Graph of sinh x (hyperbolic sine)

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The hyperbolic sine, sinh x = (eˣ − e⁻ˣ)/2, is half the difference of a growing and a decaying exponential. Its graph passes through the origin, rises everywhere, and soon looks just like eˣ/2 on the right and −e⁻ˣ/2 on the left.

Open in calculator

The graph of

y=sinh(x)

for −4 ≤ x ≤ 4. Dots mark its root.

Key features

Domain: All real numbers
Range: All real numbers
Definition: sinh x = (eˣ − e⁻ˣ)/2
Root and inflection point: (0, 0)
Monotonic: Increasing everywhere
Symmetry: Odd: sinh(−x) = −sinh x
Identity: cosh²x − sinh²x = 1
y-intercept: None
Derivative: cosh(x)
Antiderivative: cosh(x)+C

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

What the graph of sinh x looks like

Near the origin sinh x ≈ x, so the graph crosses with slope 1 and looks almost straight. Further out the eˣ term dominates on the right and the −e⁻ˣ term on the left, so the graph grows exponentially in both directions.

The name comes from the hyperbola: the point (cosh t, sinh t) traces the right half of x² − y² = 1, just as (cos t, sin t) traces the unit circle.

Unlike sin x, sinh x is not periodic and is unbounded. Its derivative is cosh x, which is always at least 1, so the curve never flattens.

Points of interest

Found numerically to about twelve digits in the window −4 ≤ x ≤ 4, then shown in exact form where one exists.

x-intercepts (roots): x = 0
Local maxima: None in this window
Local minima: None in this window
Inflection points: (0, 0)

Table of values

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

Values of sinh x
x	y = sinh x
−3	−10.0179
−2	−3.62686
−1	−1.1752
0	0
1	1.1752
2	3.62686
3	10.0179

How to graph sinh x by hand

1
Plot (0, 0) and draw a short segment of slope 1 through it.
2
Plot (1, 1.18), (2, 3.63), (3, 10.0) and the mirror points with both signs negative.
3
Draw an increasing curve, concave down on the left and concave up on the right.