3.1 3.2 Notes
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1. The document discusses coordinate planes and identifying points using ordered pairs. It provides examples of ordered pairs and asks the reader to identify other coordinates. 2. It then discusses a vertical line l that passes through the point (3,7) and asks the reader to find 6 points on the line and 6 points not on the line, as well as how to tell if a point is on the line based on its coordinates. 3. Finally, it introduces transformations on coordinate planes and asks the reader questions about finding points with specific x- and y-coordinates or where the y-coordinate is less than the x-coordinate. It provides homework problems for further practice.
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(run) between any two points.
Symbols
A line contains the
points (x1, y1) and
(x2, y2) .
m 5
rise
run 5
y2 2 y1
x2 2 x1
Diagram
(x2, y2)
(x1, y1)
O
x
y run
rise
Lesson
Vocabulary
• slope
• slope-intercept
form
• point-slope form
L
V
L
V
• s
LL
VVV
• s
hsm11gmse_NA_0307.indd 189 2/24/09 6:49:09 AM
http://media.pearsoncmg.com/aw/aw_mml_shared_1/copyright.html
Problem 1
Got It?
190 Chapter 3 Parallel and Perpendicular Lines
Finding Slopes of Lines
A What is the slope of line b?
m 5
2 2 (22)
21 2 4
5 4
25
5 245
B What is the slope of line d?
m 5
0 2 (22)
4 2 4
5 20 Undefi ned
1. Use the graph in Problem 1.
a. What is the slope of line a?
b. What is the slope of line c?
As you saw in Problem 1 and Got It 1 the slope of a line can be positive, negative,
zero, or undefi ned. Th e sign of the slope tells you whether the line rises or falls
to the right. A slope of zero tells you that the line is horizontal. An undefi ned slope
tells you that the line is vertical.
You can graph a line when you know its equation. Th e equation of a line has diff erent
forms. Two forms are shown below. Recall that the y-intercept of a line is the
y-coordinate of the point where the line crosses the y-axis.
O
y
c
b
d a
x
2
2 86
4
6
( 1, 2)
(4, 2)
(4, 0)
(5, 7)(1, 7)
(2, 3)
Positive slope
O
x
y
Negative slope
O
x
y
Zero slope
O
x
y
Undefined slope
O
x
y
Key Concept Forms of Linear Equations
Defi nition
Th e slope-intercept form of an equation of
a nonvertical line is y 5 mx 1 b, where m
is the slope and b is the y-intercept.
Symbols
Th e point-slope form of an equation of a
nonvertical line is y 2 y1 5 m(x 2 x1),
where m is the slope and (x1, y1) is a point
on the line.
y mx b
slope y-intercept
y y1 m(x x1)
slope x-coordinatey-coordinate
A
...
Coordinate System.pptx
The rectangular coordinate system (also known as Cartesian plane) was developed by René Descartes and uses two perpendicular number lines (x-axis and y-axis) that intersect at the origin (0,0) to locate points in a plane. Any point is identified with an ordered pair (x,y) denoting its distances from the x-axis and y-axis. The axes divide the plane into four quadrants, and points can be plotted or identified from their coordinates.
Coordinate System.pptx
The rectangular coordinate system (also known as the Cartesian plane) was developed by René Descartes and uses two perpendicular number lines (the x-axis and y-axis) that intersect at the origin (0,0) to locate points in a plane. Any point is identified with an ordered pair (x,y) denoting its distances from the x-axis and y-axis. The axes divide the plane into four quadrants, and points can be plotted or identified based on their coordinates.
Coordinate geometry
The branch of mathematics which deals with location of objects in 2-D (dimensional) plane is called coordinate geometry. Need to present your work in most impressive & informative manner i.e. through Power Point Presentation call us at skype Id: kumar_sukh79 or mail us: clintech2011@gmail.com for using my service.
Rectangular Coordinates System.pptx
1. The rectangular coordinate system, also known as the Cartesian plane, is composed of two perpendicular number lines that intersect at the origin (0,0).
2. It is used to plot and locate ordered pairs by measuring their distance from the origin along the x-axis and y-axis.
3. Real-world examples where the rectangular coordinate system is used include seating charts, maps, and describing the location of objects in a room.
Co ordinate geometry
This document provides information about coordinate geometry. It begins by explaining how a coordinate system allows one to precisely locate points on a plane using ordered pairs of numbers (x,y). It then defines the four quadrants defined by the x and y axes, and provides examples of plotting points from their coordinates. Finally, it derives the distance formula to calculate the distance between two points from their x and y coordinates.
4.1 4.2 Notes A
The document provides instruction and examples for calculating slope between two points on a graph. It defines slope as "rise over run", or the change in the y-values divided by the change in the x-values between two points. Examples are given for finding the slope between points, identifying positive and negative slopes, and special cases where the slope is zero or undefined. Students are asked to provide their own examples comparing slopes and complete an exit slip with slope problems.
