Parent function of $\sqrt{x^2 - 4}$?

Question

Does this particular function($\sqrt{x^2 - 4}$) have a parent, such that it can be represented as a translation, compression, rotation, stretching, etc, of the parent graph?

Answer 1

$y=\sqrt{x^2-4}$ is the top half of the two branches of the hyperbola $x^2-y^2 = 4$, with standard form $\dfrac{x^2}{2^2} - \dfrac{y^2}{2^2}= 1$. It could be thought of as a vertical and horizontal stretch by a factor of $2$ from the "parent" hyperbola $x^2-y^2 = 1$. In that case, the parent function is $f(x)=\sqrt{x^2-1}$, and your function is $g(x)=\sqrt{x^2-4}$, with $g(x)=2f(x/2)$.

Answer 2

Hint: Start from the curve $y=\sqrt{x^2-1}$. Multiply by $2$. We get $2\sqrt{x^2-1}$, which is $\sqrt{4x^2-4}$. Now replace $2x$ by $x$.

So here we can consider $y=\sqrt{x^2-1}$ as the mother curve. For some purposes, this has advantages.