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

Q: What is the area under e^(−x²)?

Over the whole real line it is √π ≈ 1.77245. The calculator computes int(exp(-x^2), x, -Infinity, Infinity) to twelve digits.

Q: Where are the inflection points of the bell curve?

At x = ±1/√2 for e^(−x²). For a normal distribution with standard deviation σ they are at the mean ± σ.

Q: Does e^(−x²) have an antiderivative?

Not one built from elementary functions. It is expressed with the error function: (√π/2) erf(x) + C.

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The graph of y = e^(−x²) is the bell curve. It peaks at (0, 1), falls away symmetrically, and is almost zero beyond |x| = 3. Rescaled, it is the normal distribution used throughout statistics.

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The graph of

y=e^−x²

for −4 ≤ x ≤ 4. Dots mark its maximum.

Key features

Domain: All real numbers
Range: 0 < y ≤ 1
Maximum: (0, 1)
Inflection points: x = ±1/√2 ≈ ±0.707, y = e^(−1/2) ≈ 0.607
Horizontal asymptote: y = 0 in both directions
Total area: ∫ e^(−x²) dx over all reals = √π ≈ 1.7725
Symmetry: Even
y-intercept: None
Derivative: −2xe^−x²
Antiderivative: √π2erf(x)+C

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

What the graph of e^(−x²) looks like

Squaring x makes the exponent −x² negative on both sides and very negative quickly, so the curve falls off far faster than e⁻ˣ: e⁻⁹ ≈ 0.00012 at x = 3.

The curve changes from concave down to concave up at x = ±1/√2. In the normal distribution with standard deviation σ, these inflection points sit exactly one σ from the mean.

e^(−x²) has no elementary antiderivative, yet its area over the whole line is exactly √π. The normal density (1/√(2π)) e^(−x²/2) is this curve stretched so the area is 1.

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): None in this window
Local maxima: (0, 1)
Local minima: None in this window
Inflection points: (−√2/2, 0.606531), (√2/2, 0.606531)

Table of values

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

Values of e^(−x²)
x	y = e^(−x²)
0	1
1/2	0.778801
√2/2	0.606531
1	0.367879
3/2	0.105399
2	0.0183156
3	0.00012341

How to graph e^(−x²) by hand

1
Plot the peak (0, 1).
2
Plot (±0.5, 0.78), (±1, 0.37), (±2, 0.018).
3
Draw a symmetric bell, changing curvature near x = ±0.71 and flattening toward the axis.

Transformations

How the graph moves when you change the formula. The original is e^(−x²) in blue.

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y = e^(-x^2/2)/sqrt(2pi) The standard normal density: wider, lower, area 1. Same as normalpdf(x).
y = e^(-(x - 1)^2) Centred at x = 1.
y = e^(-4x^2) Narrower: the width scales with 1/√4.

Where it shows up

The normal (Gaussian) distribution of measurement errors and many natural quantities.
Heat diffusion: an initial spike of heat spreads into a Gaussian.
Gaussian blur in image processing.