This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
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
...
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
Model Attribute Check Company Auto PropertyCeline George
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An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
2. Warm Up Find the slope of the line containing each pair of points. 1. (0, 2) and (3, 4) 2. (–2, 8) and (4, 2) 3. (3, 3) and (12, –15) Write the following equations in slope-intercept form. 4. y – 5 = 3( x + 2) 5. 3 x + 4 y + 20 = 0 – 2 – 1 y = 3 x + 11
3. Graph a line and write a linear equation using point-slope form. Write a linear equation given two points. Objectives
4. In lesson 5-6 you saw that if you know the slope of a line and the y- intercept, you can graph the line. You can also graph a line if you know its slope and any point on the line.
5. • • 2 Example 1A: Using Slope and a Point to Graph Graph the line with the given slope that contains the given point. slope = 2; (3, 1) Step 1 Plot (3, 1) . Step 2 Use the slope to move from (3, 1) to another point. Move 2 units up and 1 unit right and plot another point. Step 3 Draw the line connecting the two points. 1 (3, 1)
6. slope = ; (–2, 4) Step 1 Plot (–2, 4) . Step 2 Use the slope to move from (–2, 4) to another point. Move 3 units up and 4 units right and plot another point. Step 3 Draw the line connecting the two points. • • (–2, 4) 3 4 (3, 7) Example 1B: Using Slope and a Point to Graph Graph the line with the given slope that contains the given point.
7. Example 1C: Using Slope and a Point to Graph Graph the line with the given slope that contains the given point. slope = 0; (4, –3) A line with a slope of 0 is horizontal. Draw the horizontal line through (4, –3) . (4, –3) •
8. Check It Out! Example 1 Graph the line with slope –1 that contains (2, –2). Step 1 Plot (2, –2) . Step 2 Use the slope to move from (2, –2) to another point. Move 1 unit down and 1 unit right and plot another point. Step 3 Draw the line connecting the two points. • • − 1 1 (2, –2)
9. If you know the slope and any point on the line, you can write an equation of the line by using the slope formula. For example, suppose a line has a slope of 3 and contains ( 2 , 1 ). Let ( x , y ) be any other point on the line. 3( x – 2) = y – 1 y – 1 = 3 ( x – 2 ) Slope formula Substitute into the slope formula. Multiply both sides by (x – 2). Simplify.
10.
11. Example 2: Writing Linear Equations in Point-Slope Form Write an equation in point-slope form for the line with the given slope that contains the given point. A. B. C.
12. Check It Out! Example 2 Write an equation in point-slope form for the line with the given slope that contains the given point. a. b. slope = 0; (3, –4) y – ( – 4 ) = 0 ( x – 3 ) y + 4 = 0( x – 3)
13. Example 3: Writing Linear Equations in Slope-Intercept Form Write an equation in slope-intercept form for the line with slope 3 that contains (–1, 4). Step 1 Write the equation in point-slope form: y – 4 = 3 [ x – (–1) ] Step 2 Write the equation in slope-intercept form by solving for y. y – 4 = 3( x + 1) Rewrite subtraction of negative numbers as addition. Distribute 3 on the right side. y – 4 = 3 x + 3 y = 3 x + 7 Add 4 to both sides. y – y 1 = m ( x – x 1 ) + 4 + 4
14. Check It Out! Example 3 Step 1 Write the equation in point-slope form: Add 1 to both sides. y – y 1 = m ( x – x 1 ) Write an equation in slope-intercept form for the line with slope that contains (–3, 1).
15. Rewrite subtraction of negative numbers as addition. Step 2 Write the equation in slope-intercept form by solving for y. Check It Out! Example 3 Continued Add 1 to both sides. Distribute on the right side. + 1 +1 Write an equation in slope-intercept form for the line with slope that contains (–3, 1).
16. Example 4A: Using Two Points to Write an Equation Write an equation in slope-intercept form for the line through the two points. (2, –3) and (4, 1) Step 1 Find the slope. Step 2 Substitute the slope and one of the points into the point-slope form. Choose (2, –3). y – y 1 = m ( x – x 1 ) y – (–3) = 2 ( x – 2 )
17. Step 3 Write the equation in slope-intercept form. y = 2 x – 7 Example 4A Continued Write an equation in slope-intercept form for the line through the two points. (2, –3) and (4, 1) y + 3 = 2( x – 2) y + 3 = 2 x – 4 – 3 –3
18. Example 4B: Using Two Points to Write an Equation Write an equation in slope-intercept form for the line through the two points. (0, 1) and (–2, 9) Step 1 Find the slope. Step 2 Substitute the slope and one of the points into the point-slope form. Choose (0, 1). y – y 1 = m ( x – x 1 ) y – 1 = –4 ( x – 0 )
19. Example 4B Continued Write an equation in slope-intercept form for the line through the two points. (0, 1) and (–2, 9) Step 3 Write the equation in slope-intercept form. y = –4 x + 1 y – 1 = –4( x – 0) y – 1 = –4 x + 1 +1
20. Check It Out! Example 4a Write an equation in slope-intercept form for the line through the two points. (1, –2) and (3, 10) Step 1 Find the slope. Step 2 Substitute the slope and one of the points into the point-slope form. Choose (1, –2). y – y 1 = m ( x – x 1 ) y – (–2) = 6 ( x – 1 ) y + 2 = 6( x – 1)
21. Check It Out! Example 4a Continued Write an equation in slope-intercept form for the line through the two points. Step 3 Write the equation in slope-intercept form. y + 2 = 6 x – 6 y = 6 x – 8 (1, –2) and (3, 10) y + 2 = 6( x – 1) – 2 – 2
22. Check It Out! Example 4b Write an equation in slope-intercept form for the line through the two points. (6, 3) and (0, –1) Step 1 Find the slope. Step 2 Substitute the slope and one of the points into the point-slope form. Choose (6, 3). y – y 1 = m ( x – x 1 )
23. Check It Out! Example 4b Continued Step 3 Write the equation in slope-intercept form. Write an equation in slope-intercept form for the line through the two points. (6, 3) and (0, –1) + 3 +3
24. Example 5: Problem-Solving Application The cost to stain a deck is a linear function of the deck’s area. The cost to stain 100, 250, 400 square feet are shown in the table. Write an equation in slope-intercept form that represents the function. Then find the cost to stain a deck whose area is 75 square feet.
