axis of symmetry

Examples of axis of symmetry in the following topics:

• Symmetry of Functions

  • In the next graph below, quadratic functions have symmetry over a line called the axis of symmetry.  
  • The axis splits the U-shaped curve into two parts of the curve which are reflected over the axis of symmetry.  
  • The function $y=x^2+4x+3$ shows an axis of symmetry about the line $x=-2$.  
  • Notice that the $x$-intercepts are reflected points over the axis of symmetry and are equidistant from the axis.
  • This type of symmetry is a translation over an axis.

• Graphing Quadratic Equations In Standard Form

  • The coefficients $b$ and $a$ together control the axis of symmetry of the parabola and the $x$-coordinate of the vertex.
  • The axis of symmetry for a parabola is given by:
  • Because $a=2$ and $b=-4,$ the axis of symmetry is:
  • More specifically, it is the point where the parabola intercepts the y-axis.
  • The axis of symmetry is a vertical line parallel to the y-axis at  $x=1$.

• Parabolas As Conic Sections

  • The axis of symmetry is a line that is at the same angle as the cone and divides the parabola in half.
  • The point on the axis of symmetry where the right angle is located is called the focus.
  • The focal length is the leg of the right triangle that exists along the axis of symmetry, and the focal point is the vertex of the right triangle.
  • The light leaves the parabola parallel to the axis of symmetry.
  • The vertex of the parabola here is point $P$, and the diagram shows the radius $r$ between that point and the cone's central axis, as well as the angle $\theta$ between the parabola's axis of symmetry and the cone's central axis.

• Symmetry and Centricity

  • And the study of symmetry in general takes us far afield of the compositional techniques generally associated with common-practice tonal music.
  • Think of pitch symmetry in terms of a musical "mirror."
  • Pitch symmetry always implies an axis of symmetry.
  • The pitch-space line shows that it has a different axis of symmetry—around E2.
  • Pitch-class symmetry is very similar to pitch symmetry, but understood in pitch-class space.

• Curve Sketching

  • Curve sketching is used to produce a rough idea of overall shape of a curve given its equation without computing a detailed plot.
  • It is an application of the theory of curves to find their main features.
  • Determine the symmetry of the curve.
  • If the exponent of $x$ is always even in the equation of the curve, then the $y$-axis is an axis of symmetry for the curve.
  • Similarly, if the exponent of $y$ is always even in the equation of the curve, then the $x$-axis is an axis of symmetry for the curve.

• Chirality and Symmetry

  • Some examples of symmetry elementsare shown below.
  • The face playing card provides an example of a center or point of symmetry.
  • The boat conformation of cyclohexane shows an axis of symmetry (labeled C2 here) and two intersecting planes of symmetry (labeled σ).
  • The notation for a symmetry axis is Cn, where n is an integer chosen so that rotation about the axis by 360/nº returns the object to a position indistinguishable from where it started.
  • In addition to the point of symmetry noted earlier, (E)-1,2-dichloroethene also has a plane of symmetry (the plane defined by the six atoms), and a C2 axis, passing through the center perpendicular to the plane.

• Animal Characterization Based on Body Symmetry

  • Animals can be classified by three types of body plan symmetry: radial symmetry, bilateral symmetry, and asymmetry.
  • At a very basic level of classification, true animals can be largely divided into three groups based on the type of symmetry of their body plan: radially symmetrical, bilaterally symmetrical, and asymmetrical.
  • All types of symmetry are well suited to meet the unique demands of a particular animal's lifestyle.
  • Radial symmetry is the arrangement of body parts around a central axis, like rays on a sun or pieces in a pie.
  • Only members of the phylum Porifera (sponges) have no body plan symmetry.

• Trigonometric Symmetry Identities

  • The trigonometric symmetry identities are based on principles of even and odd functions that can be observed in their graphs.
  • The following symmetry identities are useful in finding the trigonometric function of a negative value.
  • Now that we know the sine, cosine, and tangent of $\displaystyle{\frac{5\pi}{6}}$, we can apply the symmetry identities to find the functions of $\displaystyle{-\frac{5\pi}{6}}$.
  • Cosine and secant are even functions, with symmetry around the $y$-axis.
  • Explain the trigonometric symmetry identities using the graphs of the trigonometric functions

• Body Plans

  • Animal body plans can have varying degrees of symmetry and can be described as asymmetrical, bilateral, or radial.
  • Radial symmetry describes an animal with an up-and-down orientation: any plane cut along its longitudinal axis through the organism produces equal halves, but not a definite right or left side.
  • Bilateral symmetry is found in both land-based and aquatic animals; it enables a high level of mobility.
  • In order to describe structures in the body of an animal it is necessary to have a system for describing the position of parts of the body in relation to other parts .
  • Animals exhibit different types of body symmetry.

• Cylindrical and Spherical Coordinates

  • Cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
  • A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis.
  • Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational symmetry about the longitudinal axis, such as water flow in a straight pipe with a round cross-section, heat distribution in a metal cylinder, electromagnetic fields produced by an electric current in a long, straight wire, and so on.
  • For the conversion between cylindrical and Cartesian coordinate co-ordinates, it is convenient to assume that the reference plane of the former is the Cartesian $xy$-plane (with equation $z = 0$), and the cylindrical axis is the Cartesian $z$-axis.
  • A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.