2 8 Bzca5e
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The document discusses distance and midpoint formulas as well as circles. It provides examples of finding the distance between two points, finding the midpoint of a line segment between two points, writing the standard form equation of a circle given its center and radius, and finding the center and radius of a circle given its equation in general form. It also discusses graphing circle equations and identifying the domain and range from the graph.
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The document defines a parabola and its key components: focus, directrix, vertex, and axis of symmetry. It explains that the standard form of a parabolic equation depends on whether the axis of symmetry is horizontal or vertical. Examples are given of identifying the focus, directrix, vertex, and axis of symmetry from standard form equations. The standard form and properties of parabolas are outlined to determine these components from equations.
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The student is able to (I can):
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The document discusses key concepts in analytic geometry including:
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- The simplest and most well-known geometric figure is the line, which can be characterized by simple parameters and describes the shortest distance between two points.
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This video illustrates how to solve problems involving the distance formula, midpoint formula, and section formula. It contains 5 example problems: 1) finding the midpoint of a line segment on the x-axis, 2) finding the ratio in which a point divides a line segment internally, 3) finding the ratio a point divides a line segment, 4) finding the area of a triangle given 3 points, and 5) finding a point that is a given distance from two other points using the distance formula. The video explains each formula and step of solving the example problems in detail.
Module#9 notes
The document discusses the standard equation forms of a hyperbola depending on whether it has a horizontal or vertical transverse axis. For a hyperbola with a horizontal transverse axis, the standard equation is (x-h)^2/a^2 - (y-k)^2/b^2 = 1. For a hyperbola with a vertical transverse axis, the standard equation is (y-k)^2/a^2 - (x-h)^2/b^2 = 1. The document also lists key properties of hyperbolas including the center, vertices, foci, and asymptotes.
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The student is able to (I can):
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• Find the distance between two points.
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Module#9 notes
The document discusses the standard equation forms of a hyperbola depending on whether it has a horizontal or vertical transverse axis. For a hyperbola with a horizontal transverse axis, the standard equation is (x-h)^2/a^2 - (y-k)^2/b^2 = 1. For a hyperbola with a vertical transverse axis, the standard equation is (y-k)^2/a^2 - (x-h)^2/b^2 = 1. The document also lists key properties of hyperbolas including the center, vertices, foci, and asymptotes.
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A hyperbola is defined as the set of all points whose distance from two fixed points (called foci) has a constant difference. The standard equation of a hyperbola depends on whether its transverse axis is horizontal or vertical. Key properties include the center, vertices, foci, and asymptotes. Examples are provided to demonstrate how to determine these properties from a given standard equation of a hyperbola.
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The document provides information about conic sections and ellipses, including:
- The standard form equation of an ellipse is given
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This document discusses the general equation of a circle and provides examples of finding the center and radius of circles given their equations.
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Here are the steps to solve these problems:
1. The points (x1, y1) and (x2, y2) form a diameter of a circle. The point (x, y) is another point on the circle.
(a) The gradient of the diameter AB is (y2-y1)/(x2-x1).
(b) The equation of AB is y-y1 = (y2-y1)/(x2-x1)(x-x1)
(c) Since P lies on AB, substitute the point (x, y) into the equation of AB to determine the value of x.
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Find the midpoint of two given points.
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This document contains examples and explanations of circles and ellipses. It defines a circle as a locus of points that are a constant distance r from a fixed center point C. It provides examples of writing equations of circles given the center and radius. It also discusses finding the center and radius from a standard circle equation. For ellipses, it explains how to write the standard equation given the vertices and covertices, and how to find the foci from the ellipse equation.
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A hyperbola is defined as the set of all points whose distance from two fixed points (called foci) has a constant difference. The standard equation of a hyperbola depends on whether its transverse axis is horizontal or vertical. Key properties include the center, vertices, foci, and asymptotes. Examples are provided to demonstrate how to determine these properties from a given standard equation of a hyperbola.
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The document analyzes the real-world applications of concepts from analytical geometry in three examples:
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1) This document discusses how to plot points on a coordinate plane given their coordinates and scale, and how to find the distance between two points.
2) It provides examples of finding the distance between points that have a common x-coordinate and between points that have a common y-coordinate.
3) An exercise is included that asks the reader to find the distance between several pairs of points with given coordinates.
