Graph of the Square Root Function $y = \sqrt{x}$

$y = \sqrt{x}$ is the non-negative number whose square is $x$¹. It can also be written $x^{1/2}$, and because it involves a radical it is called an irrational function. Its graph looks like the parabola $y = x^2$ tipped onto its side.

Domain and range

• The domain is $x \geq 0$
• The range is $y \geq 0$
• The function increases monotonically
• The graph lives entirely in the first quadrant

Squaring a real number never gives a negative result, so $\sqrt{x}$ is real only when $x \geq 0$. For negative $x$ there is no real value, and nothing to draw.

Monotonicity and shape

For $x > 0$ the derivative is $y' = \dfrac{1}{2\sqrt{x}}$, which is always positive, so the function increases. The slope shrinks as $x$ grows, however, so the climb becomes ever gentler.

The second derivative $y'' = -\dfrac{1}{4x\sqrt{x}}$ is negative, so the curve is concave down throughout. It is not bounded above: it keeps rising, slowly, without end.

The tangent at the origin

As $x \to 0^{+}$ the derivative tends to $+\infty$, so at the origin the tangent line is vertical, lying along the $y$-axis. The curve leaps out of the origin and immediately begins to level off. The origin is the endpoint of the domain, and the function is not differentiable there.

Notable points

Since $\sqrt{4x} = 2\sqrt{x}$, multiplying $x$ by $4$ only doubles $y$; multiplying $x$ by $100$ multiplies $y$ by just $10$.

Relationships with other functions

It is the inverse of $y = x^2$ restricted to $x \geq 0$, and the two graphs are reflections of each other in the line $y = x$. As the power function $x^p$ with $p = \dfrac{1}{2}$, it has $0 < p < 1$, so it rises steeply near the origin and gently far from it.

Applications

• A square of area $S$ has side $\sqrt{S}$
• Free fall from height $h$ takes time $t = \sqrt{\dfrac{2h}{g}}$
• The distance between two points, $\sqrt{x^2 + y^2}$
• The standard deviation, the square root of the variance

The error of a sample mean shrinks like $\dfrac{1}{\sqrt{n}}$ in the number of observations: halving the error costs four times the data, which is exactly what the flattening of this curve says.

1. Square root, Wikipedia