- A line is defined as a set of points on a plane where the slope between any two points is equal. A line on the Cartesian plane can be described by a linear equation.
- The slope of a line describes its direction, and can be calculated using the rise over run formula. Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals.
- The equation of a line can be written in various forms, including slope-intercept form where y = mx + b, with m being the slope and b being the y-intercept.
The document discusses the concept of slope and how it is used to describe the steepness of a line. It defines slope as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. Several forms of linear equations are presented, including point-slope form, slope-intercept form, and standard form. Relationships between parallel and perpendicular lines based on their slopes are also described. Examples are provided to demonstrate finding slopes, writing equations of lines, and determining if lines are parallel or perpendicular based on their slopes.
The document discusses point-slope form and writing linear equations. It provides examples of writing equations for lines that are parallel and perpendicular to given lines. Specifically, it explains that parallel lines have the same slope and different y-intercepts, while perpendicular lines have slopes that are negative reciprocals of each other. Examples are given of writing equations for parallel and perpendicular lines using point-slope form based on a given point and the slope of another line.
This lecture discusses distance, midpoint, slope, lines, symmetries of graphs, equations of circles, and quadratic equations. It defines distance as the square root of the sum of the squared differences of x- and y-coordinates between two points. The midpoint formula finds the midpoint of a line segment between two points. Slope is defined as the rise over the run between two points on a line. Lines can be written in point-slope form, slope-intercept form, and intercept forms. Parallel and perpendicular lines are identified based on equal or negative reciprocal slopes. Symmetries of graphs include reflections across the x-axis, y-axis, or origin. The equation of a circle is given by (x-h)2
* Find the slope of a line.
* Use slopes to identify parallel and perpendicular lines.
* Write the equation of a line through a given point
- parallel to a given line
- perpendicular to a given line
The document discusses linear equations and graphing linear relationships. It defines key terms like slope, y-intercept, x-intercept, and provides examples of writing linear equations in slope-intercept form and point-slope form given certain information like a point and slope. It also discusses using tables and graphs to plot linear equations and find slopes from graphs or between two points. Real-world examples of linear relationships are provided as well.
Lesson 3-7 Equations of Lines in the Coordinate Plane 189.docxSHIVA101531
Lesson 3-7 Equations of Lines in the Coordinate Plane 189
3-7
Objective To graph and write linear equations
Ski resorts often use steepness to rate the difficulty
of their hills. The steeper the hill, the higher the
difficulty rating. Below are sketches of three new hills
at a particular resort. Use each rating level only once.
Which hill gets which rating? Explain.
Difficulty Ratings
Easiest
Intermediate
Difficult
3300 ft 3000 ft 3500 ft
1190 ft 1180 ft 1150 ft
A B C
Th e Solve It involves using vertical and horizontal distances to determine steepness.
Th e steepest hill has the greatest slope. In this lesson you will explore the concept of
slope and how it relates to both the graph and the equation of a line.
Essential Understanding You can graph a line and write its equation when you
know certain facts about the line, such as its slope and a point on the line.
Equations of Lines in the
Coordinate Plane
Think back!
What did you
learn in algebra
that relates to
steepness?
Key Concept Slope
Defi nition
Th e slope m of a line is the
ratio of the vertical change
(rise) to the horizontal change
(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
...
Unit 4 hw 8 - pointslope, parallel & perpLori Rapp
The document discusses the point-slope formula for writing the equation of a line given a point and slope. It provides examples of using the formula, such as writing the equation of the line through point (3, -2) with slope 5. It also discusses that horizontal lines have a slope of 0 and the equation y=b, since the y-coordinate remains constant while the x-coordinate changes. The slope of a horizontal line is 0 because when calculating slope using two points, the change in y-values is 0.
Straight_Lines in mat 111 full lecture.pptElaiyarajaR1
A straight line is the shortest distance between two points. It can be defined using different forms of equations like slope-intercept, two point, and normal forms. The slope of a line is calculated as the tangent of the angle between the line and the x-axis. Parallel lines have equal slopes, while perpendicular lines have slopes that multiply to -1. A line's equation can be found using various properties like slope, two points on the line, or intercepts with the axes.
The document discusses the concept of slope and how it is used to describe the steepness of a line. It defines slope as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. Several forms of linear equations are presented, including point-slope form, slope-intercept form, and standard form. Relationships between parallel and perpendicular lines based on their slopes are also described. Examples are provided to demonstrate finding slopes, writing equations of lines, and determining if lines are parallel or perpendicular based on their slopes.