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The document provides the steps to write the equation of a parabola in standard form given certain information about its properties. It then works through two example problems:
1) Given the equation of a parabola in general form, it rewrites the equation in standard form and determines the vertex, focus, directrix, and axis of symmetry.
2) Given the vertex and directrix of a parabola, it derives the standard form equation that satisfies those conditions.
3) Given the height and base width of an arch with a parabolic shape, it illustrates the situation, determines the necessary values, and writes the standard form equation that models the arch's shape.
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The document discusses the standard form of a circle equation and how to find the center and radius from the equation. It provides examples of writing the equation of a circle given the center and/or radius or a point on the circle. The standard form is (x - h)2 + (y - k)2 = r2, where (h, k) are the coordinates of the center and r is the radius. If the equation is not in standard form, it may need to be completed into a perfect square to extract the center and radius.
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The document discusses determining the equation of a circle given its diameter or radius and center. It provides an example of finding the standard form equation of a circle given the endpoints of its diameter (-3,6) and (3,-2). It also presents a word problem about determining if a point (3,3) lies within a danger zone defined as a circle of radius 4km centered at the origin. It shows solving this graphically and by substituting the point into the standard form equation.
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The document provides information about conic sections and ellipses, including:
- The standard form equation of an ellipse is given
- Steps are shown to find the center, vertices, foci, minor axis length, and major axis length of an ellipse given in standard form
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*Correctly use notation for distance and segments
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This document discusses conic sections and circles. It defines conic sections as sections obtained when a plane cuts through a circular cone. It defines a circle as the set of all points equidistant from a fixed center point, where the fixed distance is called the radius. The document also mentions that Apollonius, a Greek mathematician, studied conic sections and gave them their names. He believed they should be studied for their mathematical beauty rather than practical applications.
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This document discusses the general equation of a circle and provides examples of finding the center and radius of circles given their equations.
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The document discusses the standard form of the equation of a circle and how to find the center and radius from the equation. It provides examples of writing the equation given the center and radius, as well as finding the center and radius from an equation. Key aspects covered include:
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- You can find the center and radius by comparing the equation to the standard form
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Equation of a circle
Here are some things you did well and could improve on:
WWW:
- You explained the key concepts around writing the equation of a circle clearly and concisely. Breaking it down step-by-step makes it easy to understand.
- Providing examples with worked solutions is very helpful for reinforcement. The visual diagrams additionally aid comprehension.
- Giving practice problems for students to try on their own, along with answers, allows for application of the material.
EBI:
- Some of the text could be formatted for easier reading (e.g. consistent formatting of equations).
- Adding brief summaries or recaps after sections of explanation may aid retention.
- Providing guidance on common errors
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Find the coordinates of a point a fractional distance from one end of a segment.
mathemaics 10 lesson about cicles. its part
grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements.grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corresponding measurements. grade 10 lesson about circle. mathematocs lesson about circles. parts of the circles and the corr
Ellipse.pptx
1. The document defines an ellipse and its key properties including its standard equation form. It discusses how an ellipse is a set of points where the sum of the distances to two fixed points (foci) is constant.
2. Parts of an ellipse like its vertices, covertices, axes, and directrices are defined. The standard equation of an ellipse centered at the origin is derived.
3. Examples are provided of determining the coordinates of foci, vertices, covertices, and directrices from equations. Problems involving finding equations or properties given certain conditions are also presented.
Circle
The document provides a history and overview of circles. It discusses:
1) Euclid defined circles in 300 BC in his work "The Elements", establishing circles as fundamental objects in geometry. He defined a point and line and established early theorems about circles.
2) Key parts of a circle include the radius, diameter, circumference, chord, secant, tangent, arc, sector, and segment. The circumference is related to the diameter by pi.
3) Circles can be defined analytically using equations in the standard "center-radius" form of (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the
Circle
The document discusses circles and their equations. It defines a circle as all points in a plane that are a fixed distance from a fixed center point. This fixed distance is called the radius. The standard form of a circle equation is (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the center and r is the radius. It also discusses converting between the standard and general forms of a circle equation.
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This document discusses completing the square to factor quadratic trinomials. It provides examples of factoring expressions of the form x^2 + bx + c by finding the constant to add inside the square. These quadratic trinomials can be written as perfect squares minus/plus a constant. The document demonstrates completing the square on sample expressions and emphasizes that this method allows solving quadratic equations using square roots rather than the quadratic formula.
