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

Square Transformations

What effects do square transformations have on the graphs?

• 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

How do I sketch the graph of the square of a function: y = [f(x)]²?

• 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