Graph of 1/x²

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The graph of y = 1/x² looks like a volcano: two branches, both above the x-axis, shooting up on either side of the y-axis and flattening toward the x-axis far away.

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The graph of

y=1x²

for −5 ≤ x ≤ 5. Dots mark nothing: it has no roots or turning points.

Key features

Domain: x ≠ 0
Range: y > 0
Vertical asymptote: x = 0
Horizontal asymptote: y = 0
Symmetry: Even: symmetric about the y-axis
Monotonic: Increasing for x < 0, decreasing for x > 0
y-intercept: None: x = 0 is an asymptote
Derivative: −2x³
Antiderivative: −1x+C

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

What the graph of 1/x² looks like

Squaring x makes the output positive on both sides, so both branches sit above the x-axis, mirror images of each other in the y-axis.

It falls off much faster than 1/x: at x = 10, 1/x is 0.1 but 1/x² is 0.01. This inverse-square shape is how gravity, light intensity and electric force weaken with distance.

Near zero it rises faster too, which is why the area under 1/x² from 1 to infinity is finite (exactly 1) while the area under 1/x is infinite.

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 1/x²
x	y = 1/x²
−2	1/4
−1	1
−1/2	4
1/2	4
1	1
2	1/4
3	1/9

How to graph 1/x² by hand

1
Draw the asymptotes x = 0 and y = 0.
2
Plot (1, 1), (2, ¼), (½, 4) and their mirror images (−1, 1), (−2, ¼), (−½, 4).
3
Draw two branches above the x-axis, rising along the y-axis and flattening toward the x-axis.

Transformations

How the graph moves when you change the formula. The original is 1/x² in blue.

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y = 1/(x - 1)^2 Shifted right: the asymptote moves to x = 1.
y = 1/x^2 - 1 Shifted down 1: now crosses the x-axis at x = ±1.
y = -1/x^2 Reflected: both branches below the x-axis.

Where it shows up

Newton’s law of gravitation and Coulomb’s law: force proportional to 1/r².
Light and sound intensity falling off with the square of the distance from a point source.