Graph of e^x

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The graph of eˣ is the exponential curve with a remarkable property: at every point its slope equals its height. It crosses the y-axis at (0, 1), creeps along just above the x-axis to the left, and shoots upward to the right.

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The graph of

y=e^x

for −5 ≤ x ≤ 3. Dots mark its y-intercept.

Key features

Domain: All real numbers
Range: y > 0
Horizontal asymptote: y = 0 as x → −∞
Notable points: (0, 1) and (1, e) with e ≈ 2.71828
Monotonic: Increasing everywhere
Concavity: Concave up everywhere
Special property: d/dx eˣ = eˣ
y-intercept: (0, 1)
Derivative: e^x
Antiderivative: e^x+C

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

What the graph of eˣ looks like

e ≈ 2.71828 is the one base for which the exponential curve has slope exactly 1 where it crosses the y-axis. That single fact makes eˣ its own derivative: the slope of the graph at any point equals the y-value there.

To the left the curve gets arbitrarily close to the x-axis without touching it, since eˣ is positive for every x: e⁻⁵ ≈ 0.0067. To the right it doubles roughly every 0.693 units (ln 2), and eventually outgrows any power of x.

Reflecting the graph in the line y = x gives ln x, its inverse. Reflecting it in the y-axis gives e⁻ˣ, the shape of exponential decay.

Points of interest

Found numerically to about twelve digits in the window −5 ≤ x ≤ 3, then shown in exact form where one exists.

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

Table of values

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

Values of eˣ
x	y = eˣ
−3	0.0497871
−2	0.135335
−1	0.367879
0	1
1	e
2	7.38906
3	20.0855

How to graph eˣ by hand

1
Draw the horizontal asymptote y = 0 (the x-axis).
2
Plot (0, 1), (1, 2.72), (2, 7.39) and (−1, 0.37).
3
Draw an increasing curve through the points that bends upward, flattening toward the x-axis on the left without touching it.
4
Check the slope at (0, 1): the curve should cross the y-axis at 45°.

Transformations

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

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y = e^(-x) Reflected in the y-axis: exponential decay.
y = e^(x - 2) Shifted right 2: crosses the y-axis at e⁻² ≈ 0.135.
y = e^x - 3 Shifted down 3: the asymptote moves to y = −3 and the curve now crosses the x-axis at x = ln 3.

Where it shows up

Continuous compound interest: P e^(rt).
Population growth, radioactive decay and cooling, wherever the rate of change is proportional to the amount.
The normal distribution, whose bell curve is built from e^(−x²/2).

Frequently asked questions

What is the domain and range of e^x?

The domain is all real numbers; the range is y > 0.

Does e^x ever equal zero?

No. It gets arbitrarily close to 0 as x → −∞, so y = 0 is a horizontal asymptote, but it is always positive.

Why is e^x its own derivative?

e is defined so that the slope of the exponential curve at x = 0 is exactly 1. Since e^(x+h) = eˣ·eʰ, the slope at any x is eˣ times the slope at 0, which is eˣ.

What is the y-intercept of e^x?

(0, 1), because e⁰ = 1.