25. • The answer will have two parts—an equation in slope-intercept form and the cost to stain an area of 75 square feet. • The ordered pairs given in the table—(100, 150), (250, 337.50), (400, 525)—satisfy the equation. Example 5 Continued Understand the Problem 1
26. You can use two of the ordered pairs to find the slope. Then use point-slope form to write the equation. Finally, write the equation in slope-intercept form. Example 5 Continued 2 Make a Plan
27. Step 1 Choose any two ordered pairs from the table to find the slope. Use (100, 150) and (400, 525). Step 2 Substitute the slope and any ordered pair from the table into the point-slope form. y – 150 = 1.25 ( x – 100 ) Use (100, 150). Example 5 Continued y – y 1 = m ( x – x 1 ) Solve 3
28. Step 3 Write the equation in slope-intercept form by solving for y. y – 150 = 1.25( x – 100) y – 150 = 1.25 x – 125 Distribute 1.25. y = 1.25 x + 25 Add 150 to both sides. Step 4 Find the cost to stain an area of 75 sq. ft. y = 1.25 x + 25 y = 1.25 (75) + 25 = 118.75 The cost of staining 75 sq. ft. is $118.75. Example 5 Continued
29. If the equation is correct, the ordered pairs that you did not use in Step 2 will be solutions. Substitute (400, 525) and (250, 337.50) into the equation. Example 5 Continued Look Back 4 y = 1.25 x + 25 337.50 1.25 (250) + 25 337.50 312.50 + 25 337.50 337.50 y = 1.25 x + 25 525 1.25 (400) + 25 525 500 + 25 525 525 y = 1.25 x + 25
30. Check It Out! Example 5 What if…? At a newspaper the costs to place an ad for one week are shown. Write an equation in slope-intercept form that represents this linear function. Then find the cost of an ad that is 21 lines long.
31. Check It Out! Example 5 Continued • The answer will have two parts—an equation in slope-intercept form and the cost to run an ad that is 21 lines long. • The ordered pairs given in the table—(3, 12.75), (5, 17.25),(10, 28.50)—satisfy the equation. Understand the problem 1
32. You can use two of the ordered pairs to find the slope. Then use the point-slope form to write the equation. Finally, write the equation in slope-intercept form. Check It Out! Example 5 Continued 2 Make a Plan
33. Step 1 Choose any two ordered pairs from the table to find the slope. Use (3, 12.75) and (5, 17.25). Check It Out! Example 5 Continued Step 2 Substitute the slope and any ordered pair from the table into the point-slope form. Use (5, 17.25). y – y 1 = m ( x – x 1 ) y – 17.25 = 2.25 ( x – 5 ) Solve 3
34. Step 3 Write the equation in slope-intercept form by solving for y. y – 17.25 = 2.25( x – 5) y – 17.25 = 2.25 x – 11.25 Distribute 2.25. y = 2.25 x + 6 Add 17.25 to both sides. Check It Out! Example 5 Continued Step 4 Find the cost for an ad that is 21 lines long. y = 2.25 x + 6 y = 2.25 (21) + 6 = 53.25 The cost of the ad 21 lines long is $53.25. Solve 3
35. If the equation is correct, the ordered pairs that you did not use in Step 2 will be solutions. Substitute (3, 12.75) and (10, 28.50) into the equation. Check It Out! Example 5 Continued Look Back 4 y = 2.25 x + 6 12.75 2.25 (3) + 6 12.75 6.75 + 6 12.75 12.75 28.50 2.25 (10) + 6 28.50 22.50 + 6 28.50 28.50 y = 2.25 x + 6
36. Lesson Quiz: Part I Write an equation in slope-intercept form for the line with the given slope that contains the given point. 1. Slope = –1; (0, 9) y = – x + 9 2. Slope = ; (3, –6) Write an equation in slope-intercept form for the line through the two points. 3. (–1, 7) and (2, 1) 4. (0, 4) and (–7, 2) y = –2 x + 5 y = x – 5 y = x + 4
37. Lesson Quiz: Part II 5. The cost to take a taxi from the airport is a linear function of the distance driven. The cost for 5, 10, and 20 miles are shown in the table. Write an equation in slope-intercept form that represents the function. y = 1.6 x + 6
