# Axial symmetry

Figures with their axes of symmetry (dashed). The figure at the bottom right is not axially symmetrical.
Axial symmetry in architecture
( Castle Howard manor )

Axial symmetry is the mirror-image arrangement of characters on both sides of an imaginary line. In the geometry are axial symmetry or axial symmetry similar designations that property. A figure is called axially symmetrical if it is mapped onto itself by the vertical axis reflection on its axis of symmetry .

In the case of a two-dimensional figure, axial symmetry is synonymous with mirror symmetry . In three-dimensional spaces, on the other hand , the axis symmetry corresponds to a rotational symmetry around 180 ° (while the mirror symmetry in three -dimensional space is a symmetry to a plane of symmetry ).

## definition

A figure is axially symmetrical if there is a straight line g, so that for each point P of the figure there is another point P 'of the figure (possibly identical to P), so that the connecting line [PP'] is bisected at right angles by this straight line .

A plane figure F is called axially or axially symmetric if a straight line g can be specified in its plane so that F is converted into itself by mirroring at g.

The straight line g is then called the axis of symmetry .

## Examples

• As you can see in the adjacent figure, the square has exactly four axes of symmetry. Quadrilaterals that are not squares have fewer or no axes of symmetry. A rectangle , for example, still has two axes of symmetry, namely the two perpendicular perpendiculars on the opposite sides and the isosceles trapezoid , the dragon square and the anti-parallelogram still have at least one axis of symmetry.
• The circle even has an infinite number of axes of symmetry, since it is symmetrical with respect to every diameter.
• Another figure with an infinite number of axes of symmetry is the straight line . It is infinitely long and therefore symmetrical with respect to each axis perpendicular to it, as well as the axis lying on itself.
• Not only 2-dimensional figures can be axially symmetrical. The sphere is axially symmetrical with respect to every straight line through the center point. This should not be confused with plane symmetry. The sphere is also plane symmetrical. That is, it is symmetrical about a reflection about a plane that contains the center of the sphere.
• The cuboid is also axially symmetrical.
• The graph of the cosine function is also axisymmetric to the y-axis . The topic of axially symmetric functions is discussed in more detail in the following section.

## Axial symmetry of function graphs

### overview

Function whose graph is axially symmetric to the line x = a

A task that is particularly popular in school is to prove the axis symmetry for the graph of a function . If the y-axis of the coordinate system is the axis of symmetry, it must be shown that the equation${\ displaystyle f \ colon D \ to \ mathbb {R}}$

${\ displaystyle f (-x) \, = \, f (x)}$

is fulfilled for all x of the domain . Then the graph of the function is said to be symmetric about the y-axis. Such functions are also called straight functions . This condition states that the function values ​​of the arguments and must match. ${\ displaystyle D}$${\ displaystyle f}$${\ displaystyle x}$${\ displaystyle -x}$

If you want to examine the axis symmetry of a function graph with respect to any straight line parallel to the y-axis with the equation , you have to test whether the function matches the equation ${\ displaystyle x = a}$${\ displaystyle f}$

${\ displaystyle f (ax) \, = \, f (a + x)}$

for a fixed and for all of the definition met. Substituting with gives the equivalent condition ${\ displaystyle a \ in \ mathbb {R}}$${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle xa}$

${\ displaystyle f (2a-x) \, = \, f (x).}$

### Examples

The quadratic function serves as an example

${\ displaystyle f (x) \, = \, x ^ {2} -1.}$

Applying the mentioned condition for the axis symmetry in relation to the y-axis results

${\ displaystyle f (-x) \, = \, (- x) ^ {2} -1 = x ^ {2} -1 = f (x).}$

The graph (a parabola) is therefore symmetrical about the y-axis.

An example of a function will now be given whose graph is not symmetric about the y-axis, but is axisymmetric. The function

${\ displaystyle f (x) = x ^ {2} -4x + 3}$

is one such example. The claim is that the graph of is axially symmetric with respect to the normal . So it is true and it follows from it ${\ displaystyle f}$${\ displaystyle x = 2}$${\ displaystyle a = 2}$

{\ displaystyle {\ begin {aligned} f (2a-x) & = f (4-x) \\ & = (4-x) ^ {2} -4 (4-x) +3 \\ & = ( 16-8x + x ^ {2}) - (16-4x) +3 \\ & = 16-8x + x ^ {2} -16 + 4x + 3 \\ & = x ^ {2} -4x + 3 \\ & = f (x) \ end {aligned}}}

This confirms the assumption of axial symmetry.

In general, the graph of a quadratic function is axisymmetric with respect to the vertical line through the vertex . This is easy to see if you paraphrase the function term in vertex form . ${\ displaystyle f (x) = ax ^ {2} + bx + c}$ ${\ displaystyle S = (x_ {s}, y_ {s})}$${\ displaystyle f (x) = a (x-x_ {s}) ^ {2} + y_ {s}}$

## Body of revolution

A class of axially symmetrical bodies in 3-dimensional space are the rotational bodies. A three-dimensional object is a solid of revolution when a rotation through any angle around a fixed axis maps the object onto itself. This axis is the axis of symmetry. The simplest example of a solid of revolution is the cylinder .

## Plane symmetry

Another generalization of the axial symmetry to 3-dimensional space is plane symmetry. A figure is precisely symmetrical, if there is a plane so that at mirroring this, the figure is mapped onto itself.

## literature

1. Axial symmetry. In: Duden online. Retrieved November 21, 2019 .
2. ^ Arnfried Kemnitz: Mathematics at the beginning of the course . Basic knowledge for all technical, mathematical, scientific and economic courses. 9th revised and expanded edition. Springer-Verlag, Wiesbaden 2010, ISBN 978-3-8348-1293-3 , pp. 144 ( limited preview in Google Book Search [accessed November 21, 2019]).