4.1 4.2 Notes A
The document provides instruction and examples for calculating slope between two points on a graph. It defines slope as "rise over run", or the change in the y-values divided by the change in the x-values between two points. Examples are given for finding the slope between points, identifying positive and negative slopes, and special cases where the slope is zero or undefined. Students are asked to provide their own examples comparing slopes and complete an exit slip with slope problems.
3D Geometry Theory 3
The document discusses determining the angle between two lines and finding the point of intersection between two lines in 3D space. It provides the equations and process for finding the angle between two lines given their direction ratios. It also outlines the steps to find the point of intersection, which involves setting the coordinates of a point on each line equal to determine values for lambda and mu, and substituting those values back into one of the line equations. An example problem demonstrates finding the intersection point of two lines.
4.1 4.2 Notes B
This document appears to be notes from a math lesson that discusses slope. It includes examples of calculating slope between different points, the definition of slope as "rise over run", and special cases like undefined and zero slopes. The homework assigned is to complete practice problems calculating and applying slopes from pages 311-312 and 317-318 of the textbook.
Von(pa savepoh...)
The linear equation for the line is y = 2x
The slope of the line is 2. This is found by taking the change in y over the change in x between any two points. For example, between (0,0) and (1,2), the change in y is 2 and the change in x is 1, so the slope is 2/1 = 2.
Any linear equation can be written in the form y = mx + b, where m is the slope and b is the y-intercept (the y value when x is 0). Plugging in the points given, we can see that b must be 0. Therefore, the linear equation that fits the data is y = 2x.
Cartesian Coordinate System
The document explains how to use a coordinate system to precisely locate points on a flat surface. It introduces the x-axis and y-axis, which intersect at the origin point (0,0) and divide the plane into four quadrants. Any point can be specified with an ordered pair of numbers representing its distance from each axis, like (4,2), which would be four units from the y-axis and two units from the x-axis. Several examples are given of plotting and translating points between different locations on the grid.
Math - analytic geometry
1) The document provides a refresher on analytic geometry concepts including the Cartesian plane, lines, parabolas, ellipses, and circles. It gives definitions, properties, and equations for these concepts.
2) Examples are worked through, such as finding the coordinates of points, slopes of lines, and equations of lines and circles. Practice problems and their solutions are also provided.
3) Key topics covered include the Cartesian plane, distance between points, slope and equations of lines, parallel and perpendicular lines, conic sections including parabolas, circles, and ellipses, and their defining properties and equations.
Astaño
This document discusses rectangular coordinate systems in three dimensions. It begins by introducing the 3D coordinate system with three perpendicular axes (x, y, z) that intersect at the origin. Points in 3D space are located using ordered triples (x, y, z). It then discusses visualizing points and objects in 3D, finding distances between points, and graphing simple objects like lines and planes. Examples are provided to illustrate key concepts like plotting points, finding distances, and graphing lines and planes defined by equations.
C08s3
This document discusses polar form and DeMoivre's theorem for complex numbers. It begins by introducing polar form, which represents a complex number z as z = r(cosθ + i sinθ) where r is the modulus and θ is the argument. It then states DeMoivre's theorem, which says that for any positive integer n, zn = rn(cosnθ + i sinnθ). An example calculates (−1 + √3i)12 by first converting to polar form and then applying DeMoivre's theorem.
Section 11 1-notes_2
1. The document introduces polar coordinates as an alternative to Cartesian coordinates for locating points on a plane. Polar coordinates use a distance (r) from a fixed point (the pole) and an angle (θ) instead of x and y values.
2. Several examples are worked through to convert between polar and Cartesian coordinates and graph basic polar equations like r = constants and r = functions of θ.
3. Properties of more complex polar equations are explored through graphing, including equations of the form r = a + bsinθ, r = acosθ, and r2 = functions of θ. Symmetry properties are also discussed.
fundamentals of 2D and 3D graphs
The document discusses coordinate systems and graphs. It describes:
1) The Cartesian coordinate system uses perpendicular x and y axes to locate points on a plane using coordinates like (x,y).
2) 3D coordinate systems extend this to locate points in space using x, y, and z coordinates like (x,y,z).
3) Graphs can show the relationship between variables by plotting their coordinates and drawing a line or curve through the points.
G6 m3-c-lesson 16-s
This document provides a lesson on symmetry in the coordinate plane. It introduces opposite numbers and extends the concept to points on the coordinate plane. Examples are given comparing the coordinates and locations of points that are reflections of each other across the x- and y-axes. Students are asked to locate points and write their coordinates that are reflections of a given point across different axes. They are then asked questions about lines of symmetry based on relationships between point coordinates.
Similar to 3.1 3.2 Notes (20)
Lesson 3-7 Equations of Lines in the Coordinate Plane 189.docx
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2.2 2.3 notes a
The document describes a numeric pattern where each row contains consecutive odd integers centered around 1. It asks students to conjecture the pattern and sum of terms in each row. It also provides homework questions on conditional statements, deductive reasoning, and analyzing the truth value of related conditional statements.
