# Reciprocal Graphs

## Understanding Reciprocal Graphs

Reciprocal graphs are graphical representations of mathematical functions in the form of y = a/x and y = x/b, where a and b are real constants and x is a variable. These graphs are useful for visualizing relationships that exhibit inverse proportionality, meaning that as one variable increases, the other decreases.

### Asymptotes

When graphing a reciprocal function, it is essential to consider its asymptotes. Asymptotes are lines that the curve approaches but never touches. The y = a/x and y = x/b graphs have asymptotes at x = 0 and y = 0 respectively. A vertical asymptote represents values of x that cannot be divided by zero, while a horizontal asymptote represents values of y that cannot equal zero.

For instance, the graph of y = a/x is symmetric to the lines y = x and y = -x, which is crucial to consider when sketching the graph.

### Types of Reciprocal Graphs

The coordinate plane is divided into four quadrants, labeled with roman numerals (I, II, III, and IV) for visualization purposes.

Reciprocal Functions of the Form y = a/x

If a is a positive value, the graph of y = a/x will be drawn in quadrants I and III. For example, y = 2/x will have a graph like the one shown below. On the other hand, if a is negative, the graph will be drawn in quadrants II and IV, as shown by y = -2/x.

• y = 2/x graph:
• <