Lesson on Reciprocal Functions

Introduction

• Discussion on plotting reciprocal functions.
• Example function: ( y = \frac{4}{x} ).

Plotting Reciprocal Function ( y = \frac{4}{x} )

• Coordinates Calculated:
  • ( x = 0.5, y = 8 )
  • ( x = 1, y = 4 )
  • ( x = 2, y = 2 )
  • ( x = 4, y = 1 )
  • ( x = 8, y = 0.5 )
  • ( x = 10, y = 0.4 )
• Graph Characteristics:
  • The curve is smooth.
  • The graph will never touch or cross the y-axis.
    • As ( x \to 0 ), ( y \to \infty ).
    • Division by zero leads to infinity, hence the graph approaches infinity.

Plotting Reciprocal Function ( y = \frac{2}{x} )

• Coordinates Calculated:
  • Negative quadrant:
    • ( x = -4, y = -0.5 )
    • ( x = -2, y = -1 )
    • ( x = -1, y = -2 )
    • ( x = -0.5, y = -4 )
  • Positive quadrant:
    • ( x = 0.5, y = 4 )
    • ( x = 1, y = 2 )
    • ( x = 2, y = 1 )
    • ( x = 4, y = 0.5 )
• Graph Characteristics:
  • Two reciprocal curves: one in the positive quadrant and one in the negative quadrant.
  • The curve approaches the axes infinitely but never touches them.

Conclusion

• Reciprocal functions exhibit asymptotic behavior where the curve approaches but never touches the axes.
• These functions can be plotted by calculating coordinates for given values of ( x ).

Tips for Plotting

• Always consider the behavior as ( x \to 0 ) and ( x \to \infty ).
• Be mindful of the signs of ( x ) and ( y ) as they affect the placement of points on the graph.

Practice

• Suggested to try plotting the graph yourself for better understanding.