Name the transformation

Describe it or find its constants

In 7 of 20 recent exams
Paper 1 or 2
2 to 6 marks

How it is set in the exam

Paper 1 (no calculator) mostly (6 of 7 exam versions), Paper 2 (calculator) in 1; 2–6 marks, usually the last part of a question about a rational, log or exponential function, after its features or its inverse. Common (7 of 20 exam versions). In 4 of them the function must be rewritten before the transformations show (the log laws, a common base, $p + \frac{q}{x - h}$ , or a composite), and the rewriting can carry most of the marks. Two of the seven are a pair of sine curves: name the translation, or find the constant in $pf(x) + q$ by matching the second curve's formula or one of its values (a greatest or least value).

Spot it in the exam

Exam phrases to spot (4)

“describe a sequence of transformations that maps the graph of $f$ onto the graph of $g$ ”
“describe fully the single transformation”
“ $g(x) = f(kx) + c$ : find the values of $k$ and $c$ ”
“the graph of $g$ is obtained from the graph of $f$ by a vertical stretch followed by a vertical translation; find the constant”

In disguise (5)

a log of a product, a quotient or a power of $x$ : the log laws turn it into $A + B\log x$ , and only then do the transformations show
an exponential with a different base: $8^x$ against $2^x$ needs a common base first ( $8^x = 2^{3x}$ )
$\frac{ax + b}{x + h}$ against $y = \frac{1}{x}$ : divide first, to $p + \frac{q}{x + h}$
a composite $h(f(x))$ , where $h$ adds a constant, asked for as a transformation
a given curve stretched vertically and then translated, with one of the two sizes unknown and fixed by one known value of the new curve (in the exams, the greatest or least value of a transformed sinusoid)

Variants

from = $\frac{1}{x}$ , an exponential with another base, a log, a general $f$ given by graphs or points, or a sinusoid; asked = describe one, two or three transformations, or find the constants.

Looks like it, but isn't (6)

the transformation is given, and the new formula, graph or range is asked for: Transform a graph
the transformations that map $y = x^2$ onto a given quadratic, or one parabola onto another: complete the square to vertex form, Find the vertex; the words of the description are this guide's (Describing a transformation
a sinusoid's constants fitted to its graph or to a context, with no second curve: Find a trig model's equation (Topic 3)
a bound on a sinusoid's values, such as $g(x) \ge 4$ for all $x$ , that fixes an unknown vertical shift, with the shift or a value such as the least $y$ -intercept asked for (whether $g$ is given by its own formula or as a translation of $f$ ): Read a sinusoid's features from its equation (Topic 3)
the constants of $\frac{ax + b}{cx + d}$ from its asymptotes or from points on its graph, with no transformation asked for: Find a rational function's constants from its features
only the asymptotes, intercepts or range of a rational function are asked for, with no transformation: Rational functions: asymptotes, intercepts, sketch, range

Worked example

From 1/x to a rational function: rewrite, then three transformations

Paper 1 (no calculator)

Question

Let $f(x) = \dfrac{3x + 4}{x + 2}$ , for $x \ne -2$ .
(a) Show that $f(x) = 3 - \dfrac{2}{x + 2}$ . [2]
(b) Hence describe a sequence of three transformations that maps the graph of $y = \frac{1}{x}$ onto the graph of $y = f(x)$ . [4]

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Which method?

The first move is to write $g$ in terms of $f$ . How depends on what is given:

a rational function and $y = \frac{1}{x}$
First move
divide: $\frac{ax + b}{x + h} = p + \frac{q}{x + h}$ (the asymptotes $x = -h$ and $y = p$ confirm it). If the bottom is $cx + d$ with $c \ne 1$ , also divide the fraction's top and bottom by $c$ , so that $x$ stands alone: $\frac{6}{3x + 1} = \frac{2}{x + \frac{1}{3}}$ (then the only horizontal change is the translation).
Example
$\frac{3x + 4}{x + 2} = 3 - \frac{2}{x + 2}$
a log of a product, a quotient or a power, and $y = \log_a x$
First move
the log laws, to $A + B\log_a x$
Example
$\log_3\frac{\sqrt{x}}{27} = \frac{1}{2}\log_3 x - 3$
an exponential with another base, or a shift in the power, and $y = a^x$
First move
a common base and the index laws
Example
$8^{x - 1} = \frac{1}{8} \times 2^{3x}$
a composite $h(f(x))$ with $h$ linear
First move
write it out: $h(x) = x + k$ gives $f(x) + k$
Example
a translation $k$ up
two formulas already in the shape $f(x)$ and $pf(x - a) + b$ (or $pf(qx) + b$ )
First move
read the constants off
Example
$2\cos\left(x + \frac{\pi}{4}\right) + 1$ from $2\cos x$
graphs, or facts about $g$ , with unknown constants
First move
match corresponding features: a turning point, an intercept, an asymptote
Example
Example 2.3

