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:

$$(1,k)\text{and}(-1,-k)$$

These points are useful in sketching the graph.