Arctan graph: y = arctan x (inverse tangent)

Q: What are the asymptotes of arctan x?

Two horizontal asymptotes: y = π/2 as x → ∞ and y = −π/2 as x → −∞. There are no vertical asymptotes.

Q: What is the range of arctan x?

−π/2 < y < π/2. The end values are never reached.

Q: What is arctan 1?

π/4 (45°), because tan(π/4) = 1.

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The arctan graph is an S-shaped curve that accepts every real number and squeezes the output between −π/2 and π/2. It passes through the origin with slope 1 and levels off toward two horizontal asymptotes.

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The graph of

y=arctan(x)

for −10 ≤ x ≤ 10. Dots mark its root at the origin.

Key features

Domain: All real numbers
Range: −π/2 < y < π/2
Horizontal asymptotes: y = −π/2 as x → −∞, and y = π/2 as x → ∞
Monotonic: Increasing everywhere
Slope at the origin: 1
Symmetry: Odd: arctan(−x) = −arctan x
y-intercept: None
Derivative: 11+x²
Antiderivative: xarctan(x)−ln(1+x²)2+C

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

What the graph of arctan x looks like

Tangent is one-to-one on −π/2 < x < π/2, where it sweeps through every real number. Its inverse, arctan, therefore takes any real input and returns an angle strictly between −π/2 and π/2.

Reflecting the central branch of tan x in y = x turns its vertical asymptotes into horizontal ones: the arctan graph approaches y = π/2 on the right and y = −π/2 on the left but never reaches them.

The slope is 1/(1 + x²): exactly 1 at the origin and shrinking toward 0 far away, which is why the curve flattens out. The origin is its only inflection point.

Points of interest

Found numerically to about twelve digits in the window −10 ≤ x ≤ 10, 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 arctan x
x	y = arctan x
−√3	−π/3
−1	−π/4
−√3/3	−π/6
0	0
√3/3	π/6
1	π/4
√3	π/3

How to graph arctan x by hand

1
Draw dashed horizontal asymptotes at y = π/2 ≈ 1.57 and y = −π/2.
2
Plot (0, 0), (1, π/4) and (−1, −π/4), with π/4 ≈ 0.785.
3
Draw an S-curve through them with slope 1 at the origin, flattening toward the asymptotes.

Transformations

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

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y = 2atan(x) Asymptotes move to y = ±π.
y = atan(3x) Steeper in the middle (slope 3 at the origin), same asymptotes.
y = atan(x - 2) + 1 Shifted right 2 and up 1: centred on (2, 1).

Where it shows up

Converting x and y coordinates to an angle (via the two-argument atan2, which also gets the quadrant right).
Smooth saturation curves, for example in control systems and in mapping an unbounded quantity to a bounded one.