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

The function $y = x \sin x$ is the product of the linear function $x$ and the sine $\sin x$. It produces a distinctive graph that oscillates with growing amplitude.

Definition

Its domain is all real numbers.

Symmetry

Both $x$ and $\sin x$ are odd, so their product is even: indeed $(-x)\sin(-x) = (-x)(-\sin x) = x \sin x$, and the graph is symmetric about the $y$-axis.

Amplitude and envelope

Since $-1 \le \sin x \le 1$, we have $|x \sin x| \le |x|$, so the graph is contained between the two lines $y = x$ and $y = -x$. These lines $y = \pm x$ form its envelope. The curve touches the envelope where $|\sin x| = 1$, at $x = \dfrac{\pi}{2} + n\pi$, and its swing grows in proportion to $|x|$. Unlike an ordinary trigonometric function it is not periodic, because the amplitude keeps increasing.

Zeros and behavior near the origin

The function is zero where $x = 0$ or $\sin x = 0$, that is at $x = n\pi$. Near the origin $\sin x \approx x$, so $x \sin x \approx x^2$: the curve meets the $x$-axis from above like a parabola. Thus the origin is a local minimum of $0$, and nearby the function stays $y \ge 0$.

Specific values and derivative

For instance $y = \dfrac{\pi}{2} \approx 1.57$ at $x = \dfrac{\pi}{2}$ (touching the envelope $y = x$), $y = 0$ at $x = \pi$, and $y = -\dfrac{3\pi}{2} \approx -4.71$ at $x = \dfrac{3\pi}{2}$ (touching $y = -x$). Its derivative is $\dfrac{d}{dx}(x \sin x) = \sin x + x \cos x$, whose zeros locate the local maxima and minima.

Range and limits

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

Applications

Expressions of the form $x \sin x$ (or $t \sin \omega t$) arise when the amplitude of a resonating oscillator grows with time, and in the analysis of beats and amplitude modulation.