Graph of exponential decay: y = (1/2)ˣ

Updated 20 October 2026 · Values computed by the ihateDesmos engine

Exponential decay is the mirror image of growth. The graph of y = (1/2)ˣ starts high on the left, passes through (0, 1), and halves every time x increases by one, sinking toward the x-axis without ever reaching it.

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The graph of

y=(12)^x

for −4 ≤ x ≤ 6. Dots mark its y-intercept.

Key features

Domain: All real numbers
Range: y > 0
y-intercept: (0, 1)
Horizontal asymptote: y = 0 as x → ∞
Half-life: 1 unit of x
Equivalent forms: (1/2)ˣ = 2⁻ˣ = e^(−x ln 2)
y-intercept: (0, 1)
Derivative: −(12)^xln(2)
Antiderivative: −(12)^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 (1/2)ˣ looks like

Each step to the right halves the height: (0, 1), (1, 0.5), (2, 0.25), (3, 0.125). Each step to the left doubles it, so the curve climbs steeply on the left.

The graph is 2ˣ reflected in the y-axis. Any decaying exponential can be written as e^(−kx) with k > 0, and its half-life, the distance over which it halves, is ln 2 / k.

Decay never reaches zero in finite time, which is why radioactive samples are described by half-lives rather than “times to disappear”.

Points of interest

Found numerically to about twelve digits in the window −4 ≤ x ≤ 6, 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 (1/2)ˣ
x	y = (1/2)ˣ
−3	8
−2	4
−1	2
0	1
1	1/2
2	1/4
3	1/8

How to graph (1/2)ˣ by hand

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

Transformations

How the graph moves when you change the formula. The original is (1/2)ˣ in blue.

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y = (1/3)^x Faster decay: the height divides by 3 at each step.
y = 8(1/2)^x Starts at 8 instead of 1: the y-intercept is the initial amount.
y = (1/2)^x + 2 Settles toward 2 instead of 0, like a cooling object approaching room temperature.

Where it shows up

Radioactive decay with a given half-life.
Drug concentration in the bloodstream after a dose.
Newton’s law of cooling: the temperature difference decays exponentially.

Frequently asked questions

What does an exponential decay graph look like?

A decreasing curve that passes through its starting value at x = 0 and approaches a horizontal asymptote (usually y = 0) on the right without touching it.

How do you find the half-life from a graph?

Read the starting value, then find how far you must move right for the height to fall to half of it. For y = (1/2)ˣ that distance is exactly 1.

Is (1/2)^x the same as 2^(−x)?

Yes. (1/2)ˣ = 1/2ˣ = 2⁻ˣ, so the graph is 2ˣ reflected in the y-axis.

y = aˣ: growth when a > 1, decay when 0 < a < 1.

Graph of e^x

Through (0, 1), always positive; its slope equals its height.

Graph of ln x (natural logarithm)

Defined for x > 0, through (1, 0) and (e, 1).

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

The bell curve: peak (0, 1), area √π.

Graph of 1/x (the reciprocal function)