1}{\pi}$ there will be no further oscillations, as we will have $| \frac{1}{x} | \in \left(0 , \pi\right)$.
The resulting graph will have oscillations which grow in amplitude and are stretched further apart as $x$ is further from $0$, soon ceasing to oscillate at all, and which have smaller amplitudes and wilder oscillations closer to $0$.