y = \sqrt{x^2+1}

A Hyperbola Branch $y = \sqrt{x^2+1}$

$y = \sqrt{x^2+1}$ places $x^2+1$ under a square root. Squaring gives the equation of a hyperbola¹.

y^2 - x^2 = 1

Since a square root is non-negative, this function represents the upper branch of that hyperbola.

Domain and range

The domain is all real numbers
The range is $y \geq 1$
Even function
The minimum point is $(0, 1)$

The radicand is always at least $1$ , so the function is defined everywhere and never dips below $1$ .

Monotonicity

The derivative is $f'(x) = \dfrac{x}{\sqrt{x^2+1}}$ , negative for $x < 0$ and positive for $x > 0$ . Hence the lowest point is the minimum, and the curve increases on either side of it.

Asymptotes

For large $|x|$ we have $\sqrt{x^2+1} \approx |x|$ .

Direction	Asymptote	Limit of $f'$
$x \to +\infty$	$y = x$	$1$
$x \to -\infty$	$y = -x$	$-1$

These are its two slant asymptotes.

Concavity

The second derivative $f''(x) = \dfrac{1}{(x^2+1)^{3/2}}$ is always positive, so the graph is concave up everywhere and has no inflection points.

Relation to the hyperbolic functions

Setting $x = \sinh t$ gives $\sqrt{\sinh^2 t + 1} = \cosh t$ , so the curve admits the parametrisation $(\sinh t, \cosh t)$ , reflecting the identity $\cosh^2 t - \sinh^2 t = 1$ .

Curve	Behaviour for large $x$
$y = \sqrt{x^2+1}$	approaches the line $y = x$
$y = \cosh x$	grows like $\dfrac{e^x}{2}$

Although it resembles the catenary, it is a different curve.

Applications

In relativity, the energy-momentum relation has exactly this form.

E = \sqrt{(pc)^2 + (mc^2)^2}

The expression $\sqrt{x^2 + \varepsilon}$ is also used as a smooth approximation that rounds off the sharp corner of $|x|$ , which is valuable in optimisation.

Hyperbola, Wikipedia

A Hyperbola Branch y=x2+1y = \sqrt{x^2+1}y=x2+1​