hyperbolas
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The document discusses how to graph hyperbolas when the center is not at the origin. It provides an example of a hyperbola with center (2, -1), which shifts the original graph 2 units right and 1 unit down. It then discusses key features of the hyperbola like vertices, foci, asymptotes, and their locations relative to the shifted center point.
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A hyperbola is a set of points where the difference between the distances from two fixed points (foci) is constant. The constant is called the focal length. A hyperbola has two branches and two vertices, and is defined by its foci, center, transverse axis, and conjugate axis. The standard form of a hyperbola equation relates its focal length c to its transverse axis length a and conjugate axis length b using the formula a^2 + b^2 = c^2.
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hyperbola.pptx
A hyperbola is defined by two foci points, where the absolute value of the difference between the distance from a point to each foci is a constant. The key parts of a hyperbola include the center, foci, vertices, transverse and conjugate axes, asymptotes, and latus recta. The document provides examples of horizontal and vertical transverse axis hyperbolas with equal a and b values, giving the equations, asymptotes, lengths of axes, coordinates of vertices and foci.
Lecture 2.1 shortest derivation of equation of ellipse and meaning of eccentr...
1) An ellipse can be obtained by compressing the y-coordinates of a circle by a factor of b/a, where b < a.
2) The eccentricity, e, of an ellipse is defined as the ratio of the distance from the focus (F) to a point on the ellipse and the distance from the directrix to the same point.
3) The sum of the distances from any point on the ellipse to the two foci is equal to the major axis of the ellipse.
Parabola
The document discusses parabolas and their key properties:
- A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
- The vertex is the point where the axis of symmetry intersects the parabola. The focus and directrix are a fixed distance (p) from the vertex.
- The latus rectum is the line segment from the focus to the parabola, perpendicular to the axis of symmetry. Its length is determined by the equation of the parabola.
Transformations
This document discusses different types of geometric transformations including reflections, rotations, translations, and enlargements. It provides examples and explanations of each type of transformation. Reflections require a mirror line and produce a image that is the same distance from the mirror line as the original shape. Rotations occur around a center of rotation when a shape is turned a specified angle. Translations move a shape without changing its size or orientation through a movement described by a vector. Enlargements produce similar shapes that are enlarged or reduced versions of the original through a scale factor from a center of enlargement.
Fundamentals of Physics "VECTORS"
3-1 VECTORS AND THEIR COMPONENTS
After reading this module, you should be able to . . .
3.01 Add vectors by drawing them in head-to-tail arrangements, applying the commutative and associative laws.
3.02 Subtract a vector from a second one.
3.03 Calculate the components of a vector on a given coordinate system, showing them in a drawing.
3.04 Given the components of a vector, draw the vector
and determine its magnitude and orientation.
3.05 Convert angle measures between degrees and radians.
3-2 UNIT VECTORS, ADDING VECTORS BY COMPONENTS
After reading this module, you should be able to . . .
3.06 Convert a vector between magnitude-angle and unit vector notations.
3.07 Add and subtract vectors in magnitude-angle notation
and in unit-vector notation.
3.08 Identify that, for a given vector, rotating the coordinate
system about the origin can change the vector’s components but not the vector itself.
etc...
returika
This document introduces three-dimensional coordinate systems and graphs in three dimensions. It defines the three perpendicular axes (x, y, z) used to locate points in three-dimensional space. Points are represented by ordered triples (a,b,c) indicating their distances from each axis. Equations in three variables determine surfaces in three-dimensional space that can be graphed and analyzed. The document also introduces the distance formula for calculating distances between points in three-dimensional space and defines spheres using this formula. It provides examples of describing, sketching, and finding equations for various three-dimensional surfaces and regions.
Plano numérico
The document discusses key concepts about the Cartesian coordinate plane including:
- The coordinate plane consists of two perpendicular number lines that intersect at the origin point.
- The coordinate plane is used to describe the position of points and analyze geometric shapes like circles, parabolas, ellipses, and hyperbolas.
- Formulas are provided to calculate the distance between two points on the coordinate plane and to find the midpoint of a line segment between two points.
- Equations are given for circles, parabolas, ellipses, and hyperbolas centered at various positions on the coordinate plane.
- An example problem is included to find the coordinates of points and distances between points on a coordinate plane diagram.
