Transformations: Part A — Core transformations and equations of curves

Remark: Initially, I had only planned to discuss the order of transformations, but it eventually made more sense to cover the whole topic properly — which is why this article took ages to write! Another reason for covering the whole topic is that most of the material is highly relevant to the TMUA, while some parts involve subtle conceptual points and useful ideas that are not particularly well known. My aim is not just to give you the results and tools, but to help you understand why they work.

I have eventually decided to cover this topic in two parts, since it is a rather long topic, but a very relevant and important one. Part A — this article — covers the standard vertical and horizontal transformations, including reflections in the lines $x=a$ and $y=b$, establishes the three principles that allow transformations to be combined in any valid order, and ends with a general first-principles method for transforming the equation of any curve. Part B then treats the two modulus transformations, develops a direct method for finding corresponding points without drawing every intermediate graph, and finishes with a bonus section on rotating a curve through $90^\circ$ about an arbitrary point.

What you will get from Part A. A secure understanding of what a transformation actually does to a curve, why the order in which transformations are applied matters, and a method that produces the transformed equation even when the curve is not written as $y=f(x)$.

TMUA Relevance Score: 10/10

Suppose the graph of $y=f(x)$ contains the point $(p,q)$, so $f(p)=q$. Every standard function transformation changes either the output of the function or the input given to the function:

• A vertical transformation changes the output, so it changes the $y$-coordinate of each point.

• A horizontal transformation changes the input, so it changes the $x$-coordinate at which the same function value appears.

This distinction is the central idea behind the entire topic.

Vertical transformations

There are three basic vertical transformations. In every case, the $x$-coordinate is unchanged and the transformation acts on the $y$-coordinate.

Vertical translations

Consider the specific transformation from $y=f(x)$ to $y=f(x)+2$. Some corresponding points are shown below.

What association can you spot between the points in the two rows?

Every $y$-coordinate has increased by $2$, while every $x$-coordinate is unchanged. The whole graph is therefore translated vertically by $+2$ units. This example shows how examining corresponding points reveals the effect of a transformation. We can summarise as follows.

In terms of an individual point,

$$(p,q)\longmapsto(p,q+a).$$

In terms of the whole graph,

$y=f(x)+a$ is obtained by translating the graph of $y=f(x)$ vertically by $a$ units.

Reflection in the $x$-axis

The transformation from $y=f(x)$ to $y=-f(x)$ changes every output $q$ into $-q$. You could construct a table of corresponding points yourself to see this more clearly.

In terms of an individual point,

$$(p,q)\longmapsto(p,-q).$$

In terms of the whole graph,

$y=-f(x)$ is obtained by reflecting the graph of $y=f(x)$ in the $x$-axis.

Vertical stretches

For $k>0$, the transformation from $y=f(x)$ to $y=kf(x)$ multiplies every $y$-coordinate by $k$.

In terms of an individual point,

$$(p,q)\longmapsto(p,kq).$$

In terms of the whole graph,

$y=kf(x)$ is obtained by stretching the graph of $y=f(x)$ vertically by scale factor $k$.

If $k>1$, this is a genuine stretch, while if $0<k<1$, the graph is compressed towards the $x$-axis.

Applying a vertical transformation correctly

The phrase apply to the whole current function matters. Suppose the current function is $g(x)=f(x)-4$. Stretching its graph vertically by scale factor $3$ gives

It does not give $3f(x)-4$, because that would multiply only one part of the current function rather than the whole output.

The fundamental principle is:

A vertical transformation must be applied to the whole current function.

Horizontal transformations

There are also three basic horizontal transformations. In every case, the $y$-coordinate remains unchanged. We instead ask which new $x$-coordinate gives $f$ the same input, and hence the same output.

Horizontal translations

Why does $y=f(x-2)$ move the graph to the right rather than to the left? This result is often well remembered but not well understood. The following corresponding points reveal what is happening.

Again, what association can you spot between the two rows?

Compare the corresponding points. For example, $(1,f(1))$ has become $(3,f(1))$, while $(2,f(2))$ has become $(4,f(2))$. Each $x$-coordinate has increased by $2$, while the $y$-coordinate is unchanged.

We can also explain this algebraically. If $(p,f(p))$ lies on $y=f(x)$, then to obtain the same output from $f(x-2)$, we need $x-2=p$, so $x=p+2$. Therefore, the graph moves horizontally to the right by $2$ units. More generally, the situation can be summarised as follows.

In terms of an individual point,

$$(p,q)\longmapsto(p+a,q).$$

In terms of the whole graph,

$y=f(x-a)$ is obtained by translating the graph of $y=f(x)$ horizontally by $a$ units.

Reflection in the $y$-axis

For $y=f(-x)$, the same output $q=f(p)$ occurs when $-x=p$, so $x=-p$.

