Graph of the Function $y = x \sin x$

$y = x \sin x$ is the product of the linear function $x$ and the sine $\sin x$. It draws a distinctive graph that oscillates with a swing that keeps growing.

Domain and symmetry

The domain is all real numbers. Both $x$ and $\sin x$ are odd, so their product is even.

The graph is therefore symmetric about the $y$-axis.

The envelope

Since $-1 \leq \sin x \leq 1$, we have $|x \sin x| \leq |x|$, so the graph is caught between the two lines $y = x$ and $y = -x$. Those lines are its envelope.

The curve touches the envelope at $x = \dfrac{\pi}{2} + n\pi$, where $|\sin x| = 1$, and its swing grows in proportion to $|x|$. Unlike an ordinary trigonometric function it has no fixed amplitude.

Zeros

The value is $0$ where $x = 0$ or $\sin x = 0$, that is at $x = n\pi$. At $x = 0$ the zeros of $x$ and of $\sin x$ coincide, so that root is a double one.

Near the origin

Near the origin $\sin x \approx x$, so the following holds.

The curve meets the $x$-axis from above like a parabola, so the origin is a local minimum of $0$, and nearby the function stays $y \geq 0$.

Monotonicity

The product rule gives the derivative.

Its zeros locate the local maxima and minima. For large $x$ the term $x\cos x$ dominates, so those extrema drift toward the points where $\cos x = 0$, which are exactly where the curve touches the envelope.

Notable values

Range and limits

Because the swing grows without bound, $y$ takes arbitrarily large positive and negative values, so the range is all real numbers. As $x \to \pm\infty$ it does not converge but keeps oscillating ever more widely.

Applications

The form $x \sin x$ appears when the amplitude of a resonating system grows with time, and in the analysis of beats and amplitude modulation. When a driving frequency matches a system's natural frequency, the amplitude of the response grows in proportion to time¹.

1. Resonance, Wikipedia