Combining Transformations (College Board AP® Precalculus): Revision Note

Combining transformations

What happens when transformations are combined?

• The previous two study guides covered individual transformations

  • translations and dilations (including reflections)

• In practice, functions are often transformed by several of these different transformations at once

• E.g. the function $g\left(x\right)=-f(2x+3)-1$

  • This involves

    • a horizontal translation

    • a horizontal compression

    • a reflection over the $x$-axis

    • and a vertical translation

  • all applied to $f$

• To handle combined transformations correctly, you need to

  • identify the individual transformations

  • and apply them in the correct order

What are the preimage and image of a transformation?

• Two terms describe the sets of points involved when transformations are applied

• The preimage of the transformations is the set of points before the transformations are applied

  • I.e. the points on the graph of the original function

• The image of the transformations is the set of points after the transformations are applied

  • I.e. the points on the graph of the transformed function

• E.g. for the combined transformation $g\left(x\right)=-f(2x+3)-1$

  • the preimage is the set of points on the graph of $f$

  • the image is the set of points on the graph of $g$

What is the factored form of a combined transformation?

• The factored form of a combined transformation is

  • $af\left(b\left(x+h\right)\right)+k$

• When a transformation is written in that form, the constants $a$, $b$, $h$ and $k$ produce the same effects that you saw in the study guides for the individual transformations

  • $a$ causes a vertical dilation with a scale factor of $\left|a\right|$

    • If $a<0$ this also causes a reflection over the $\mathit{x}$-axis

  • $b$ causes a horizontal dilation with a scale factor of $\frac{1}{\left|b\right|}$

    • If $b<0$ this also causes a reflection over the $\mathit{y}$-axis

  • $h$ causes a horizontal translation by $\left|h\right|$ units

    • If $h>0$ it is a translation to the left

    • If $h<0$ it is a translation to the right

  • $k$ causes a vertical translation by $\left|k\right|$ units

    • If $k>0$ it is a translation up

    • If $k<0$ it is a translation down

What order should transformations be applied in?

• Make sure your combined transformation is written in factored form $af\left(b\left(x+h\right)\right)+k$

  • Rewrite the expression if need be

• Separate the transformations into two groups

  • Horizontal transformations

    • caused by $b$ and $h$

    • these come from changes to the input (inside the function)

  • Vertical transformations

    • caused by $a$ and $k$

    • these come from changes to the output (outside the function)

• Horizontal and vertical transformations don't affect each other, so you can do either group first

• Within each group, however, the order does matter

  • Dilations and reflections are applied first

  • followed by translations

• So if you start with the horizontal transformations, the effects of $af\left(b\left(x+h\right)\right)+k$ proceed as follows

  • first the horizontal dilation (and possibly reflection)

    • This takes $f\left(x\right)$ to $f\left(bx\right)$

  • followed by the horizontal translation

    • This takes $f\left(bx\right)$ to $f\left(b\left(x+h\right)\right)$

  • then the vertical dilation (and possibly reflection)

    • This takes $f\left(b\left(x+h\right)\right)$ to $af\left(b\left(x+h\right)\right)$

  • and finally the vertical translation

    • This takes $af\left(b\left(x+h\right)\right)$ to $af\left(b\left(x+h\right)\right)+k$

• If you start with vertical transformations first, the end result will be the same

  • as long as you do the dilations and reflections before the translations in each case

How does this work in practice?

• Consider the example $g\left(x\right)=-f(2x+3)-1$

• Start by rewriting in factored form

  • $g\left(x\right)=-f\left(2\left(x+1.5\right)\right)-1$

  • That is in factored form with

    • $a=-1$

    • $b=2$

    • $h=1.5$

    • $k=-1$

• Starting with the horizontal transformations

• First comes the horizontal dilation caused by $b=2$, with scale factor $\frac{1}{2}$

  • next comes the horizontal translation caused by $h=1.5$, which translates by $1.5$ units to the left

• Moving on to the vertical translations

• First comes the vertical reflection caused by $a=-1$ (the scale factor is $\left|a\right|=1$, so this doesn't affect the scale of the graph, but the negative sign causes a reflection over the $x$-axis)

  • and finally the vertical translation caused by $k=-1$

What if I need to apply transformations in a different order?

• As long as dilations and reflections occur before translations

  • you can use the information above to write down a transformation in factored form

• Similarly, a given combined transformation can always be rewritten in factored form

  • allowing you to describe the transformation with dilations and reflections preceding translations

• But what if you need to find the analytical form of a transformation where translations are to be applied before dilations/reflections?

• Any transformation can be created simply by

  • applying the appropriate individual transformations

  • in the specified order

    • Remember that horizontal and vertical transformations are independent of each other

    • So it is only within each category (horizontal or vertical) that the order makes a difference

• E.g., the following transformations are to be applied to a function $f\left(x\right)$:

  • A horizontal translation by 5 units to the right

    • followed by a horizontal dilation with a scale factor of 2 and a reflection over the $\mathit{y}$-axis

  • And a vertical translation by 4 units up

    • followed by a vertical dilation with a scale factor of 3

• Start by subtracting 5 from the input variable $x$

  • $f\left(x-5\right)$

    • this applies the horizontal translation by 5 to the right

• Then multiply the input variable $x$ by $-\frac{\mathbf{1}}{\mathbf{2}}$

  • $f\left(-\frac{1}{2}x-5\right)$

    • $\frac{1}{2}$ applies the horizontal dilation by a factor of $\frac{1}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}=2$

      • and the minus sign gives the reflection over the $y$-axis

    • Note that this doesn't affect the $-5$ at all

• Add 4 to the output (i.e., to the entire function)

  • $f\left(-\frac{1}{2}x-5\right)+4$

    • this applies the vertical translation up by 4

• Then multiply the output (i.e. the entire function) by 3

  • $3\left(f\left(-\frac{1}{2}x-5\right)+4\right)=3f\left(-\frac{1}{2}x-5\right)+12$

    • this applies the vertical dilation by a scale factor of 3

• Note that that transformed function could now be rewritten in factored form

  • $3f\left(-\frac{1}{2}\left(x+10\right)\right)+12$

• This shows that the combined transformation described above is equivalent to

  • a horizontal dilation by scale factor 2 with a reflection over the $y$-axis

    • followed by a horizontal translation 10 units to the left

  • and a vertical dilation by scale factor 3

    • followed by a vertical translation up by 12 units

  • This is just another way of describing the same transformation!