Lecture Notes: Reciprocal Graphs

Introduction to Reciprocal Graphs

• A reciprocal graph is in the form ( y = \frac{k}{x} ) where ( k ) is a constant (e.g., 5, 7, -2).

Plotting ( y = \frac{1}{x} )

• Create an XY table with chosen values for x:
  • ( x = 10, y = 0.1 )
  • ( x = 5, y = 0.2 )
  • ( x = 2, y = 0.5 )
  • ( x = 1, y = 1 )
  • ( x = 0.5, y = 2 )
  • ( x = 0.1, y = 10 )
  • ( x = 0, y ) is undefined (division by zero is a math error)
• Plot these points to form the graph.
• Characteristics:
  • Approaches but never touches the x-axis or y-axis (known as asymptotes).

Reciprocal Graph Characteristics

• Asymptotes: Lines the graph approaches but never reaches.
  • X-axis (( y = 0 )) and Y-axis (( x = 0 )) are asymptotes.

Plotting Negative Fractions

• For negative ( x ):
  • ( x = -0.1, y = -10 )
  • ( x = -0.5, y = -2 )
  • ( x = -1, y = -1 )
  • ( x = -2, y = -0.5 )
  • ( x = -5, y = -0.2 )
  • ( x = -10, y = -0.1 )
• Resulting graph is in the opposite quadrants compared to positive fractions.

Plotting ( y = \frac{k}{x} ) for other values of ( k )

• ( y = \frac{2}{x} ), ( y = \frac{4}{x} ):
  • Graphs move further out as ( k ) increases.
• Sketching multiple graphs on the same diagram:
  • Demonstrate how larger ( k ) values result in graphs that are further out from the origin.

Reciprocal Graph for Negative ( y = \frac{-k}{x} )

• Use an XY table for negative values.
• Graphs appear in alternate quadrants due to negative values.
• Example: ( y = \frac{-1}{x}, y = \frac{-2}{x} ).

Recap of Reciprocal Graph Features

• Form: ( y = \frac{k}{x} )
• Positive ( k ): Graphs in quadrants I and III.
• Negative ( k ): Graphs in quadrants II and IV.
• Asymptotes: Graph approaches x-axis and y-axis but never touches them.
• Larger ( k ) results in graphs further from the origin.