Reciprocal Transformations (DP IB Analysis & Approaches (AA)): Revision Note

Reciprocal transformations

What is a reciprocal transformation?

• For the graph the reciprocal transformation is

• It transforms points on the graph

  • by changing their -coordinates

    • from a height of to a height of

• All -coordinates stay the same

  • Points that lie on the line y = 1 or the line y = -1 stay the same

How do I sketch a reciprocal transformation y = 1/f(x)?

• To sketch a reciprocal transformation, you need to know

  • how key features on the original graph

    • transform to different key features on the graph of

• In general

  • the larger the heights of points on

    • the closer to the -axis they become on

    • and vice versa

  • If is positive

    • then  is positive

  • If is negative

    • then  is negative

  • If is increasing

    • then  is decreasing

  • If is decreasing

    • then  is increasing

• More specifically

  • If has a -intercept at where

    •  has a -intercept at 

  • If has an -intercept (root)at

    •  has a vertical asymptote at 

  • If has a vertical asymptote at 

    • has a discontinuity at

    • The discontinuity looks like a root when you sketch

  • If has a local maximum at where

    •  has a local minimum at 

  • If has a local minimum at where

    •  has a local maximum at 

  • If has a horizontal asymptote at

    • if then has a horizontal asymptote at

    • if then

  • If as 

    • has a horizontal asymptote at