What is given	First move	Example
a rational function and $y = \frac{1}{x}$	divide: $\frac{ax + b}{x + h} = p + \frac{q}{x + h}$ (the asymptotes $x = -h$ and $y = p$ confirm it). If the bottom is $cx + d$ with $c \ne 1$ , also divide the fraction's top and bottom by $c$ , so that $x$ stands alone: $\frac{6}{3x + 1} = \frac{2}{x + \frac{1}{3}}$ (then the only horizontal change is the translation).	$\frac{3x + 4}{x + 2} = 3 - \frac{2}{x + 2}$
a log of a product, a quotient or a power, and $y = \log_a x$	the log laws, to $A + B\log_a x$	$\log_3\frac{\sqrt{x}}{27} = \frac{1}{2}\log_3 x - 3$
an exponential with another base, or a shift in the power, and $y = a^x$	a common base and the index laws	$8^{x - 1} = \frac{1}{8} \times 2^{3x}$
a composite $h(f(x))$ with $h$ linear	write it out: $h(x) = x + k$ gives $f(x) + k$	a translation $k$ up
two formulas already in the shape $f(x)$ and $pf(x - a) + b$ (or $pf(qx) + b$ )	read the constants off	$2\cos\left(x + \frac{\pi}{4}\right) + 1$ from $2\cos x$
graphs, or facts about $g$ , with unknown constants	match corresponding features: a turning point, an intercept, an asymptote	Example 2.3

Steps

Rewrite $g$ in terms of $f$
(the table), in the shape $pf(x - a) + b$ or $pf(qx) + b$ .
Read off the transformations.
Inside the bracket is horizontal and works the opposite way ( $x - a$ is $a$ to the right; $qx$ has scale factor $\frac{1}{q}$ ; $-x$ is a reflection in the $y$ -axis). Outside is vertical and does what it says. A minus sign outside is a reflection in the $x$ -axis.
Describe each one fully
, with its type, direction and size, in an order that works: a vertical stretch or reflection before a vertical translation. If the question names the kinds ("a stretch and a translation"), use exactly those kinds.
Constants:
match a feature of $f$ with its image on $g$ . Horizontal constants come from $x$ -coordinates, vertical ones from $y$ -coordinates: one equation per unknown.
Check
with one point: follow it through your sequence and substitute into $g$ .

Common mistakes

"Move", or half a description
"move" is not accepted. Each transformation needs its type, direction and size: "translation $3$ units down" (or a vector), "reflection in the $x$ -axis", "horizontal stretch with scale factor $\frac{1}{4}$ ". "A translation of $3$ " or "a reflection" alone scores .
A composite in the wrong order
with $h(x) = x + 6$ , $h(f(x)) = f(x) + 6$ is $6$ units up, but $f(h(x)) = f(x + 6)$ is $6$ units left. $(h \circ f)(x)$ means $h(f(x))$ : apply $f$ first.
Scale factor $q$ for $f(qx)$
it is $\frac{1}{q}$ . $f(4x)$ is a horizontal stretch with scale factor $\frac{1}{4}$ .

More mistakes (6)

The wrong sign on a horizontal translation
$\frac{1}{x + 2}$ is $2$ units left.
A vertical translation before a vertical stretch, with the same numbers
$2f(x) + 6$ is "stretch by $2$ , then up $6$ ". With the translation first it would be "up $3$ , then stretch by $2$ ", because $2\left(f(x) + 3\right) = 2f(x) + 6$ (correct, but give the stretch first); "up $6$ , then stretch by $2$ " gives $2f(x) + 12$ .
Stretching only part of $f(x)$
a vertical stretch multiplies the whole formula, its constant term included: $\frac{1}{2}(4\sin x + 8) = 2\sin x + 4$ , not $2\sin x + 8$ . If a constant is fixed by a greatest or least value, stretch that value first (the greatest value $12$ of $4\sin x + 8$ becomes $6$ ), then translate.
Not rewriting first
$\frac{3x + 4}{x + 2}$ shows its transformations only as $3 - \frac{2}{x + 2}$ ; a log needs the log laws; different bases need a common base.
Two stretches when "a stretch and a translation" are asked for
(or the reverse): the maths is right, but the question fixes the kinds, and the final mark is lost.
$k$ upside down
in $g(x) = f(4x)$ a point at $x = 2$ moves to $x = \frac{1}{2}$ , so $k = \frac{\text{old } x}{\text{new } x}$ , not the other way round.

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