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Combining Transformations (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

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 $g (x) = - f (2 x + 3) - 1$
- 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 is the factored form of a combined transformation?

The factored form of a combined transformation is
- $a f (b (x + h)) + k$
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 $|a|$
  - If $a < 0$ this also causes a reflection over the $x$ -axis
- $b$ causes a horizontal dilation with a scale factor of $\frac{1}{|b|}$
  - If $b < 0$ this also causes a reflection over the $y$ -axis
- $h$ causes a horizontal translation by $|h|$ 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 $|k|$ 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 $a f (b (x + h)) + k$
- 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 $a f (b (x + h)) + k$ proceed as follows
- first the horizontal dilation (and possibly reflection)
  - This takes $f (x)$ to $f (b x)$
- followed by the horizontal translation
  - This takes $f (b x)$ to $f (b (x + h))$
- then the vertical dilation (and possibly reflection)
  - This takes $f (b (x + h))$ to $a f (b (x + h))$
- and finally the vertical translation
  - This takes $a f (b (x + h))$ to $a f (b (x + h)) + k$
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 $g (x) = - f (2 x + 3) - 1$
Start by rewriting in factored form
- $g (x) = - f (2 (x + 1.5)) - 1$
- 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

Graph transformation from y=f(x) to y=-f(2x+3)-1 with dilation of scale factor 1/2 then horizontal translation to the left by 1.5 units. — **Applying the horizontal transformations in a combined transformation**

Moving on to the vertical translations
First comes the vertical reflection caused by $a = - 1$ (the scale factor is $|a| = 1$ , 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 $k = - 1$

Final steps for graph transformation -f(2x+3)-1: reflect y=f(2x+3) in the x-axis to get y=-f(2x+3), then translate y=-f(2x+3) down by 1 unit to get -f(2x+3)-1 — **Applying the vertical transformations in a combined transformation**

Examiner Tips and Tricks

If a combined transformation on the exam is not in factored form, rewrite it in factored form first. This helps makes the different parts of the transformation more clear.

If the question gives you a specific sequence of transformations and asks you to find the values of constants like $a$ , $b$ , $h$ and $k$ , work through each transformation one at a time and match it to the corresponding part of the expression.

Remember that, for a combined transformation in factored form, dilations and reflections are applied before translations

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 $f (x)$ :
- 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 $y$ -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 $x$
- $f (x - 5)$
  - this applies the horizontal translation by 5 to the right
Then multiply the input variable $x$ by $- \frac{1}{2}$
- $f (- \frac{1}{2} x - 5)$
  - $\frac{1}{2}$ applies the horizontal dilation by a factor of $\frac{1}{\frac{1}{2}} = 2$
    - 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)
- $f (- \frac{1}{2} x - 5) + 4$
  - this applies the vertical translation up by 4
Then multiply the output (i.e. the entire function) by 3
- $3 (f (- \frac{1}{2} x - 5) + 4) = 3 f (- \frac{1}{2} x - 5) + 12$
  - this applies the vertical dilation by a scale factor of 3
Note that that transformed function could now be rewritten in factored form
- $3 f (- \frac{1}{2} (x + 10)) + 12$
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!

Worked Example

The table gives values for a polynomial function $f$ at selected values of $x$ .

$x$	$- 6$	$- 3$	0	3	6
$f (x)$	8	$- 4$	5	$- 1$	6

Let $g (x) = a f (b x) + c$ , where $a$ , $b$ , and $c$ are positive constants. In the $x y$ -plane, the graph of $g$ is constructed by applying three transformations to the graph of $f$ in this order: a horizontal dilation by a factor of 3, a vertical dilation by a factor of 2, and a vertical translation by 5 units. What is the value of $g (- 9)$ ?

(A) $- 8$

(B) $- 4$

(D) $1$

Answer:

First, determine the constants from the given transformations:

$b > 0$ and the scale factor of the horizontal dilation is 3

so $\frac{1}{b} = 3 \Rightarrow b = \frac{1}{3}$

$a > 0$ and the scale factor of the vertical dilation is 2

so $a = 2$

A vertical translation by 5 units means 5 is added to the output

So $c = 5$

Putting that all together

$g (x) = 2 f (\frac{1}{3} x) + 5$

Now find $g (- 9)$

$\begin{array}{rcl} g (- 9) & = & 2 f (\frac{1}{3} (- 9)) + 5 \\ = & 2 f (- 3) + 5 \end{array}$

From the table $f (- 3) = - 4$ , so

$\begin{array}{rcl} g (- 9) & = & 2 (- 4) + 5 \\ = & - 8 + 5 \\ = & - 3 \end{array}$

That is answer (C); but it's worth considering the incorrect answers as well

Option (A) is $2 f (- 3) = - 8$
- this applies the vertical dilation but forgets the vertical translation
Option (B) is $f (- 3) = - 4$
- this forgets both the vertical dilation and the vertical translation
Option (D) is $f (- 3) + 5 = 1$
- this applies the vertical translation but forgets the vertical dilation

Worked Example

The functions $f$ and $g$ are defined for all real numbers such that $g (x) = f (3 (x + 5))$ . Which of the following sequences of transformations maps the graph of $f$ to the graph of $g$ in the same $x y$ -plane?

(A) A horizontal dilation of the graph of $f$ by a factor of 3, followed by a horizontal translation of the graph of $f$ by $- 15$ units

(B) A horizontal dilation of the graph of $f$ by a factor of 3, followed by a horizontal translation of the graph of $f$ by 15 units

(C) A horizontal dilation of the graph of $f$ by a factor of $\frac{1}{3}$ , followed by a horizontal translation of the graph of $f$ by 5 units

(D) A horizontal dilation of the graph of $f$ by a factor of $\frac{1}{3}$ , followed by a horizontal translation of the graph of $f$ by $- 5$ units

Answer:

The expression $g (x) = f (3 (x + 5))$ is in factored form

Reading the transformations from this form:

The factor of 3 multiplying the input means a horizontal dilation by a factor of $\frac{1}{3}$
The $+ 5$ inside the inner brackets means a horizontal translation by $- 5$ units (i.e. 5 units to the left)

With the expression in factored form, the correct order is: dilate first, then translate

You can verify this

Starting from $f (x)$ , compress horizontally by $\frac{1}{3}$ to get $f (3 x)$
then translate left by 5 to get $f (3 (x + 5))$ ✓

So (D) is the correct answer; but it's worth considering the incorrect answers as well

Options (A) and (B) use a dilation scale factor of 3 instead of $\frac{1}{3}$
- a horizontal dilation by a factor of 3 would mean the multiplier on $x$ is $\frac{1}{3}$ , not 3
Option (C) has the correct dilation but translates in the wrong direction (right instead of left)

(D) A horizontal dilation of the graph of $f$ by a factor of $\frac{1}{3}$ , followed by a horizontal translation of the graph of $f$ by $- 5$ units

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Combining Transformations (College Board AP® Precalculus): Study Guide

Combining transformations

What happens when transformations are combined?

What is the factored form of a combined transformation?

What order should transformations be applied in?

How does this work in practice?

Examiner Tips and Tricks

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

Worked Example

Worked Example

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Author: Roger B

Reviewer: Mark Curtis