What shape does the graph of y = 1/x have, particularly as x approaches zero?

The graph of the function y = 1/x is characterized by its hyperbolic shape. As x approaches zero from the positive side (positive values of x), the value of y increases without bound; that is, it goes towards positive infinity. Conversely, as x approaches zero from the negative side (negative values of x), y decreases without bound, tending towards negative infinity.

This behavior creates two distinct branches of the hyperbola located in the first and third quadrants of the Cartesian plane, and the graph never touches or crosses the axes. This distinct curvature and the absence of intersection with the axes is what classifies the graph as a hyperbola, rather than a straight line or other shapes.

The attributes of the function’s asymptotic behavior—where it approaches the axes but never reaches them—further reinforce the hyperbolic nature of its graph. Thus, the correct interpretation of the graph of y = 1/x reveals a hyperbola, particularly highlighted by its extreme behavior as x nears zero.