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.
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.
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.
