By completing the square we get $f(x) = \sqrt{ (2x + 1/4)^2 + 95/16}$.
This means (after squaring both sides and taking $(2x + 1/4)^2$ to the left hand side and factoring) that $( f(x) - (2x + 1/4) ) ( f(x) + (2x + 1/4) ) = 95/16$ and hence $f(x) - (2x + 1/4) = \frac{95/16}{f(x) + (2x + 1/4)}.$
But $f(x) + (2x + 1/4) \rightarrow \infty$ as $x \rightarrow \infty$. This implies that $f(x) - (2x + 1/4) \rightarrow 0$ as $x \rightarrow \infty$.
Remark: Likewise $f(x) + (2x + 1/4) = \frac{95/16}{f(x) - (2x + 1/4) }$ and hence as $x \rightarrow -\infty, f(x) - (2x + 1/4) \rightarrow \infty$ and hence $\frac {95/16}{f(x) - (2x + 1/4)} \rightarrow 0.$ This implies that $f(x) + (2x + 1/4) \rightarrow 0$ and thus $y = -(2x + 1/4)$ is another asymptote.