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The Square Root Function

Simple Explanation

The basic square root function, y = √x, traces exactly one half of a sideways parabola, starting at the origin and extending only to the right, since negative numbers have no real square root — its domain is restricted to x ≥ 0.

Why Do We Need It?

Recognizing this restricted-domain shape is essential — unlike most other elementary functions, a square root function simply does not exist for part of the number line.

See It

Graph of y = √x

Formula

General Square Root Function

y = a√(x − h) + k (x ≥ h)

A square root function traces half of a sideways parabola, starting at a fixed point (h, k) and extending in only one horizontal direction, since the domain is restricted to x ≥ h.

a: — scales the curve, and flips it if negative
h, k: — the coordinates of the curve's starting point

When to use it: Whenever you need to recognize or sketch the shape of a square root function, and identify its domain restriction.

Worked Example

Find the domain and evaluate a square root function

For y = √(x − 3), state the domain and find y when x = 12.

Why Does This Work?

The square root of a negative number is not a real number, so any x-value that would make the expression under the root negative simply has no valid y-value — this forces the domain restriction, x ≥ h.

Real-Life Example

Pendulum period versus length

A pendulum's swing period depends on the square root of its length: T = 2π√(L/g).

Since length L cannot be negative, this formula naturally has the same domain restriction (L ≥ 0) as the basic square root function.

Practice

What is the domain of y = √(x + 5)?

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Common mistake

Assuming a square root function is defined everywhere, like most other elementary functions — always check its domain restriction first.

Quick Review

y = a√(x−h)+k starts at (h,k) and extends in one direction only.
Domain: x ≥ h (the expression under the root cannot be negative).
Shaped like half of a sideways parabola.