Reciprocal graphs

Reciprocal graphs have a hyperbolic shape and asymptotes. They can take the form \(f(x) = \frac{k}{x}\). The graph has two distinct branches: one in the first and third quadrants (for positive \(k\) or the second and fourth quadrants (for negative \(k\)).

Shape of a reciprocal graph

When \(k > 0\), the graph is in the first and third quadrants, with one branch approaching the positive y-axis and the other approaching the negative x-axis.

When \(k < 0\), the graph is in the second and fourth quadrants, with one branch approaching the negative y-axis and the other approaching the positive x-axis.

Asymptotes

Vertical Asymptote: The line \(x = 0\) is a vertical asymptote since the function is undefined at \(x = 0\).

Horizontal Asymptote: As \(x\) increases or decreases without bound, \(f(x)\) approaches 0, so the horizontal asymptote is \(y = 0\).

Key Points

For \(f(x) = \frac{k}{x}\)​, the graph passes through the points:

\[(1, k) \quad \text{and} \quad (-1, -k)\]

These points are useful in sketching the graph.