In terms of an individual point,

$$(p,q)\longmapsto(-p,q).$$

In terms of the whole graph,

$y=f(-x)$ is obtained by reflecting the graph of $y=f(x)$ in the $y$-axis.

Horizontal stretches

For $k>0$, consider $y=f(kx)$. To recover the old input $p$, we require $kx=p$, so the new $x$-coordinate is $p/k$.

In terms of an individual point,

$$(p,q)\longmapsto\left(\frac{p}{k},q\right).$$

In terms of the whole graph,

$y=f(kx)$ is obtained by stretching the graph of $y=f(x)$ horizontally by scale factor $1/k$.

Applying a horizontal transformation correctly

Suppose the current function is $g(x)=f(2x+1)-4$. Translating its graph $3$ units to the right means replacing every $x$ in $g(x)$ with $x-3$:

We do not simply replace the existing expression $2x+1$ with $2x+1-3$. A horizontal transformation acts by replacing only $x$ throughout the current function. Any subsequent simplification takes place only after that substitution has been made.

The fundamental principle is:

A horizontal transformation must replace every occurrence of $x$ in the whole current function, and nothing else.

Order of the six basic transformations

From the way these six transformations work, we can identify three high-level principles governing the order in which they may be applied.

1. Vertical and horizontal transformations are independent of each other.

2. A vertical transformation must act on the whole current function.

3. A horizontal transformation must replace every occurrence of $x$, and nothing else.

Vertical and horizontal transformations may therefore be interleaved freely. Transformations in the same direction may also be performed in different orders, provided each transformation is applied correctly and its parameters are adjusted for the chosen order. These high-level principles are best understood through an example in which we construct the same final function in several different ways.

Example 1

The graph of $y=10-2f(2x-1)$ is obtained by transforming the graph of $y=f(x)$. Give a possible sequence of transformations that produces the required graph.

There are many valid answers. We will consider three contrasting possibilities.

Possible sequence 1:

The transformations are:

• a vertical stretch by scale factor $2$;
• a reflection in the $x$-axis;
• a vertical translation by $+10$ units;
• a horizontal translation by $+1$ unit;
• a horizontal stretch by scale factor $1/2$.

In this sequence, we applied all the vertical transformations before the horizontal ones.

Possible sequence 2:

The transformations are:

• a horizontal translation by $+1$ unit;
• a horizontal stretch by scale factor $1/2$;
• a vertical stretch by scale factor $2$;
• a reflection in the $x$-axis;
• a vertical translation by $+10$ units.

In this sequence, we applied all the horizontal transformations before the vertical ones.

Possible sequence 3:

The transformations are:

• a vertical stretch by scale factor $2$;
• a horizontal stretch by scale factor $1/2$;
• a reflection in the $x$-axis;
• a horizontal translation by $+\frac12$ unit;
• a vertical translation by $+10$ units.

In this sequence, we alternated between vertical and horizontal transformations. In sequence 2, the horizontal transformations consisted of a translation by $+1$ followed by a stretch by scale factor $1/2$. Here, we perform them in the reverse order: the stretch comes first, so the required translation changes to $+\frac{1}{2}$. This illustrates why transformations in the same direction cannot simply be reordered while leaving their parameters unchanged.

A convenient order

One particularly convenient order is:

• For horizontal transformations, apply addition before multiplication.

• For vertical transformations, apply multiplication before addition.

Using this order usually makes the required transformation parameters immediately apparent. For example, to obtain $f(10-2x)$, we may use

These correspond to a translation by $-10$, a reflection in the $y$-axis, and a horizontal stretch by scale factor $1/2$. However, this is only a convenient order, not a compulsory one. When considering the order of transformations, the important point is to understand and apply the three fundamental principles above.

Reflections in $y=b$ and $x=a$

Reflections in arbitrary horizontal and vertical lines may appear to be two additional types of transformation, but each can be constructed entirely from transformations we already know. Students therefore already possess all the required knowledge: the only ingredients are translations and reflections in the coordinate axes. However, reflections in the lines $x=a$ and $y=b$ are rarely taught explicitly or used in standard school mathematics courses.

This combination is precisely what makes them highly relevant to the TMUA. The necessary knowledge is already available, but students must apply it in a creative and perhaps unexpected way rather than reproduce a familiar method. Any topic with these two characteristics is natural TMUA territory: the individual ideas are known, but the way in which they must be combined may not be.

Understanding these reflections before the TMUA may therefore give you a small but useful advantage. 🙂

Reflection in $y=b$

Rather than treating reflection in $y=b$ as a completely new transformation, we can construct it using two familiar vertical transformations.

First reflect the graph of $y=f(x)$ in the $x$-axis, and then translate the resulting graph vertically by $+2b$ units:

Notice that the vertical translation is applied to the whole current function $-f(x)$, giving $2b-f(x)$.

