Graph of x cos x

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The graph of y = x cos x is an oscillation whose amplitude grows with distance from the origin, trapped between the lines y = x and y = −x. Unlike x sin x, it is odd, so it passes through the origin with slope 1.

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The graph of

y=xcos(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 = 0 and x = π/2 + nπ
Envelope: −|x| ≤ y ≤ |x|
Symmetry: Odd
Slope at the origin: 1
y-intercept: None
Derivative: cos(x)−xsin(x)
Antiderivative: xsin(x)+cos(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 cos x looks like

cos x is even and x is odd, so x cos x is odd: the graph has rotational symmetry about the origin. It starts out like y = x, since cos x ≈ 1 near 0.

It touches the upper envelope y = x where cos x = 1 (x = 2nπ) and the lower one where cos x = −1, with zeros halfway between.

The area under x cos x has the closed form x sin x + cos x, found by integration by parts.

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 = −14.1372, x = −10.9956, x = −7.854, x = −4.7124, x = −1.5708, x = 0 … (11 in view)
Local maxima: (−3.4256, 3.2884), (0.8603, 0.5611), (6.4373, 6.361)
Local minima: (−6.4373, −6.361), (−0.8603, −0.5611), (3.4256, −3.2884)
Inflection points: (−14.2764, 1.98066), (−11.1727, −1.96871), (−8.09616, 1.94163), (−5.08699, −1.86131), (−2.28893, 1.50607), (0, 0) … (11 in view)