The gentle curve rising from zero — the inverse of squaring, used in distances, physics, and statistics.
The square root function f(x) = √x returns the non-negative value whose square equals x. It is the inverse of the squaring function (x²) restricted to non-negative values.
Because you cannot take the square root of a negative number in the real number system, the domain is restricted to x ≥ 0. The graph starts at the origin (0, 0) and curves upward to the right, rising steeply at first and then more and more gradually — it grows, but at a decreasing rate.
The square root function is a specific case of the power function f(x) = x^(1/2).
f(x) = a√(x - h) + k
Vertical stretch/compression and reflection (a < 0 flips it)
Horizontal shift — moves the starting point left or right
Vertical shift — moves the starting point up or down
Domain
[0, +∞) — only defined for non-negative x
Range
[0, +∞) — always non-negative output
Starting Point
(0, 0) for basic form; (h, k) for transformed form
Inverse
Inverse of f(x) = x² (for x ≥ 0)
Growth Rate
Increasing but decelerating — grows slower and slower
Concavity
Concave down — the curve bends downward throughout
f(x) = √xBasic square root: starts at origin, rises then flattens.
f(x) = √(x - 4)Shifted right 4 units — starts at (4, 0).
f(x) = 2√xVertically stretched — rises twice as fast.
Explore how shifting and stretching transforms the curve
In the real number system, there is no real number whose square is negative. For example, there is no real y such that y² = -4. Therefore, √x is only defined for x ≥ 0. (In the complex number system, √(-1) = i, but that is a different context.)