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This document discusses two objectives related to calculating areas and integrals. The first objective covers finding the area under a curve using integrals, while the second objective likely expands on this topic or covers related concepts.
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The document discusses finding the equation of the tangent line to the graph of f(x)=2x^2 at the point (1,2). It also discusses the motion of a ball thrown straight up in the air, with the equation h(t)=-16t^2 + 64t, where t is time. It asks for the time when the ball strikes the ground, passes the rooftop on the way down, and its instantaneous speed at various times.
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This section discusses one-sided limits and continuous functions. It has two objectives: 1) To define and evaluate one-sided limits and use them to determine if a function is continuous at a point. 2) To use limits to analyze the graph of a function near any discontinuities and determine if they are removable, jump, or essential discontinuities.
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This document discusses finding limits using tables and graphs. The first objective relates to using tables and graphs to find limits, while the second objective likely continues the discussion on finding limits through tables and graphs. In summary, the document covers finding limits through tables and graphs across two objectives.
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This document discusses permutations and combinations. It contains examples of calculating the number of possible codes, lineups, and committees given different conditions like allowing or not allowing repetition. It also provides examples of listing combinations and calculating the number of arrangements of different objects like letters, flags, and people on committees.
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The document discusses three counting objectives: 1) Listing all subsets of the set {1,2,3}, 2) Determining the number of students in a survey of 200 that had been to Florida or New York or neither, and 3) Calculating the number of possible three-symbol code words where the first symbol is an uppercase letter, second is a digit, and third is a lowercase letter.
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Mathematical induction is a method of proof typically used to establish that a property holds for all natural numbers. The principle of mathematical induction states that to show a statement is true for all natural numbers n, it must be shown that the statement holds true for the base case n=1, and also that if the statement holds true for any natural number k, then it must also hold true for k+1. This allows proving statements about an infinite set by considering only a finite number of cases.
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This document discusses geometric sequences and series. It lists 4 objectives: 1) Identifying geometric sequences by listing the first term and common ratio, 2) and 3) Contain no information but indicate additional objectives, 4) Contains no information. The example geometric sequence given is 2, 8, 32, 128, ... with a first term of 2 and common ratio of 4.
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This document contains objectives and problems regarding arithmetic sequences. Objective 1 introduces arithmetic sequences and asks to identify the first term and common difference of sequences. Objective 2 asks to find specific terms in arithmetic sequences and identify the first term and common difference given additional information. Objective 3 asks to find sums of terms in arithmetic sequences and evaluate numeric expressions that represent arithmetic sequences. The final problem asks to calculate the total number of seats in a stadium section where each row has two more seats than the row in front of it.
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2 8 Bzca5e
- 1. Section 2.8 Distance and Midpoint Formulas; Circles
- 5. Example Find the distance between (4,-5) and (9,-2).
- 7. (-1,5) (6,8)
- 8. Example Find the midpoint of the segment whose endpoints are (-1,7) and (-5,9).
- 9. Circles
- 12. Write the standard form of the equation of the circle with center (-4,1) and radius of 3. Standard Form (-4,1) 3
- 13. Find the center and radius of the circle whose equation is Graph the equation. Use the graph to identify the relation’s domain and range. Why is it a relation and not a function? 3 (-3,4) Domain: [-6,0]; Range:[1,7] Center(-3,4); radius=3
- 14. Example Write the standard form of the equation of the circle with center at (-2,7) and a radius of 5.
- 15. Example Find the center and radius of the circle whose equation is below. Graph the equation. Use the graph to identify the relation’s domain and range.
- 16. General Form
- 17. ? ?
- 18. Example Complete the square and write the equation in standard form. Then give the center and radius of the circle and graph the equation.
- 19. Example Complete the square and write the equation in standard form. Then give the center and radius of the circle and graph the equation.
- 20. (a) (b) (c) (d) Find the distance between the points (-1,8) and (9,5).
- 21. (a) (b) (c) (d) Find the midpoint of the line segment with the endpoints of (-1,-1) and (- 5,8).
- 22. (a) (b) (c) (d) Write the equation of the line with center at (0,7) and a radius of 4.