Graph of an exponential function: y = aˣ

Updated 20 October 2026 · Values computed by the ihateDesmos engine

An exponential function multiplies by the same factor each time x goes up by one. The graph of y = 2ˣ doubles at every step: 1, 2, 4, 8, 16. Every exponential graph y = aˣ passes through (0, 1) and has the x-axis as an asymptote.

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The graph of

y=2^x

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

Key features

Domain: All real numbers
Range: y > 0
y-intercept: (0, 1) for every base a
Horizontal asymptote: y = 0
Growth or decay: Growth when a > 1, decay when 0 < a < 1
Doubling time of 2ˣ: Exactly 1 unit of x
y-intercept: (0, 1)
Derivative: 2^xln(2)
Antiderivative: 2^xln(2)+C

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

What the graph of 2ˣ looks like

For y = 2ˣ, each step of 1 to the right doubles the height and each step to the left halves it: (−2, ¼), (−1, ½), (0, 1), (1, 2), (2, 4), (3, 8). The left side approaches the x-axis forever without touching it.

Changing the base changes how fast the curve climbs, but every graph of aˣ goes through (0, 1), because a⁰ = 1. Bases above 1 give rising curves; bases between 0 and 1 give falling ones, and (1/a)ˣ is the mirror image of aˣ in the y-axis.

All exponential graphs are horizontal stretches of one another: 2ˣ = e^(x ln 2), so the graph of 2ˣ is the graph of eˣ stretched horizontally by a factor of 1/ln 2 ≈ 1.44.

Points of interest

Found numerically to about twelve digits in the window −5 ≤ x ≤ 5, 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 2ˣ
x	y = 2ˣ
−3	1/8
−2	1/4
−1	1/2
0	1
1	2
2	4
3	8

How to graph 2ˣ by hand

1
Draw the asymptote y = 0.
2
Plot (0, 1) and then double going right and halve going left: (1, 2), (2, 4), (3, 8), (−1, ½), (−2, ¼).
3
Join the points with a smooth curve that bends upward and flattens toward the x-axis on the left.

Transformations

How the graph moves when you change the formula. The original is 2ˣ in blue.

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y = 3^x A larger base: steeper on the right, closer to the axis on the left. Still through (0, 1).
y = (1/2)^x A base between 0 and 1 gives decay. This is 2ˣ reflected in the y-axis.
y = 2^(x - 1) + 1 Shifted right 1 and up 1: the asymptote moves to y = 1.

Common mistakes

Drawing the curve crossing the x-axis. aˣ is always positive for a > 0.
Confusing 2ˣ (exponential) with x² (a parabola). In an exponential, the variable is in the exponent.

Where it shows up

Compound interest: P(1 + r)ᵗ grows exponentially with base 1 + r.
Doubling processes such as cell division and the early spread of an epidemic.
Half-lives: a quantity with half-life T follows (1/2)^(t/T).

Frequently asked questions

What does the graph of an exponential function look like?

A curve that passes through (0, 1), stays above the x-axis, and either rises ever more steeply (base above 1) or falls toward zero (base between 0 and 1). The x-axis is a horizontal asymptote.

What is the asymptote of an exponential function?

For y = aˣ it is y = 0. For y = aˣ + k it is y = k.

How can I tell growth from decay on a graph?

Growth curves rise from left to right; decay curves fall. Algebraically, growth means the base is above 1 and decay means it is between 0 and 1.

What is the domain and range of an exponential function?

For y = aˣ with a > 0, the domain is all real numbers and the range is y > 0.