Composite Transformations of Graphs (DP IB Analysis & Approaches (AA): HL): Revision Note

Composite transformations of graphs

What transformations do I need to know?

• $y=f(x-a)$

  • is horizontal translation by vector $\left(\begin{array}{c}a\\ 0\end{array}\right)$

    • If $a$ is positive then the graph moves right

    • If $a$ is negative then the graph moves left

• $y=f(x+a)$

  • is horizontal translation by vector $\left(\begin{array}{c}-a\\ 0\end{array}\right)$

    • If $a$ is positive then the graph moves left

    • If $a$ is negative then the graph moves right

• $y=f\left(x\right)+a$

  • is vertical translation by vector $\left(\begin{array}{c}0\\ a\end{array}\right)$

    • If $a$ is positive then the graph moves up

    • If $a$ is negative then the graph moves down

• $y=f\left(ax\right)$

  • is a horizontal stretch by scale factor $\frac{1}{a}$ parallel to the $x$-axis

    • If $a>1$ then the graph gets closer to the $y$-axis

    • If $0<a<1$ then the graph gets further from the $y$-axis

• $y=af\left(x\right)$

  • is a vertical stretch by scale factor $a$ parallel to the $y$-axis

    • If $a>1$ then the graph gets further from the $x$-axis

    • If $0<a<1$ then the graph gets closer to the $x$-axis

• $y=f(-x)$

  • is a horizontal reflection about the $y$-axis

• $y=-f\left(x\right)$

  • is a vertical reflection about the $x$-axis

Does the order of transformations matter?

• The order of applying transformations does matter

  • In general

    • different horizontal transformations need to be applied in order

    • different vertical transformations need to be applied in order

  • but horizontal and vertical transformations can swap orders

    • They are independent of each other

• e.g. if there are two horizontal transformations H₁ then H₂ and two vertical transformations V₁ then V₂

  • then the following orders are all acceptable

    •  Horizontal then vertical: H₁ H₂ V₁ V₂

    • Vertical then horizontal: V₁ V₂ H₁ H₂

    • Mixed up (but H₁ before H₂ and V₁ before V₂):

    • H₁ V₁ H₂ V₂

    • H₁ V₁ V₂ H₂

    • V₁ H₁ V₂ H₂

    • V₁ H₁ H₂ V₂

Composite vertical transformations af(x)+b

What does the transformation af(x)+b represent?

• The transformation $af\left(x\right)+b$ represents, in order:

  • a vertical stretch of $y=f\left(x\right)$ by scale factor $a$

    • $y=af\left(x\right)$

  • followed by a translation of $\left(\begin{array}{c}0\\ b\end{array}\right)$ 

    • $y=\left[af\right(x\left)\right]+b$

    • giving $y=af\left(x\right)+b$

• It is not a translation of $\left(\begin{array}{c}0\\ b\end{array}\right)$ followed by a vertical stretch by scale factor $a$

  • because that would give

    • translation: $y=f\left(x\right)+b$

    • then stretch: $y=a\left[f\right(x)+b]$

    • final equation:$y=af\left(x\right)+ab$

• If $a$ is negative, then there is also a reflection in the $x$-axis

  • either before or after the vertical stretch

Composite horizontal transformations f(ax+b)

What does the transformation f(ax+b) represent?

• The transformation $f\left(ax+b\right)$ represents, in order:

  • a translation of $y=f\left(x\right)$ by $\left(\begin{array}{c}-b\\ 0\end{array}\right)$

    • $y=f\left(x+b\right)$

  • followed by a horizontal stretch by scale factor $\frac{1}{a}$

    • $y=f\left(\left(ax\right)+b\right)$

    • giving $y=f\left(ax+b\right)$

• It is not a horizontal stretch by scale factor $\frac{1}{a}$ followed by a translation of $\left(\begin{array}{c}-b\\ 0\end{array}\right)$

  • because that would give

    • stretch: $y=f\left(ax\right)$

    • then translation: $y=f\left(a\right(x+b\left)\right)$

    • final equation: $y=f(ax+ab)$

• If $a$ is negative, then there is also a reflection in the $y$-axis

  • either before or after the horizontal stretch