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Graph of x² (the parabola y = x²)

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The graph of y = x² is the simplest parabola: a symmetric U with its lowest point, the vertex, at the origin. Every quadratic function is this same shape, stretched, flipped or moved.

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The graph of
y=x2
 for −5 ≤ x ≤ 5. Dots mark its vertex.

Key features

Domain
All real numbers
Range
y ≥ 0
Vertex
(0, 0), a minimum
Axis of symmetry
x = 0
Opens
Upward
Symmetry
Even: (−x)² = x²
Focus and directrix
Focus (0, 1/4), directrix y = −1/4
y-intercept
None
Derivative
2x
Antiderivative
13x3+C

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

What the graph of x² looks like

Squaring makes negative and positive inputs give the same output, so the graph is a mirror image of itself across the y-axis: (−2, 4) matches (2, 4). The only point where the output is zero is the vertex, (0, 0).

The parabola gets steeper as it climbs. Its slope is 2x: zero at the vertex, 2 at x = 1, 4 at x = 2. The heights 1, 4, 9, 16 grow by the odd numbers 3, 5, 7, which is a quick check when plotting.

Geometrically, a parabola is the set of points the same distance from a fixed point (the focus) and a fixed line (the directrix). For y = x² those are (0, 1/4) and y = −1/4, which is why parabolic dishes focus incoming parallel rays.

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)
x = 0
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 x²
xy = x²
−39
−24
−11
00
11
24
39

How to graph x² by hand

  1. 1
    Plot the vertex (0, 0).
  2. 2
    Plot (1, 1) and (−1, 1), then (2, 4) and (−2, 4), then (3, 9) and (−3, 9).
  3. 3
    Draw a smooth U through the points. The bottom should be rounded, not a V.

Transformations

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

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  • y = (x - 2)^2 + 1 Vertex form a(x − h)² + k with h = 2, k = 1: the vertex moves to (2, 1).
  • y = -0.5x^2 a = −0.5: flipped upside down and widened.
  • y = 3x^2 a = 3: narrower, climbing three times as fast.

Common mistakes

  • Drawing a V with a sharp point at the vertex. A parabola is smooth everywhere.
  • Shifting the wrong way: (x − 2)² moves the graph right by 2, not left.

Where it shows up

  • Projectile motion: height against time under gravity is a downward parabola.
  • Area: the area of a square of side x is x².
  • Parabolic reflectors in satellite dishes, car headlights and telescopes.

Frequently asked questions

What is the vertex of y = x²?
The origin, (0, 0). It is the minimum point of the graph.
What is the range of x²?
y ≥ 0. Squares of real numbers are never negative.
What is the axis of symmetry of y = x²?
The y-axis, x = 0.
How do I graph y = a(x − h)² + k?
Start from y = x², stretch it by a (flipping it if a is negative), then move the vertex to (h, k). See the vertex form guide for worked examples.

Related graphs

Graph of x³ (the cubic y = x³)

An S-shape through the origin, flat at its inflection point.

Square root graph: y = √x

Starts at the origin; half of a sideways parabola.

Absolute value graph: y = |x|

A V with its vertex at the origin, slopes ±1.

Graph of a linear function: y = mx + b

A straight line: slope m, y-intercept b.

Graph of 1/x²

Always positive; both branches above the x-axis.

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

The bell curve: peak (0, 1), area √π.

Graph x² your way

Change the window, add sliders for a, b, h and k, trace the curve with your keyboard, or overlay its derivative.

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