Lecture Notes: Reciprocal Graphs

Overview

• Reciprocal graphs are characterized by the equation (y = \frac{a}{x}).
• Common form: (y = \frac{1}{x}).
• The graph is a curve with sections in the top right and bottom left quadrants.

Key Characteristics

• Symmetrical along the lines (y = x) and (y = -x).
• If unsure about the graph, create a table of (x) and (y) values to verify.

Table of Values

• Example values:
  • (x = -2) results in (y = -\frac{1}{2}).
  • (x = 0) leads to (y) being undefined (infinity), hence the curve doesn’t cross the y-axis.
  • (x = 2) results in (y = \frac{1}{2}).

Variations in Reciprocal Graphs

When (a > 1)

• Example: (y = \frac{4}{x}).
• The graph shifts outward, away from the axes.
• Larger (a) values lead to a greater shift.

When (0 < a < 1)

• Example: (y = \frac{0.5}{x}).
• The graph moves inward, towards the axes.

When (a < 0)

• Example: (y = \frac{-1}{x}).
• The graph swaps quadrants to top left and bottom right.

Conclusion

• Understanding different forms of reciprocal graphs helps in recognizing their behavior based on the value of (a).
• This knowledge aids in graph plotting and analysis.