## Content

### Geometric transformations of graphs of functions

The graph of a function can be changed to produce the graph of a new function using:

- vertical or horizontal translations
- reflections in either axis
- stretching from either axis.

For example, if \(f(x) = \sin x\) and \(g(x) = x+2\), then

\[ (f\circ g)(x) = \sin(x+2) \qquad\text{and}\qquad (g\circ f)(x) = \sin(x) + 2. \]Both of these can be thought of as translations of \(f(x) = \sin x\), so clearly we cannot talk about the translation of a function by 2.

On the other hand, the image of the graph of a function under a rotation is usually not the graph of a function. Rotations of graphs of quadratic functions are discussed briefly in the module Quadratics .

#### Translations

##### Vertical translations in general

Consider the graph of \(y_1 = f(x)\) and the graph of \(y_2 = f(x)+7\), as shown in the following diagram.

We call \(y_2 = f(x)+7\) the vertical translation 7 units up of \(y_1 = f(x)\). Similarly, \(y_3 = f(x)-7\) is the vertical translation 7 units down of \(y_1 = f(x)\).

##### Vertical translations of parabolas

The standard parabola \(y=x^2\) in shown on the left in the following diagram. We can describe each point on the parabola in terms of a **parameter** \(p\); the corresponding point is \((p,p^2)\).

If \(y=x^2\) is translated 7 units up, we obtain \(y=x^2+7\), which is also parametrised by \(p\); the corresponding point is \((p,p^2+7)\).

##### Horizontal translations of parabolas

Suppose that the standard parabola \(y=x^2\) is translated 3 units to the right. The vertex \((0,0)\) is moved to \((3,0)\), which is the vertex of the parabola \(y=(x-3)^2\). Their graphs are as follows.

Clearly, the point \((p,p^2)\) is moved to the point \((p+3,p^2)\). If \(x=p+3\) and \(y=p^2\), then \(y=(x-3)^2\). In the same way, a translation 5 units to the left maps \(y=x^2\) to \(y=(x+5)^2\), with the point \((p,p^2)\) moved to the point \((p-5,p^2)\).

##### Horizontal translations in general

Now we consider what happens to the graph of an arbitrary function \(y = f(x)\) when it is translated \(a\) units to the right.

The translation moves the point \((x,y)\) to the point \((x+a,y)\). We can view this translation as a mapping from the coordinate plane to itself, given by

\[ (x,y) \mapsto (x',y'), \qquad\text{where}\quad x' = x + a \quad\text{and}\quad y' = y. \]Here we have \(x = x' - a\) and \(y = y'\). So the graph \(y = f(x)\) maps to \(y' = f(x' - a)\). That is, \(y = f(x)\) maps to \(y = f(x - a)\).

#### Example

Consider the graph of \(y=x^3\).

- If the graph is translated 5 units to the right, then what are the coordinates of the inflexion point and what is the equation of the image?
- If the graph is translated 4 units to the left and 7 units up, then what are the coordinates of the inflexion point and what is the equation of the image?

#### Solution

- The point of inflexion \((0,0)\) is translated to \((5,0)\). The equation of the image of the graph is \(y=(x-5)^3\).
- The point of inflexion \((0,0)\) is translated to \((-4,7)\). Translating the graph \(y=x^3\) to the left by 4 units gives \(y=(x+4)^3\). Then translating 7 units up yields \(y=(x+4)^3+7\).

##### General translations

A general translation \(T\) of the plane is given by

\[ T(x,y)=(x+a,y+b), \]for some real numbers \(a,b\). In particular, we have \(T(0,0)=(a,b)\). Clearly, this mapping is the composite of a horizontal translation through \(a\) and a vertical translation through \(b\). Thus, for example, it maps the graph of \(y=x^2\) to the graph of \(y=(x-a)^2+b\).

##### Exercise 3

Find the translation which maps the graph of \(y=x^2\) to \(y=x^2+4x+10\). Sketch the graph of \(y=x^2+4x+10\).

##### Summary for translations

The following table gives the image of the graph of \(y=f(x)\) under various translations, with \(a\) and \(b\) positive constants.

Translation | Image |
---|---|

\(a\) units up | \(y=f(x)+a\) |

\(a\) units down | \(y=f(x)-a\) |

\(b\) units to the right | \(y=f(x-b)\) |

\(b\) units to the left | \(y=f(x+b)\) |

\(a\) units up and \(b\) units to the right | \(y=f(x-b)+a\) |

\(a\) units down and \(b\) units to the left | \(y=f(x+b)-a\) |

We note that sometimes:

- 'up' is expressed as 'in the positive direction of the \(y\)-axis'
- 'down' is expressed as 'in the negative direction of the \(y\)-axis'
- 'to the right' is expressed as 'in the positive direction of the \(x\)-axis'
- 'to the left' is expressed as 'in the negative direction of the \(x\)-axis'.

#### Reflections

Assume we are given a line \(l\) in the plane. We can reflect any point \(P\) in the line \(l\) to obtain a point \(Q=R_l(P)\), as shown in the following diagram.

