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

Square transformations

What is a square transformation?

• For the graph the square transformation is

• It transforms points on the graph

  • by keeping the -coordinates the same

  • but squaring their -coordinates

    • increasing them from a height of to a height of

• Any points below the -axis (where is negative)

  • transform to being above the -axis

    • since is always positive

• Points that have -coordinates of or 1 stay in the same position

How do I sketch a square transformation y = [f(x)]²?

• To sketch a square transformation, you need to know

  • how key features on the original graph

    • transform to different key features on the graph of

• In general

  • any points on with heights that satisfy

    • move further away from the -axis on

  • any points on with heights that satisfy

    • move closer to the the -axis on

  • where negative heights are reflected in the -axis then

    • increased if

    • or decreased if

• More specifically

  • If has a -intercept at

    •  has a -­intercept at 

  • If has an -intercept (root)at

    •  has a root and minimum point at

  • If has a vertical asymptote at 

    •  has a vertical asymptote at 

  • If has a local maximum at

    •   has a local maximum at if

    •   has a local minimum at if

  • If has a local minimum at

    •   has a local minimum at if

    •   has a local maximum at if

What happens to x-intercepts under a square transformation?

• The square transformation affects -intercepts in different ways

  • If the graph touches the -axis via a turning point

    • the turning point becomes flatter / more bucket-like on

  • e.g. compare to

    • Small heights less than 1 around the turning point get smaller when squared (not bigger)

    • e.g. the point becomes (i.e. flattened)

• If the graph of cuts the -axis

  • the -intercept turns into a smooth minimum point

    • e.g. compare to