The document discusses point-slope form and writing linear equations. It provides examples of writing equations for lines that are parallel and perpendicular to given lines. Specifically, it explains that parallel lines have the same slope and different y-intercepts, while perpendicular lines have slopes that are negative reciprocals of each other. Examples are given of writing equations for parallel and perpendicular lines using point-slope form based on a given point and the slope of another line.
This lecture discusses distance, midpoint, slope, lines, symmetries of graphs, equations of circles, and quadratic equations. It defines distance as the square root of the sum of the squared differences of x- and y-coordinates between two points. The midpoint formula finds the midpoint of a line segment between two points. Slope is defined as the rise over the run between two points on a line. Lines can be written in point-slope form, slope-intercept form, and intercept forms. Parallel and perpendicular lines are identified based on equal or negative reciprocal slopes. Symmetries of graphs include reflections across the x-axis, y-axis, or origin. The equation of a circle is given by (x-h)2
* Find the slope of a line.
* Use slopes to identify parallel and perpendicular lines.
* Write the equation of a line through a given point
- parallel to a given line
- perpendicular to a given line
The document discusses linear equations and graphing linear relationships. It defines key terms like slope, y-intercept, x-intercept, and provides examples of writing linear equations in slope-intercept form and point-slope form given certain information like a point and slope. It also discusses using tables and graphs to plot linear equations and find slopes from graphs or between two points. Real-world examples of linear relationships are provided as well.
Lesson 3-7 Equations of Lines in the Coordinate Plane 189.docxSHIVA101531
Lesson 3-7 Equations of Lines in the Coordinate Plane 189
3-7
Objective To graph and write linear equations
Ski resorts often use steepness to rate the difficulty
of their hills. The steeper the hill, the higher the
difficulty rating. Below are sketches of three new hills
at a particular resort. Use each rating level only once.
Which hill gets which rating? Explain.
Difficulty Ratings
Easiest
Intermediate
Difficult
3300 ft 3000 ft 3500 ft
1190 ft 1180 ft 1150 ft
A B C
Th e Solve It involves using vertical and horizontal distances to determine steepness.
Th e steepest hill has the greatest slope. In this lesson you will explore the concept of
slope and how it relates to both the graph and the equation of a line.
Essential Understanding You can graph a line and write its equation when you
know certain facts about the line, such as its slope and a point on the line.
Equations of Lines in the
Coordinate Plane
Think back!
What did you
learn in algebra
that relates to
steepness?
Key Concept Slope
Defi nition
Th e slope m of a line is the
ratio of the vertical change
(rise) to the horizontal change
(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
...
Unit 4 hw 8 - pointslope, parallel & perpLori Rapp
The document discusses the point-slope formula for writing the equation of a line given a point and slope. It provides examples of using the formula, such as writing the equation of the line through point (3, -2) with slope 5. It also discusses that horizontal lines have a slope of 0 and the equation y=b, since the y-coordinate remains constant while the x-coordinate changes. The slope of a horizontal line is 0 because when calculating slope using two points, the change in y-values is 0.
Straight_Lines in mat 111 full lecture.pptElaiyarajaR1
A straight line is the shortest distance between two points. It can be defined using different forms of equations like slope-intercept, two point, and normal forms. The slope of a line is calculated as the tangent of the angle between the line and the x-axis. Parallel lines have equal slopes, while perpendicular lines have slopes that multiply to -1. A line's equation can be found using various properties like slope, two points on the line, or intercepts with the axes.
This document contains information about straight lines and their various forms of equations. It discusses the slope-intercept form, intercept form, two-point form and general form of a straight line. It provides examples of finding the equation of a line given certain conditions like two points, slope and intercept, perpendicular or parallel lines etc. There are also examples of problems involving finding slope, intercepts, perpendicular or parallel lines to a given line. The document contains the necessary formulas and steps to solve such problems.
This document provides a lesson on identifying and writing equations for parallel and perpendicular lines. It includes examples of identifying parallel lines based on their having the same slope. Perpendicular lines are defined as having slopes whose product is -1. The document also demonstrates how to write equations for lines parallel or perpendicular to a given line based on their point and slope. Examples are provided to illustrate finding slopes from graphs and using them to determine if lines are parallel or perpendicular.
Here are the steps to solve these problems:
1. Find the slopes of the two lines:
m1 = (8-2)/(5--2) = 6/3 = 2 (slope of r)
m2 = (7-0)/(-8--2) = 7/-6 = -1 (slope of s)
The slopes are negative reciprocals, so r ⊥ s.
