axis

Examples of axis in the following topics:

• Regional Terms and Axes

  • The Dorsoventral axis (DV axis) is formed by the connection of the dorsal and ventral points of a region.
  • The Anterioposterior axis (AP axis) is the axis formed by the connection of the anterior (top) and posterior (bottom) ends of a region.
  • The AP axis of a region is by definition perpendicular to the DV axis and vice-versa.
  • The Left-to-right axis is the axis connecting the left and right hand sides of a region.
  • Axis (A) (in red) shows the AP axis of the tail, (B) shows the AP axis of the neck, and (C) shows the AP axis of the head.

• 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.
  • 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 perpendicular thin black line through the center is the conjugate axis.
  • The two thick black lines parallel to the conjugate axis (thus, perpendicular to the transverse axis) are the two directrices, D1 and D2.

• The Cartesian System

  • The horizontal axis is known as the x-axis and the vertical axis is known as the y-axis.
  • The non-integer coordinate, $(-1.5,-2.5)$ is in the middle of -1 and -2 on the x-axis and -2 and -3 on the y-axis.
  • The revenue is plotted on the y-axis and the number of cars washed is plotted on the x-axis.  
  • Point $(4,0)$ is on the x-axis and not in a quadrant.  
  • Point $(0,-2)$ is on the y-axis and also not in a quadrant.

• Cylindrical Shells

  • In the shell method, a function is rotated around an axis and modeled by an infinite number of cylindrical shells, all infinitely thin.
  • The volume of the solid formed by rotating the area between the curves of $f(x)$ and $g(x)$ and the lines $x=a$ and $x=b$ about the $y$-axis is given by:
  • The volume of solid formed by rotating the area between the curves of $f(y)$ and and the lines $y=a$ and $y=b$ about the $x$-axis is given by:
  • Each segment located at $x$, between $f(x)$and the $x$-axis, gives a cylindrical shell after revolution around the vertical axis.
  • Use shell integration to create a cylindrical shell and calculate the volume of a "solid of revolution" perpendicular to the axis of revolution.

• Symmetry of Functions

  • Functions and relations can be symmetric about a point, a line, or an axis.  
  • The image below shows an example of a function and its symmetry over the $x$-axis (vertical reflection) and over the $y$-axis (horizontal reflection).  
  • The axis splits the U-shaped curve into two parts of the curve which are reflected over the axis of symmetry.  
  • 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.

• Reflections

  • The reflection of a function can be performed along the $x$-axis, the $y$-axis, or any line.  
  • A vertical reflection is a reflection across the $x$-axis, given by the equation:
  • The result is that the curve becomes flipped over the $x$-axis.  
  • The result is that the curve becomes flipped over the $y$-axis.  
  • Calculate the reflection of a function over the $x$-axis, $y$-axis, or the line $y=x$

• Area of a Surface of Revolution

  • A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane, known as the axis .
  • Examples of surfaces generated by a straight line are cylindrical and conical surfaces when the line is co-planar with the axis, as well as hyperboloids of one sheet when the line is skew to the axis.
  • If the curve is described by the parametric functions $x(t)$, $y(t)$, with $t$ ranging over some interval $[a,b]$ and the axis of revolution the $y$-axis, then the area $A_y$ is given by the integral:
  • Likewise, when the axis of rotation is the $x$-axis, and provided that $y(t)$ is never negative, the area is given by:
  • A portion of the curve $x=2+\cos z$ rotated around the $z$-axis (vertical in the figure).

• Parts of an Ellipse

  • The major axis of the ellipse is the longest width across it.
  • For a horizontal ellipse, that axis is parallel to the x-axis.
  • The major axis has length $2a$.
  • For a horizontal ellipse, it is parallel to the y-axis.
  • The minor axis has length $2b$.

• Volumes of Revolution

  • The disc method is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when integrating parallel to the axis of revolution.
  • The shell method is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution.
  • Disc integration about the $y$-axis.
  • Integration is along the axis of revolution ($y$-axis in this case).
  • The integration (along the $x$-axis) is perpendicular to the axis of revolution ($y$-axis).

• Zeroes of Linear Functions

  • Graphically, where the line crosses the $x$-axis, is called a zero, or root.  
  • If there is a horizontal line through any point on the $y$-axis, other than at zero, there are no zeros, since the line will never cross the $x$-axis.  
  • If the horizontal line overlaps the $x$-axis, (goes through the $y$-axis at zero) then there are infinitely many zeros, since the line intersects the $x$-axis multiple times.  
  • Because the $x$-intercept (zero) is a point at which the function crosses the $x$-axis, it will have the value $(x,0)$, where $x$ is the zero.
  • To find the zero of a linear function, simply find the point where the line crosses the $x$-axis.