UKMT Senior Mathematical Challenge, 2016

A square has vertices at $(0, 0)$, $(1, 0)$, $(1, 1)$ and $(0, 1)$. Graphs of the following equations are drawn on the same set of axes as the square. $x^2 + y^2 = 1$, $y = x + 1$, $y = -x^2 + 1$, $y = x$, $y = \frac{1}{x}$. How many of the graphs pass through exactly two of the vertices of the square?

(A) $1$ (B) $2$ (C) $3$ (D) $4$ (E) $5$