How to linearize a hyperbola

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If a hyperbola is given by

$$\frac{y^2}{a^2}-\frac{x^2}{b^2} = 1$$

rewriting it as a function of x we have that

$$y(x) = a \sqrt{1+\frac{x^2}{b^2}}$$

is there a function $f(y)$ for which when I use it I will get a linear function on the graph $f(y)\space vs\space x$?

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