1.6 1.7 notes b
This document provides examples and explanations of the midpoint and distance formulas, as well as transformations in the coordinate plane. It includes 4 examples of using the midpoint and distance formulas to find midpoints and lengths of segments. It also defines different types of transformations, such as translations, reflections, and rotations. Students are given homework problems applying these concepts, and examples of identifying transformations and describing them with arrow notation.
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This document provides homework questions and a review packet for a chapter. It lists 7 numbered sections that appear to be questions or tasks related to reviewing material from the first chapter. The document aims to help students review and reinforce their understanding of the concepts covered in the initial chapter through completing the homework and review activities.
1.6 1.7 notes b
This document provides examples and explanations of the midpoint and distance formulas, as well as transformations in the coordinate plane. It includes 4 examples of using the midpoint and distance formulas to find midpoints and lengths of segments. It also defines different types of transformations, such as translations, reflections, and rotations. Students are given homework problems applying these concepts, and examples of identifying transformations and describing them with arrow notation.
1.6 1.7 notes a
This document provides examples and explanations of the midpoint and distance formulas, as well as transformations in the coordinate plane. It includes 4 examples of using the midpoint and distance formulas to find midpoints and lengths of line segments. It also discusses the different types of transformations (translations, reflections, rotations, and dilations) and provides examples of identifying each type using arrow notation. The homework assignments are to complete practice problems from sections 1.6 and 1.7 in the textbook, as listed.
1.6 1.7 notes a
This document provides examples and explanations of the midpoint and distance formulas, as well as transformations in the coordinate plane. It includes 4 examples of using the midpoint and distance formulas to find midpoints and lengths of line segments. It also discusses the different types of transformations - translations, reflections, rotations, and dilations - and provides examples of identifying each type of transformation using arrow notation. The homework assignments are to complete practice problems from sections 1.6 and 1.7 in the textbook, as listed.
1.5 1.6 notes b
1. The document provides instructions for a quiz and homework assignments on geometry chapters 1.5 and 1.6.
2. Students are asked to work on their vocabulary packet silently after finishing the quiz. Then the class will review geometry formulas.
3. The homework assignments are to complete problems on page 38 from section 1.5 and page 47 from section 1.6 in the textbook. These cover midpoints, distances, and the midpoint and distance formulas.
1.5 1.6 notes b
1. The document provides instructions for a quiz and homework assignments on geometry chapters 1.5 and 1.6.
2. It includes examples of using the midpoint formula to find the midpoint of a segment and using given midpoints and endpoints to find missing endpoints.
3. There are also examples of using the Pythagorean theorem and distance formula to find the length of a segment between two points.
2.6 2.7 notes a
1. The document provides instructions and tasks for students to complete mathematical expressions, homework questions, and a lesson on reversible and non-reversible operations.
2. Students are asked to simplify expressions, complete homework problems, and determine whether example operations are reversible by considering if the starting number can be determined.
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2.4 notes b
1. Students were asked to put math problems on the board from previous homework. The document then provides examples of expressions and teaches how to simplify them using order of operations and properties like the distributive property. Students are asked to simplify sample expressions involving variables.
2. The document reviews that expressions need to have "like terms" to be simplified, such as terms with the same variables. Students practice simplifying expressions with multiple variables and terms by combining like terms.
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2.3 notes b
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8.7 notes b
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8.7 notes b
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8.7 notes
The document discusses a 4 step process but provides no details on the actual steps or content of the process. It references numbered sections but provides no information within those sections. Overall, the document does not contain any substantive information that could be summarized due to the lack of details provided within the numbered sections.
8.7 d2 a
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3.1 3.2 Notes
- 1. Day 31: 3.1 and 3.2 Getting Started & Transformations Launch: Overview of Coordinate Plane y‐axis origin x‐axis y‐coordinate ordered pair x‐coordinate 1
- 2. Identifying Ordered Pairs: I. (1, 2) II. (‐1, 2) III. (1, ‐2) IV. (0, 0) F B What are the other E A O coordinates? C D 2
- 3. Other Coordinate l Plane Practice: (3, 7) Line l is vertical and passes through (3, 7). a. Find 6 points on l. b. Find 6 points not on l. c. How can you tell if a point is or isn't on l by looking at its coordinates? 3
- 4. For you to think 1. The y‐axis is the only vertical line that passes about: through (0, 0). What do the coordinates of the points on the y‐axis have in common? Do the points on the x‐axis follow the same pattern? 2. Graph a horizontal line through (‐2, 1) in a coordinate plane. Name two other points on the line. 4
- 5. What about this? 3. Shade the region or set of points on a coordinate plane where you find (a,b) in the following cases: a. 3≤a≤4 and 0≤b≤1 b. 2≤a≤5 and 0≤b≤1 a. b. Complete #8 on page 191 on your own. (3 minutes) 5
- 8. 8
- 9. Homework: pg. 193 #11, 15, 19 pg. 200 #6‐8, 10, 12, 13 9
- 10. 10