The graph of the function $y = \frac{1}{x}$ is characterized as a hyperbola consisting of two distinct branches, each located in separate quadrants of the Cartesian coordinate system.

To gain a clearer understanding, let’s analyze the function in more detail. The equation $y = \frac{1}{x}$ defines a reciprocal function, indicating that for every value of $x$, the corresponding $y$ value is the reciprocal of $x$. When this function is plotted, it results in two separate curves that form the hyperbola.

In the first quadrant, where both $x$ and $y$ are positive, the behavior of the function is as follows: as $x$ increases, $y$ decreases, approaching zero but never actually reaching it. Conversely, as $x$ approaches zero from the positive side, $y$ increases rapidly towards positive infinity. This behavior creates a curve that asymptotically approaches both the x-axis and y-axis without ever touching them.

In the third quadrant, where both $x$ and $y$ are negative, the function exhibits similar behavior, but it is mirrored. As $x$ decreases (becoming more negative), $y$ also decreases (becoming more negative), approaching zero from the negative side. As $x$ nears zero from the negative side, $y$ decreases rapidly towards negative infinity. This results in a second curve that also gets infinitely close to the x-axis and y-axis without intersecting them.

These two branches of the hyperbola are asymptotic to both the x-axis and y-axis, meaning they approach these axes infinitely closely but never actually intersect them. This property is a fundamental characteristic of hyperbolas and is essential for identifying the graph of the function $y = \frac{1}{x}$.