We can verify that these transformations indeed give the desired reflection:

The original and final $y$-coordinates are $q$ and $2b-q$, whose midpoint is

They therefore lie the same vertical distance on opposite sides of the line $y=b$, confirming that the combined transformation is a reflection in $y=b$. We can therefore summarise the result in the same way as before.

In terms of an individual point,

$$(p,q)\longmapsto(p,2b-q).$$

In terms of the whole graph,

$y=2b-f(x)$ is obtained by reflecting the graph of $y=f(x)$ in the line $y=b$.

Reflection in $x=a$

Again, rather than treating reflection in $x=a$ as a completely new transformation, we can construct it using two familiar horizontal transformations.

First reflect the graph of $y=f(x)$ in the $y$-axis, and then translate the resulting graph horizontally by $+2a$ units:

Notice that the horizontal translation replaces $x$ with $x-2a$ in the current function $f(-x)$, giving $f(-(x-2a))$.

We can verify that these transformations indeed give the desired reflection:

The original and final $x$-coordinates are $p$ and $2a-p$, whose midpoint is

They therefore lie the same horizontal distance on opposite sides of the line $x=a$, confirming that the combined transformation is a reflection in $x=a$. We can again summarise the result in the same way.

In terms of an individual point,

$$(p,q)\longmapsto(2a-p,q).$$

In terms of the whole graph,

$y=f(2a-x)$ is obtained by reflecting the graph of $y=f(x)$ in the line $x=a$.

Since these two reflections can be constructed from the six basic transformations, they also obey the three high-level ordering principles.

Transforming the equation of a curve from first principles

So far, we have written curves in the form $y=f(x)$ and transformed them by changing the input or output of the function. However, transformations ultimately act on points, so we can instead track the successive images of an arbitrary point $(p,q)$. If its final image is $(X,Y)$, we express $p$ and $q$ in terms of $X$ and $Y$, then impose the relation originally satisfied by $(p,q)$. This produces a relation between $X$ and $Y$, and hence the equation of the transformed curve. We call this the first-principles point-tracking method. It is best understood through a couple of examples.

Example 2: comparing the two methods

The curve $y=x^2$ is stretched horizontally by scale factor $2$, translated horizontally by $+3$, and then reflected in the line $y=2$. Find the equation of the resulting curve.

Method 1: Transforming the function

Writing $f(x)=x^2$, the function-transformations developed earlier give

Since $f(x)=x^2$, the resulting curve has equation

Method 2: Tracking an arbitrary point

Let $(p,q)$ be an arbitrary point on the original curve, so $q=p^2$. Track this point through the three transformations:

Therefore, $p=(X-3)/2$ and $q=4-Y$. Substituting these into $q=p^2$ gives

and hence $Y=4-(X-3)^2/4$. Relabelling the final coordinates as $x$ and $y$, we again obtain

The two methods give exactly the same result, as expected.

Example 3: an implicitly defined curve

The curve $x^2y+2x-y^2=10$ is stretched horizontally by scale factor $2$, translated horizontally by $+3$, and then reflected in the line $y=2$. Find the equation of the resulting curve.

The entire curve cannot be expressed as a single function $y=f(x)$ or $x=g(y)$. For example, setting $x=3$ gives $y^2-9y+4=0$, which has two distinct real solutions, while setting $y=1$ gives $x^2+2x-11=0$, which also has two distinct real solutions. We could separate the curve into different branches, but the function-transformation method could not transform the whole curve in a single step.

The first-principles point-tracking method does not have this limitation. Let $(p,q)$ be an arbitrary point on the original curve, so $p^2q+2p-q^2=10$. Its successive images are

Thus $p=(X-3)/2$ and $q=4-Y$. Substituting these expressions into the original equation gives

Multiplying by $4$, expanding and relabelling $X$ and $Y$ as $x$ and $y$, the transformed curve has equation

The point-tracking method is therefore more general than the function-transformation method: it can be applied directly to any curve defined by a relation between $x$ and $y$, without requiring either variable to be expressed as a function of the other.

For the transformations considered so far, each step is reversible, so we can recover $(p,q)$ uniquely from $(X,Y)$. Modulus transformations require additional care because this one-to-one correspondence may be lost.

Summary of Part A

If you take away one thing from Part A, let it be the three high-level principles:

1. Vertical and horizontal transformations are independent of each other.

2. A vertical transformation must act on the whole current function.

3. A horizontal transformation must replace every occurrence of $x$, and nothing else.

Everything above follows from them. The six basic transformations are these principles applied one at a time, and the reflections in $y=b$ and $x=a$ are each built from two of the six. When the function form becomes awkward — or the curve is not of the form $y=f(x)$ at all — the first-principles point-tracking method of the previous section will always produce the transformed equation.

Continue to Part B, where the same three principles are applied to the modulus transformations $|f(x)|$ and $f(|x|)$, and where the one-to-one correspondence between a point and its image can be lost.

PDF version — Questions · Solutions

PDF version — Questions · Solutions