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Graph of x sin x

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The graph of y = x sin x is a sine wave whose amplitude grows with |x|. It is trapped between the envelope lines y = x and y = −x, touching them whenever sin x = ±1, and crosses zero at every multiple of π.

−10−5510−550
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The graph of
y=xsin(x)
 for −15 ≤ x ≤ 15. Dots mark its roots, turning points and y-intercept.

Key features

Domain
All real numbers
Range
All real numbers
Zeros
x = nπ
Envelope
−|x| ≤ y ≤ |x|
Symmetry
Even: (−x) sin(−x) = x sin x
Touches the envelope
At x = π/2 + nπ, where sin x = ±1
y-intercept
None
Derivative
sin(x)+xcos(x)
Antiderivative
sin(x)−xcos(x)+C

Derivative and antiderivative from the calculator’s algebra system. Domain, range and asymptotes are stated for real x.

What the graph of x sin x looks like

Multiplying sin x by x scales each wave by its distance from the origin. The result is an oscillation that grows linearly, bounded by the lines y = ±x.

Since both x and sin x are odd, their product is even, so the graph is a mirror image across the y-axis. Near the origin it behaves like x², with a minimum at (0, 0).

The peaks are not exactly on the envelope lines: they occur slightly after π/2 + 2nπ, where tan x = −x. The curve touches the envelope at x = π/2 + nπ but crosses through its turning points just beyond.

Points of interest

Found numerically to about twelve digits in the window −15 ≤ x ≤ 15, then shown in exact form where one exists.

x-intercepts (roots)
x = −12.5664, x = −9.4248, x = −6.2832, x = −3.1416, x = 0, x = 3.1416 … (9 in view)
Local maxima
(−7.9787, 7.9167), (−2.0288, 1.8197), (2.0288, 1.8197), (7.9787, 7.9167)
Local minima
(−4.9132, −4.8145), (4.9132, −4.8145)
Inflection points
(−12.7223, 1.97574), (−9.62956, −1.95821), (−6.57833, 1.91352), (−3.6436, −1.75324), (−1.07687, 0.948166), (1.07687, 0.948166) … (10 in view)

Table of values

A few points to plot, with exact values where they are simple.

Values of x sin x
xy = x sin x
00
π/2π/2
π0
3π/2−3π/2
2π0
5π/25π/2
3π0

How to graph x sin x by hand

  1. 1
    Draw the envelope lines y = x and y = −x lightly.
  2. 2
    Mark the zeros at multiples of π.
  3. 3
    Draw waves that touch y = x at π/2, 5π/2, … and y = −x at 3π/2, 7π/2, …, crossing zero in between.
  4. 4
    Mirror the right half in the y-axis.

Transformations

How the graph moves when you change the formula. The original is x sin x in blue.

−10−5510−10100
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  • y = x The upper envelope.
  • y = -x The lower envelope.
  • y = x^2 sin(x) Faster growth, now between ±x².

Frequently asked questions

Is x sin x even or odd?
Even: it is the product of two odd functions.
Where is x sin x zero?
At every multiple of π, x = nπ.
What is the derivative of x sin x?
sin x + x cos x, by the product rule.

Related graphs

Graph of x cos x

An odd, growing oscillation between y = ±x.

Sine graph: y = sin x

Period 2π, amplitude 1, zeros at every multiple of π.

Graph of sin x / x (the sinc function)

A decaying wave with limit 1 at the origin.

Graph of cosine: y = cos x

The sine wave shifted left by π/2; starts at its peak.

Graph of a linear function: y = mx + b

A straight line: slope m, y-intercept b.

Graph of e^(−x²): the Gaussian bell curve

The bell curve: peak (0, 1), area √π.

Graph x sin x your way

Change the window, add sliders for a, b, h and k, trace the curve with your keyboard, or overlay its derivative.

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