Examples of Axis Expansion in the following topics:
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- 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.
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- 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.
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- 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.
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- 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.
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- 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.
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- The implementation of a military dictatorship and territorial expansionism were considered the best ways to protect the Yamato-damashii, or what Japanese
saw as their spiritual and cultural values.
- Totalitarianism, militarism, and expansionism were to become the rule, with fewer voices able to speak against it.
- The signatories of this alliance become known as the Axis Powers.
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- 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
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- 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 Ay 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+cosz rotated around the z-axis (vertical in the figure).