The point \(Q\) is found by dropping a perpendicular from \(P\) to \(l\), which meets \(l\) at \(X\). We take \(Q\) on this perpendicular such that \(PX = XQ\), with \(Q \neq P\) unless \(P\) lies on \(l\). This describes a mapping \(R_l\) from the plane \(\mathbb{R}^2\) to itself. It is an **involution**, meaning that if you perform the mapping twice you get the identity.

There are three important special cases:

- when \(l\) is the \(x\)-axis, \(y=0\)
- when \(l\) is the \(y\)-axis, \(x=0\)
- when \(l\) is the line \(y=x\).

The third case is important when determining inverses of functions, and will be discussed in the next section of this module.

##### Reflection in the \(x\)-axis

Reflection in the \(x\)-axis maps the point \((a,b)\) to the point \((a,-b)\). Thus, if we consider the graph of \(y=f(x)\) and reflect it in the \(x\)-axis, we obtain the graph of \(y=-f(x)\).

For example, the standard parabola \(y=x^2\) is reflected to the parabola \(y=-x^2\). By applying suitable horizontal and vertical translations to \(y=-x^2\), we can obtain any parabola of the form \(y=-x^2+ax+b\).

##### Reflection in the \(y\)-axis

Reflection in the \(y\)-axis maps the point \((a,b)\) to the point \((-a,b)\). Thus, if we consider the graph of \(y=f(x)\) and reflect it in the \(y\)-axis, we obtain the graph of \(y=f(-x)\).

For example, the graph \(y=x^2\) is mapped to itself, and the graph \(y=x^3\) is mapped to \(y=-x^3\).

##### Summary for reflections in an axis

We summarise reflections of a function \(y=f(x)\) in the axes in the following table.

Reflection | Image |
---|---|

Reflection in the \(x\)-axis | \(y=-f(x)\) |

Reflection in the \(y\)-axis | \(y=f(-x)\) |

##### Exercise 4

- Show that a function \(f(x)\) is even if and only if the graph of \(y=f(x)\) is mapped to itself by reflection in the \(y\)-axis.
- Show that a function \(y=f(x)\) is odd if and only if the graph of \(y=f(x)\) is mapped to itself by the composition of the reflections in the two coordinate axes.

#### Isometries

An **isometry** of the plane is a transformation which preserves distances between points. Isometries include rotations, translations, reflections, glide reflections and the identity map. In fact, it can be proved that every isometry can be obtained as a composition of at most three reflections.

Isometries are sometimes also called **congruence transformations**. Two figures that can be transformed into each other by an isometry are said to be **congruent**.

So far in this section, we have looked at translations and reflections in the axes. Now we consider a family of transformations which are not isometries: the transformations which are 'stretches' from one of the axes.

#### Some transformations which are not isometries

##### Stretches from the \(x\)-axis

The mapping \((x,y) \mapsto (x,3y)\) is called a **stretch from the \(x\)-axis**. Under this mapping, the point \((a,a^2)\) on the graph of \(y = x^2\) is sent to the point \((a,3a^2)\) on the graph of \(y = 3x^2\). This is shown in the following diagram.

We can describe this mapping as

\[ (x,y) \mapsto (x',y'), \qquad\text{where}\quad x'=x \quad\text{and}\quad y'=3y. \]Thus \(x=x'\) and \(y=\dfrac{y'}{3}\), and so the graph of \(y = x^2\) maps to \(\dfrac{y'}{3}=(x')^2\). That is, \(y=x^2\) maps to \(y=3x^2\).

##### Stretches from the \(y\)-axis

The mapping \((x,y) \mapsto (3x,y)\) is called a **stretch from the \(y\)-axis**. We can describe this mapping as
\[
(x,y) \mapsto (x',y'), \qquad\text{where}\quad x'=3x \quad\text{and}\quad y'=y.
\]
Thus \(x=\dfrac{x'}{3}\) and \(y=y'\), and so the graph of \(y = x^2\) maps to \(y'= \Bigl(\dfrac{x'}{3}\Bigr)^2\). That is, \(y=x^2\) maps to \(y = \dfrac{x^2}{9}\).

##### Summary for stretches from the axes

We have seen that \((x,y) \mapsto (x,3y)\) under a stretch from the \(x\)-axis by the factor 3. In general, the mapping \((x,y) \mapsto (x,ky)\) is the stretch from the \(x\)-axis by the factor \(k\), where \(k>0\). Similarly, the mapping \((x,y) \mapsto (kx,y)\) is the stretch from the \(y\)-axis by the factor \(k\), where \(k>0\). We summarise this in the following table.

Stretch | Transformation | Image |
---|---|---|

Stretch from \(x\)-axis by the factor \(k\), for \(k > 0\) | \((x,y) \mapsto (x,ky)\) | \(y=kf(x)\) |

Stretch from \(y\)-axis by the factor \(k\), for \(k > 0\) | \((x,y) \mapsto (kx,y)\) | \(y=f\Bigl(\dfrac{x}{k}\Bigr)\) |

Note. The composition of these two stretches gives a transformation \((x,y) \mapsto (kx,ky)\). This is called a **dilation**. A dilation is an example of a similarity transformation.