2. The slopes are m1 = 2 and m2 = -1/2. Since m1 × m2 = -1, the lines are perpendicular.
3. The given line has slope 3. The perpendicular line will have slope -1/3. Plug into the point-slope form
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document defines key concepts related to straight lines, including their various forms of equations, slope, intercepts, and how to calculate them. It discusses the general, standard, point-slope, and intercept forms of linear equations. It also explains how to find the slope between two points using the slope formula, and how to determine the x-intercept and y-intercept from a line's equation. Examples are provided to illustrate these concepts.
1. This document discusses linear equations, slope, graphing lines, writing equations in slope-intercept form, and solving systems of linear equations.
2. Key concepts explained include slope as rise over run, the different forms of writing a linear equation, finding the x- and y-intercepts, and using two points to write the equation of a line in slope-intercept form.
3. Examples are provided to demonstrate how to graph lines based on their equations in different forms, find intercepts, write equations from two points, and solve systems of linear equations.
The document discusses equations of lines. It introduces the linear equation y = 2x - 1 and shows how to graph it by substituting values for x and finding the corresponding y-values. This forms the line's points (ordered pairs). It explains that a linear equation can be written in the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. It provides examples of writing equations from their graphs in slope-intercept form and graphing lines from equations written in this form by using the slope and y-intercept. Finally, it gives examples of writing equations of lines parallel or perpendicular to given lines that pass through a given point.
The document discusses various concepts relating to straight lines in mathematics including:
1) Calculating the gradient of a straight line between two points.
2) Horizontal and vertical lines having gradients of 0 or being undefined.
3) The relationship between gradient and angle of a line.
4) Finding the midpoint, collinearity of points, and gradients of perpendicular lines.
The document discusses slope and the slope-intercept form of linear equations. It defines slope as the ratio of the rise over the run between two points on a line. Slope can be calculated using two points or using the difference of the y-coordinates over the difference of the x-coordinates. Horizontal and vertical lines have special cases for slope calculations. The slope-intercept form is defined as y=mx+b, where m is the slope and b is the y-intercept. Using this form, lines can be graphed by plotting the y-intercept and using the slope to find the second point and draw the line.
This document provides instruction on calculating and understanding slope. It defines slope as the steepness of a line and discusses how to calculate slope given two points on a line using the rise over run formula or the slope formula. It provides examples of finding the slope of lines from their graphs or given two points and discusses the slopes of horizontal and vertical lines.
This document provides instruction on calculating and understanding slope. It defines slope as the steepness of a line and discusses how to calculate slope given two points on a line using the rise over run formula or the slope formula. It provides examples of finding the slope of lines from their graphs or given two points and discusses the slopes of horizontal and vertical lines.
The document discusses linear equations and slope. It covers plotting points on a coordinate plane, calculating slope using the rise over run formula, writing equations in slope-intercept form, and graphing lines by making a table or using the slope and y-intercept. Methods are provided for determining the equation of a line given two points, the slope and a point, or by finding values from a graph.
The document discusses equations of lines, including:
1) The gradient-point form of a straight line equation which uses the gradient and coordinates of one point to determine the equation.
2) Calculating the gradient from two points on a line and using it to find the angle of inclination.
3) Determining the equation of a line parallel to another line, by setting their gradients equal since parallel lines have the same gradient.
This document provides an overview of linear equations and graphing lines. It covers basic coordinate plane information, plotting points, finding slopes, x- and y-intercepts, the slope-intercept form of a line, graphing lines using tables and slope-intercept form, and determining the equation of a line given different information like two points or a slope and point. The assignments section indicates students will apply these concepts to practice problems.
The document discusses linear equations in two variables. It will cover writing linear equations in standard and slope-intercept form, graphing linear equations using two points, intercepts and slope/point, and describing graphs by their intercepts and slope. Key topics include defining the standard form as Ax + By = C, rewriting equations between the two forms, using two points, x-intercept, y-intercept or slope/point to graph, and describing graphs by their slope and intercepts.
This document contains information about straight lines and their various forms of equations. It discusses the slope-intercept form, intercept form, two-point form and general form of a straight line. It provides examples of finding the equation of a line given certain conditions like two points, slope and intercept, perpendicular or parallel lines etc. There are also examples of problems involving finding slope, intercepts, perpendicular or parallel lines to a given line. The document contains the necessary formulas and steps to solve such problems.
