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

Square transformations

What is a square transformation?

• For the graph $y=f\left(x\right)$ the square transformation is $y={\left[f\left(x\right)\right]}^2$

• It transforms points on the graph $y=f\left(x\right)$

  • by keeping the $x$-coordinates the same

  • but squaring their $y$-coordinates

    • increasing them from a height of $y$ to a height of $y^2$

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

  • transform to being above the $x$-axis

    • since $y^2$ is always positive

• Points that have $y$-coordinates of $0$ 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 $y=f\left(x\right)$

    • transform to different key features on the graph of $y={\left[f\left(x\right)\right]}^2$

• In general

  • any points on $y=f\left(x\right)$ with heights that satisfy $\left|y\right|>1$

    • move further away from the $x$-axis on $y={\left[f\left(x\right)\right]}^2$

  • any points on $y=f\left(x\right)$with heights that satisfy $\left|y\right|<1$

    • move closer to the the $x$-axis on $y={\left[f\left(x\right)\right]}^2$

  • where negative heights are reflected in the $x$-axis then

    • increased if $\left|y\right|>1$

    • or decreased if $\left|y\right|<1$

• More specifically

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

    • $y={\left[f\left(x\right)\right]}^2$ has a $y$-­intercept at $(0,c^2)$

  • If $y=f\left(x\right)$ has an $x$-intercept (root)at $\left(a,0\right)$

    • $y={\left[f\left(x\right)\right]}^2$ has a root and minimum point at $\left(a,0\right)$

  • If $y=f\left(x\right)$ has a vertical asymptote at $x=a$

    • $y={\left[f\left(x\right)\right]}^2$ has a vertical asymptote at $x=a$

  • If $y=f\left(x\right)$ has a local maximum at $\left(x_1,y_1\right)$

    •  $y={\left[f\left(x\right)\right]}^2$ has a local maximum at $\left(x_1,{y_1}^2\right)$ if $y_1>0$

    • $y={\left[f\left(x\right)\right]}^2$  has a local minimum at $\left(x_1,{y_1}^2\right)$ if $y_1\le 0$

  • If $y=f\left(x\right)$ has a local minimum at $\left(x_1,y_1\right)$

    • $y={\left[f\left(x\right)\right]}^2$  has a local minimum at $\left(x_1,{y_1}^2\right)$ if $y_1\ge 0$

    •  $y={\left[f\left(x\right)\right]}^2$ has a local maximum at $\left(x_1,{y_1}^2\right)$ if $y_1<0$

What happens to x-intercepts under a square transformation?

• The square transformation affects $x$-intercepts in different ways

  • If the graph $y=f\left(x\right)$ touches the $x$-axis via a turning point

    • the turning point becomes flatter / more bucket-like on $y={\left[f\left(x\right)\right]}^2$

  • e.g. compare $y=x^2$ to $y=x^4$

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

    • e.g. the point $\left(0.1,0.01\right)$ becomes $\left(0.1,0.0001\right)$ (i.e. flattened)

• If the graph of $y=f\left(x\right)$ cuts the $x$-axis

  • the $x$-intercept turns into a smooth minimum point

    • e.g. compare $y=x-1$ to $y={\left(x-1\right)}^2$