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

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

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

The effects are similar to the transformation y = |f(x)|

• The parts below the x-axis are reflected

• The vertical distance between a point and the x-axis is squared

  • This has the effect of smoothing the curve at the x-axis

 is never below the x-axis

The x-coordinates stay the same

The y-coordinates change

• Their values are squared

The coordinates (x, y) become (x, y²)

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

Sketch the square 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)

  •   is a point on 

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

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

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

  • The square graph  has a y-­intercept at 

• If y = f(x) has a root at (a, 0)

  • The square graph   has a root and turning point at (a, 0)

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

  • The square graph  has a vertical asymptote at 

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

  • The square graph   has a local maximum at (x₁, y₁²) if y₁ > 0

  • The square graph   has a local minimum at (x₁, y₁²) if y₁ ≤ 0

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

  • The square graph   has a local minimum at (x₁, y₁²) if y₁ ≥ 0

  • The square graph   has a local maximum at (x₁, y₁²) if y₁ < 0