This document provides a lesson on identifying and writing equations for parallel and perpendicular lines. It includes examples of identifying parallel lines based on their having the same slope. Perpendicular lines are defined as having slopes whose product is -1. The document also demonstrates how to write equations for lines parallel or perpendicular to a given line based on their point and slope. Examples are provided to illustrate finding slopes from graphs and using them to determine if lines are parallel or perpendicular.
Here are the steps to solve these problems:
1. Find the slopes of the two lines:
m1 = (8-2)/(5--2) = 6/3 = 2 (slope of r)
m2 = (7-0)/(-8--2) = 7/-6 = -1 (slope of s)
The slopes are negative reciprocals, so r ⊥ s.
2. The slopes are m1 = 2 and m2 = -1/2. Since m1 × m2 = -1, the lines are perpendicular.
3. The given line has slope 3. The perpendicular line will have slope -1/3. Plug into the point-slope form
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document defines key concepts related to straight lines, including their various forms of equations, slope, intercepts, and how to calculate them. It discusses the general, standard, point-slope, and intercept forms of linear equations. It also explains how to find the slope between two points using the slope formula, and how to determine the x-intercept and y-intercept from a line's equation. Examples are provided to illustrate these concepts.
1. This document discusses linear equations, slope, graphing lines, writing equations in slope-intercept form, and solving systems of linear equations.
2. Key concepts explained include slope as rise over run, the different forms of writing a linear equation, finding the x- and y-intercepts, and using two points to write the equation of a line in slope-intercept form.
3. Examples are provided to demonstrate how to graph lines based on their equations in different forms, find intercepts, write equations from two points, and solve systems of linear equations.
The document discusses equations of lines. It introduces the linear equation y = 2x - 1 and shows how to graph it by substituting values for x and finding the corresponding y-values. This forms the line's points (ordered pairs). It explains that a linear equation can be written in the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. It provides examples of writing equations from their graphs in slope-intercept form and graphing lines from equations written in this form by using the slope and y-intercept. Finally, it gives examples of writing equations of lines parallel or perpendicular to given lines that pass through a given point.
The document discusses various concepts relating to straight lines in mathematics including:
1) Calculating the gradient of a straight line between two points.
2) Horizontal and vertical lines having gradients of 0 or being undefined.
3) The relationship between gradient and angle of a line.
4) Finding the midpoint, collinearity of points, and gradients of perpendicular lines.
The document discusses slope and the slope-intercept form of linear equations. It defines slope as the ratio of the rise over the run between two points on a line. Slope can be calculated using two points or using the difference of the y-coordinates over the difference of the x-coordinates. Horizontal and vertical lines have special cases for slope calculations. The slope-intercept form is defined as y=mx+b, where m is the slope and b is the y-intercept. Using this form, lines can be graphed by plotting the y-intercept and using the slope to find the second point and draw the line.
This document provides instruction on calculating and understanding slope. It defines slope as the steepness of a line and discusses how to calculate slope given two points on a line using the rise over run formula or the slope formula. It provides examples of finding the slope of lines from their graphs or given two points and discusses the slopes of horizontal and vertical lines.
This document provides instruction on calculating and understanding slope. It defines slope as the steepness of a line and discusses how to calculate slope given two points on a line using the rise over run formula or the slope formula. It provides examples of finding the slope of lines from their graphs or given two points and discusses the slopes of horizontal and vertical lines.
The document discusses linear equations and slope. It covers plotting points on a coordinate plane, calculating slope using the rise over run formula, writing equations in slope-intercept form, and graphing lines by making a table or using the slope and y-intercept. Methods are provided for determining the equation of a line given two points, the slope and a point, or by finding values from a graph.
The document discusses equations of lines, including:
1) The gradient-point form of a straight line equation which uses the gradient and coordinates of one point to determine the equation.
2) Calculating the gradient from two points on a line and using it to find the angle of inclination.
3) Determining the equation of a line parallel to another line, by setting their gradients equal since parallel lines have the same gradient.
This document provides an overview of linear equations and graphing lines. It covers basic coordinate plane information, plotting points, finding slopes, x- and y-intercepts, the slope-intercept form of a line, graphing lines using tables and slope-intercept form, and determining the equation of a line given different information like two points or a slope and point. The assignments section indicates students will apply these concepts to practice problems.
