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

Reciprocal Transformations

What effects do reciprocal transformations have on the graphs?

• The x-coordinates stay the same

• The y-coordinates change

  • Their values become their reciprocals

• The coordinates (x, y) become  where y ≠ 0

  • If y = 0 then a vertical asymptote goes through the original coordinate

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

How do I sketch the graph of the reciprocal of a function: y = 1/f(x)?

• Sketch the reciprocal transformation by considering the different features of the original graph

• Consider key points on the original graph

  • If (x₁, y₁) is a point on y = f(x) where y₁ ≠ 0

    • is a point on 

    • If |y₁| < 1 then the point gets further away from the x-axis

    • If |y₁| > 1 then the point gets closer to the x-axis

  • If y = f(x) has a y-intercept at (0, c) where c ≠ 0

    • The reciprocal graph  has a y-­intercept at 

  • If y = f(x) has a root at (a, 0)

    • The reciprocal graph  has a vertical asymptote at 

  • If y = f(x) has a vertical asymptote at 

    • The reciprocal graph has a discontinuity at (a, 0) 

    • The discontinuity will look like a root

  • If y = f(x) has a local maximum at (x₁, y₁) where y₁ ≠ 0

    • The reciprocal graph  has a local minimum at 

  • If y = f(x) has a local minimum at (x₁, y₁) where y₁ ≠ 0

    • The reciprocal graph  has a local maximum at 

• Consider key regions on the original graph

  • If y = f(x) is positive then  is positive

    • If y = f(x) is negative then  is negative

  • If y = f(x) is increasing then  is decreasing

    • If y = f(x) is decreasing then  is increasing

  • If y = f(x) has a horizontal asymptote at y = k 

    • has a horizontal asymptote at  if k ≠ 0

    •  tends to ± ∞ if k = 0

  • If y = f(x) tends to ± ∞ as x tends to +∞ or -∞

    • has a horizontal asymptote at