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.

• Parts of a Parabola

  • Parabolas also have an axis of symmetry, which is parallel to the y-axis.
  • The axis of symmetry is a vertical line drawn through the vertex.
  • The y-intercept is the point at which the parabola crosses the y-axis.
  • The x-intercepts are the points at which the parabola crosses the x-axis.
  • Due to the fact that parabolas are symmetric, the $x$-coordinate of the vertex is exactly in the middle of the $x$-coordinates of the two roots.

• Types of Conic Sections

  • An axis of symmetry, which is a line connecting the vertex and the focus which divides the parabola into two equal halves
  • Its intersection with the cone is therefore a set of points equidistant from a common point (the central axis of the cone), which meets the definition of a circle.
  • The general form of the equation of an ellipse with major axis parallel to the x-axis is:
  • where $(h,k)$ are the coordinates of the center, $2a$ is the length of the major axis, and $2b$ is the length of the minor axis.
  • It is the axis length connecting the two vertices.

• 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

• Standard Equations of Hyperbolas

  • Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the x-axis, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±b⁄a, where b=a×tan(θ) and where θ is the angle between the transverse axis and either asymptote.
  • A conjugate axis of length 2b, corresponding to the minor axis of an ellipse, is sometimes drawn on the non-transverse principal axis; its endpoints ±b lie on the minor axis at the height of the asymptotes over/under the hyperbola's vertices.
  • Because of the minus sign in some of the formulas below, it is also called the imaginary axis of the hyperbola.
  • If the transverse axis of any hyperbola is aligned with the x-axis of a Cartesian coordinate system and is centered on the origin, the equation of the hyperbola can be written as:
  • The asymptotes of the hyperbola (red curves) are shown as blue dashed lines and intersect at the center of the hyperbola, C.

• Sine and Cosine as Functions

  • Plotting the points from the table and continuing along the $x$-axis gives the shape of the sine function.
  • for all values of $x$ in the domain of $f$.
  • Looking again at the sine and cosine functions on a domain centered at the $y$-axis helps reveal symmetries.
  • The graph of the cosine function shows that it is symmetric about the y-axis.
  • The cosine function is even, meaning it is symmetric about the $y$-axis.

• Even and Odd Functions

  • Functions that have an additive inverse can be classified as odd or even depending on their symmetry properties.
  • These labels correlate with symmetry properties of the function.
  • We can confirm this graphically: functions that satisfy the requirements of being even are symmetric about the $y$-axis.
  • To check if a function is odd, the negation of the function (be sure to negate all terms of the function) must yield the same output as substituting the value $-x$.
  • The function $f(x)=x^4+2x$ pictured above is not even because the graph is not symmetric about the $y$-axis.  

• Reflections

  • Reflections are a type of transformation that move an entire curve such that its mirror image lies on the other side of the $x$ or $y$-axis.
  • Reflections produce a mirror image of a function.  
  • The reflection of a function can be performed along the $x$-axis, the $y$-axis, or any line.  
  • This reflection has the effect of swapping the variables $x$and $y$, which is exactly like the case of an inverse function.  
  • Calculate the reflection of a function over the $x$-axis, $y$-axis, or the line $y=x$