The document discusses linear equations in two variables. It will cover writing linear equations in standard and slope-intercept form, graphing linear equations using two points, intercepts and slope/point, and describing graphs by their intercepts and slope. Key topics include defining the standard form as Ax + By = C, rewriting equations between the two forms, using two points, x-intercept, y-intercept or slope/point to graph, and describing graphs by their slope and intercepts.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
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A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
Copy_of_slopeofaline.ppt
1. What is a Line?
• A line is the set of points forming a straight
path on a plane
• The slant (slope) between any two points on
a line is always equal
• A line on the Cartesian plane can be
described by a linear equation
x-axis
y-axis
2. Definition - Linear Equation
• Any equation that can be put into the form
Ax + By C = 0, where A, B, and C are
Integers and A and B are not both 0, is called
a linear equation in two variables.
• The graph will be a straight line.
• The form Ax + By C = 0 is called standard
form (Integer coefficients all on one side = 0)
3. Definition - Linear Equation
• The equation of a line describes all of the
points on the line
• The equation is the rule for any ordered pair
on the line
1. 3x + 2y – 8 = 0
(4, -2) is on the line
(5, 1) is not on the line
2. x – 7y + 2 = 0
(4, -2) is not on the line
(5, 1) is on the line
Examples:
Test the point by plugging the x and y into the equation
5. Guard against 0 in
the denominator
Slope
If x1 x2, the slope of the line
through the distinct points P1(x1, y1)
and P2(x2, y2) is:
1
2
1
2
x
x
y
y
x
in
change
y
in
change
run
rise
slope
Why is
this
needed
?
7. Calculate the slope between (-3, 6) and (5, 2)
1
2
1
2
x
x
y
y
m
)
3
-
(
)
5
(
)
6
(
)
2
(
m
8
4
-
2
1
-
x1 y1 x2 y2
We use the letter m
to represent slope
m
8. Find the Slopes
(5, -2)
(11, 2)
(3, 9)
1
2
1
2
x
x
y
y
m
3
11
9
2
1
m
Yellow
5
11
)
2
-
(
2
2
m
Blue
3
5
9
2
-
3
m
Red
8
7
-
3
2
2
11
-
9. Find the slope between (5, 4) and (5, 2).
1
2
1
2
x
x
y
y
m
)
5
(
)
5
(
)
4
(
)
2
(
m
0
2
-
STOP
This slope is undefined.
x1 y1 x2 y2
10. x
y
Find the slope between (5, 4) and (5, 2).
Rise
Run
-2
0
Undefined
= =
11. Find the slope between (5, 4) and (-3, 4).
1
2
1
2
x
x
y
y
m
)
5
(
)
3
-
(
)
4
(
)
4
(
m
8
-
0
This slope is zero.
x1 y1 x2 y2
0
13. From these results we
can see...
•The slope of a vertical
line is undefined.
•The slope of a
horizontal line is 0.
14. Find the slope of the line
4x - y = 8
)
0
(
)
2
(
)
8
-
(
)
0
(
m
2
8
Let x = 0 to
find the
y-intercept.
8
-
8
-
8
)
0
(
4
y
y
y Let y = 0 to
find the
x-intercept.
2
8
4
8
)
0
(
4
x
x
x
(0, -8) (2, 0)
4
First, find two points on the line
x1 y1 x2 y2
15. Find the slope of the line
4x y = 8 Here is an easier way
Solve
for y.
8
4
y
x
8
4
-
-
x
y
8
4
x
y
When the equation is solved for y the
coefficient of the x is the slope.
We call this the slope-intercept form
y = mx + b
m is the slope and b is the y-intercept
17. Sign of the Slope
Which have a
positive slope?
Green
Blue
Which have a
negative slope?
Red
Light Blue
White
Undefined
Zero
Slope
18. Slope of Parallel Lines
• Two lines with the
same slope are parallel.
• Two parallel lines have
the same slope.
19. Are the two lines parallel?
L1: through (-2, 1) and (4, 5) and
L2: through (3, 0) and (0, -2)
)
0
(
)
3
(
)
2
-
(
)
0
(
2
m
)
2
-
(
)
4
(
)
1
(
)
5
(
1
m
6
4
3
2
3
2
2
1
2
1
L
L
m
m
This symbol means Parallel
21. Slopes of Perpendicular Lines
• If neither line is vertical then the slopes of
perpendicular lines are negative reciprocals.
• Lines with slopes that are negative
reciprocals are perpendicular.
• If the product of the slopes of two lines is -1
then the lines are perpendicular.
• Horizontal lines are perpendicular to
vertical lines.
22. Write parallel, perpendicular or neither for the
pair of lines that passes through (5, -9) and (3, 7)
and the line through (0, 2) and (8, 3).