Unit 13.1
This document provides an overview of conic sections including circles, ellipses, hyperbolas, and parabolas. It discusses how conic sections are formed by the intersection of a double right cone and a plane. Examples are provided on graphing conic sections on a calculator and identifying their properties such as center, vertices, and intercepts. The document also covers using the midpoint and distance formulas to find the center and radius of a circle from its diameter endpoints.
Chapter 7.2 parabola
The document discusses parabolas, including their key properties and equations. It defines a parabola as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The document derives the standard equation of a parabola from this definition and discusses how to graph parabolas based on their equations. It also covers transformations, latus rectum, and other geometric properties of parabolas.
Conic_Sections_Hyperbolas FCIT compat.ppt
The document discusses hyperbolas. It begins by providing an algebraic definition of a hyperbola as the set of points where the difference between the distances to two fixed points (foci) is a constant. It then provides steps for graphing a hyperbola from its standard form equation, including identifying the center, vertices, transverse/conjugate axes, asymptotes, and foci. Examples of graphing hyperbolas are shown.
Transformations
The document defines and provides examples of translations and reflections in a coordinate plane. A translation moves a figure in a straight line by adding or subtracting from its coordinates. Two examples show translating a triangle by subtracting from x-coordinates to move left and adding to y-coordinates to move up. Another translates the same triangle by adding to x-coordinates to move right and subtracting from y-coordinates to move down. A reflection produces a mirror image of a figure by using the opposite value of one coordinate, leaving the other coordinate alone depending on whether reflecting across the x-axis or y-axis.
Lesson 10 conic sections - hyperbola
The document discusses properties of hyperbolas including:
- A hyperbola is defined as the set of points where the difference between distances to two fixed points (foci) is constant.
- The standard equation of a horizontal hyperbola is (x-h)2/a2 - (y-k)2/b2 = 1 and a vertical hyperbola is (y-k)2/a2 - (x-h)2/b2 = 1.
- Examples are given to find the equation of hyperbolas given properties like the center, foci, vertices or asymptotes and to draw the graph of hyperbolas given their equation.
PreCalc Section 1.4.ppt
1. The document discusses different types of transformations of graphs including vertical and horizontal shifts, reflections, and non-rigid transformations like stretching and shrinking.
2. Vertical and horizontal shifts move a graph up/down or left/right along an axis by adding or subtracting a constant value. Reflections flip the graph across an axis.
3. Non-rigid transformations like stretching and shrinking change the shape of the graph by multiplying the original function by a constant value, stretching or shrinking the y-values for vertical transformations and x-values for horizontal transformations.
Stolyarevska_IES
The document discusses using the GeoGebra mathematical software package in mathematics teaching. It presents 14 propositions about ellipses, hyperbolas, and parabolas. Students used GeoGebra's interactive features to construct the curves as loci of points and solve problems of varying complexity based on the propositions. This reinforced their understanding of concepts from the "Second-Order Curves" course material from sources like textbooks and papers.
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Lecture 2.1 shortest derivation of equation of ellipse and meaning of eccentr...
Lecture 2.1 shortest derivation of equation of ellipse and meaning of eccentr...
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The browning version by terrence rattingan
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The poem "Father to Son" by Elizabeth Jennings describes a disconnect between a father and son. There is a generation gap between the two and they have become like strangers to each other despite living together for years. They speak like strangers and there is no sign of understanding between them. Both the father and son long to reconnect but are unable to bridge the gap between them.
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ellipse
An ellipse is a closed curve where the sum of the distances from two fixed points (foci) to any point on the curve is a constant. It can be defined parametrically using angles or implicitly as a second-degree equation. Key properties include the major and minor axes, which are lines of symmetry, and the eccentricity, which is a measure of how non-circular the ellipse is. The area of an ellipse is πab and its circumference can be calculated using elliptic integrals.
problems on parabola
The document contains 5 problems that involve finding the vertex, focus, axis, and directrix of different parabolas defined by equations of the form x^2 + Ax + By + C = 0. For each problem, the solution provides the vertex (focal point), focus, axis of symmetry, and directrix of the parabola defined by that equation.
parabolas
This document discusses how to find the equation of a parabola given its focus and vertex. It explains that the distance from the vertex to the focus allows you to calculate the p term in the quadratic equation. Two examples are provided, one with the vertex at the origin and one with the vertex shifted. In the second example, the focus is (2,3) and the vertex is (2,6), so p=-3 and the equation is (x-2)2 = -12(y-2).