)
5
(
)
3
(
)
9
-
(
)
7
(
1
m
)
0
(
)
8
(
)
2
(
)
3
(
2
m
2
-
16
8
-
8
1
1
8
-
8
1
8
8
-
1
-
2
1
2
1 1
-
L
L
m
m
This symbol means Perpendicular
24. Objectives
• Write the equation of a line, given its
slope and a point on the line.
• Write the equation of a line, given two
points on the line.
• Write the equation of a line given its
slope and y-intercept.
25. Objectives
• Find the slope and the y-intercept of a
line, given its equation.
• Write the equation of a line parallel or
perpendicular to a given line through a
given point.
27. Write the equation of the line
with slope m = 5 and y-int -3
Take the slope intercept form y = mx + b
Replace in the m and the b y = mx + b
y = 5x + -3
y = 5x – 3
Simplify
That’s all there is to it… for this easy question
28. Find the equation of the line
through (-2, 7) with slope m = 3
Take the slope intercept form y = mx + b
Replace in the y, m and x y = mx + b
7 = mx + b
x y m
7 = 3x + b
7 = 3(-2) + b
7 = -6 + b
Solve for b
7 + 6 = b
13 = b
Replace m and b back into
slope intercept form y = 3x + 13
29. Write an equation of the line
through (-1, 2) and (5, 7).
First calculate the slope.
b
)
1
-
(
2 6
5
1
2
1
2
x
x
y
y
m
)
1
-
(
5
2
7
6
5
Now plug into y, m and x into
slope-intercept form.
(use either x, y point)
Solve for b
Replace back into slope-intercept form
b
mx
y
b
6
5
-
2
b
6
5
2
b
6
17
6
17
6
5
x
y
Only replace
the m and b
30. Horizontal and
Vertical Lines
• If a is a constant,
the vertical line
though (a, b) has
equation x = a.
• If b is a constant,
the horizontal line
though ( a, b,) has
equation y = b.
(a, b)
31. Write the equation of the line
through (8, -2); m = 0
2
-
y
Slope = 0 means the line is horizontal
That’s all there is!
32. Find the slope and
y-intercept of
2x – 5y = 1
Solve for y and
then we will be
able to read it from
the answer.
1
5
2
y
x
y
x 5
1
2
y
x
5
1
5
2
5
1
x
5
2
y
5
2
m
5
1
-
5 5 5
Slope: y-int:
33. Write an equation for the line
through (5, 7) parallel to 2x – 5y = 15.
5
2
m
15
5
2
y
x
y
x 5
15
2
5
5
5
15
5
2 y
x
y
x
3
5
2
34. We know the slope and
we know a point.
)
7
,
5
(
5
2
m
b
)
5
(
7 5
2 b
mx
y
7 = 2 + b
7 – 2 = b
5 = b
5
5
2
x
y
Write an equation for the line
through (5, 7) parallel to 2x – 5y = 15.
36. The slope of the perpendicular.
• The slope of the perpendicular line is the
negative reciprocal of m
• Flip it over and change the sign.
3
2
Examples of slopes of perpendicular lines:
-2
5
1
2
7
-
2.4
Note: The product of perpendicular slopes is -1
2
3
1
5
= -5
-2
1 2
1
12
5
-7
2 7
2
37. What about the special cases?
• What is the slope of
the line perpendicular
to a horizontal line?
1
0
Well, the slope of a
horizontal line is 0…
So what’s the negative
reciprocal of 0?
0
0
1
Anything over
zero is undefined
The slope of a line
to a horizontal
line is undefined.
38. Write an equation in for the line through (-8, 3)
perpendicular to 2x – 3y = 10.
We know the perpendicular
slope and we know a point.
3
2
slope
)
3
,
8
-
(
2
3
-
2
m
Isolate y to find the slope: 2x – 3y = 10
2x = 10 + 3y
2x – 10 = 3y
3 3 3
b
)
8
-
(
3 2
3
- b
mx
y
3 = 12 + b
3 – 12 = b
-9 = b
9
2
-3
:
x
y
answer
39. Write an equation in standard form for the line
through (-8, 3) perpendicular to
2x - 3y = 10.
3
10
3
2
x
y
9
2
3
-
x
y
41. Summary
• Vertical line
– Slope is undefined
– x-intercept is (a, 0)
– no y-intercept
• Horizontal line
– Slope is 0.
– y-intercept is (0, b)
– no x-intercept
a
x
b
y