ellipses and orbits
The orbits of the planets in our solar system are ellipses, as described by Kepler's First Law. The document provides the orbital parameters for the major planets, including the semi-major axis, eccentricity, and inclination for each planet's orbit. It derives Kepler's Third Law, which states that the square of a planet's orbital period is proportional to the cube of its semi-major axis. The problem of determining a planet's position based on time is called Kepler's Problem, and its solution involves Kepler's Equation, which relates the mean and eccentric anomalies.
parabolas
The document discusses parabolas and their key properties. It begins by introducing the standard form of a quadratic function and the steps to find the vertex, line of symmetry, and maximum/minimum value. It then provides examples of completing the square to derive the standard form (y = (x - h)2 + k) from other forms. Using this standard form, the vertex is identified as (h, k), the line of symmetry is defined as x = h, and the maximum/minimum is determined by the sign of the parabola based on whether k represents the highest or lowest point.
properties of circle
The circle has the highest area for a given perimeter. It is highly symmetric, with every line through the center forming a reflection symmetry. All circles are similar, with circumference and radius, and area and the square of the radius, being proportional using constants of 2π and π. Chords equidistant from the center are equal in length, and the perpendicular bisector of a chord passes through the center. Tangents drawn from a point outside the circle are equal in length, and a line perpendicular to a tangent at the point of contact passes through the center. The inscribed angle theorem states that an inscribed angle is half of the corresponding central angle, with inscribed angles on the same arc having the same value and
problems on conics
This document contains 5 problems identifying different conic sections from their equations. Problem 1 identifies -3x^2 + y + 2 = 0 as a parabola. Problem 2 identifies 2x^2 +3xy - 4y^2 + 2x - 3y + 1 = 0 as a hyperbola. Problem 3 identifies 2x^2 -3y^2 = 0 as a hyperbola. Problem 4 identifies -3x^2 + xy - 2y^2 + 4 = 0 as either an ellipse or circle. Problem 5 identifies x = 0 as a degenerate conic that is a line.
introduction to conics
A conic is the intersection of a plane and a right circular cone. The four basic types of conics are parabolas, ellipses, circles, and hyperbolas. All conics can be represented by the standard quadratic equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 and classified based on the values of the coefficients. Analytic geometry studies geometric shapes using the coordinate plane and formulas for their graphs, with a focus on the conics represented by the standard quadratic equation.
circles
A circle is defined as the set of all points in a plane that are equidistant from a fixed point, called the center. The distance from the center to any point on the circle is the radius. A diameter is a line segment that passes through the center containing two points on the circle. The circumference is the perimeter or distance around the circle, while the area is the region within the circle boundary. Circles have analytic definitions and properties that can be described through equations.
history of conics
- The conic sections (circles, ellipses, parabolas, and hyperbolas) have been studied for over 2000 years, with Apollonius of Perga making major contributions in the 2nd century BC by rigorously studying them and applying the work to astronomy.
- Apollonius was the first to note that conic sections can be constructed by cutting a circular cone with a plane in different ways: a circle from a perpendicular cut, a parabola from a parallel cut, a hyperbola from a cut through both parts of the cone, and an ellipse from a cut through one part of the cone not parallel to its side.
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hyperbolas
- 1. Hyperbolas translated away from the center
- 2. Example : Suppose the center is not at the origin (0, 0) but is at some other point such as (2, -1). To graph this hyperbola requires us to remember how graphs are moved horizontally and vertically by a change in the equation. Using Example #1 above, we have This will move the graph in our previous example 2 units right and 1 unit down. Both graphs are shown below.
- 3. Note that a = 3, b = 4, In this graph the transverse axis is horizontal. Thus each focus is a distance of 5 horizontally from the center. One focus is at (7, -1) and one is at (-3, -1). Note that the graphing calculator does not do a good job of showing the top and bottom halves of the branches of the hyperbola joining at the vertices which are located at (-3, -1) and (5, -1). Using this as a model, other equations describing hyperbolas with centers at (2, -1) can be written. and the slope of the asymptotes is If a = 3 and b = 2, and the transverse axis is horizontal, the equation is The slope of the asymptotes is and the vertices are located at (-1, -1) and (5, -1).
- 4. The foci are located at If a = 2 and b = 4, and the transverse axis is vertical, the equation is