2.5: Parallel and Perpendicular Vectors, The Unit Vector
Parallel and Orthogonal Vectors
Two vectors \(\vec{u}=\left\langle u_x, u_y\right\rangle\) and \(\vec{v}=\left\langle v_x, v_y\right\rangle\) are parallel if the angle between them is \(0^{\circ}\) or \(180^{\circ}\).
Also, two vectors \(\overrightarrow{u}\boldsymbol{=}\left\langle u_x,\left.u_y\right\rangle \right.\) and \(\overrightarrow{v}\boldsymbol{=}\left\langle v_x,\left.v_y\right\rangle \right.\) are parallel to each other if the vector \(\overrightarrow{u}\) is some multiple of the vector \(\overrightarrow{v}\) . That is, they will be parallel if the vector \(\overrightarrow{u}=c\overrightarrow{v}\) , for some real number \(c\) . That is, \(\overrightarrow{u}\) is some multiple of \(\overrightarrow{v}\) .
Two vectors \(\vec{u}=\left\langle u_x, u_y\right\rangle\) and \(\vec{v}=\left\langle v_x, v_y\right\rangle\) are orthogonal (perpendicular to each other) if the angle between them is \(90^{\circ}\) or \(270^{\circ}\).
Use this shortcut: Two vectors are perpendicular to each other if their dot product is 0.
The two vectors \(\overrightarrow{u}\boldsymbol{=}\left.\boldsymbol{\langle }2,-3\right\rangle\) and \(\overrightarrow{v}\boldsymbol{=}\left\langle -8,\left.12\right\rangle \right.\) are parallel to each other since the angle between them is \(180{}^\circ\) .
Solution
\(\begin{aligned}
& \theta=\cos ^{-1} \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\| \cdot\|\vec{v}\|} \\
& \theta=\cos ^{-1} \frac{\langle 2,-3\rangle \cdot\langle-8,12\rangle}{\sqrt{2^2+(-3)^2} \cdot \sqrt{(-8)^2+12^2}} \\
& \theta=\cos ^{-1} \frac{2 \cdot(-8)+(-3) \cdot 12}{\sqrt{4+9} \cdot \sqrt{64+144}} \\
& \theta=\cos ^{-1} \frac{-52}{\sqrt{13} \cdot \sqrt{208}} \\
& \theta=180^{\circ}
\end{aligned}\)
To show that the two vectors \(\overrightarrow{u}\boldsymbol{=}\left.\boldsymbol{\langle }5,10\right\rangle\) and \(\overrightarrow{v}\boldsymbol{=}\left\langle 6,\left.-3\right\rangle \right.\) are orthogonal (perpendicular to each other), we just need to show that their dot product is 0.
Solution
\[\left.\boldsymbol{\langle }5,10\right\rangle \boldsymbol{\bullet }\left\langle 6,\left.-3\right\rangle \right.\boldsymbol{=}5\bullet 6+10\bullet \left(-3\right)=30-30=0 \nonumber \]
The Unit Vector
A unit vector is a vector of length 1.
A unit vector in the same direction as the vector \(\overrightarrow{v}\) is often denoted with a “hat” on it as in \(\hat{v}\) . We call this vector “v hat.”
The unit vector \(\hat{v}\)corresponding to the vector \(\vec{v}\)is defined to be \[\hat{v}=\frac{\vec{v}}{\|\vec{v}\|} \nonumber \]
The unit vector corresponding to the vector \(\overrightarrow{v}\boldsymbol{=}\left\langle -8,\left.12\right\rangle \right.\).
Solution
\(\begin{gathered}
\hat{v}=\frac{\vec{v}}{\|\vec{v}\|} \\
\hat{v}=\frac{\langle-8,12\rangle}{\sqrt{(-8)^2+(12)^2}} \\
\hat{v}=\frac{\langle-8,12\rangle}{\sqrt{64+144}} \\
\hat{v}=\frac{\langle-8,12\rangle}{\sqrt{208}} \\
\hat{v}=\left\langle\frac{-8}{\sqrt{208}}, \frac{12}{\sqrt{208}}\right\rangle
\end{gathered}\)
Using Technology
We can use technology to find the angle \(\theta\) between two vectors.
Go to www.wolframalpha.com.
To show that the vectors \(\overrightarrow{u}\boldsymbol{=}\left.\boldsymbol{\langle }2,-3\right\rangle\) and \(\overrightarrow{v}\boldsymbol{=}\left\langle -8,\left.12\right\rangle \right.\) are parallel, enter angle between the vectors \(\mathrm{<}\) 2, -3 \(\mathrm{>}\) and \(\mathrm{<}\) -8, 12 \(\mathrm{>}\) in the entry field. Wolframalpha tells you what it thinks you entered, then tells you its answer. In this case, \(\theta =180{}^\circ\) , indicating the two vectors are parallel.
and
Try These
Determine if the vectors \(\vec{u}=\langle 2,1\rangle\) and \(\vec{v}=\langle 3,-6\rangle\) are parallel to each other, perpendicular to each other, or neither parallel nor perpendicular to each other.
- Answer
-
Parallel
Determine if the vectors \(\vec{u}=\langle 2,16\rangle\) and \(\vec{v}=\left\langle\frac{1}{2}, 4\right\rangle\) are parallel to each other, perpendicular to each other, or neither parallel nor perpendicular to each other.
- Answer
-
Perpendicular
Determine if the vectors \(\vec{u}=\langle 7,6\rangle\) and \(\vec{v}=\langle 2,-1\rangle\) are parallel to each other, perpendicular to each other, or neither parallel nor perpendicular to each other.
- Answer
-
Neither parallel nor perpendicular
Find the unit vector corresponding to the vector \(\vec{v}=\langle 2,-1\rangle\).
- Answer
-
\(\hat{v}\boldsymbol{=}\left\langle \frac{2}{\sqrt{5}},\left.\frac{-1}{\sqrt{5}}\right\rangle \right.\)


27.6: Douglass- Heroic or Cowardly?
After having read Benito Cereno and The Heroic Slave, I can identify the differences between the novellas. It is arguable that Melville and Douglass are both trying to bring the same point across; that slavery is not a good thing, and showing black rebellion as a direct consequence of slavery. Frederick Douglass and Herman Melville are bringing up the same point, but in very different ways.
Herman Melville more focuses on the idea of black resistance than anything. The black resistance then is shown through Babo’s later freedom, but the majority of the story didn’t reside around freedom, when instead it was residing around the rebellion. It’s very possible that this is because it’s from the point of view of someone who is not a slave, but that was done for a reason, to be able to show the rebellion in full force. Frederick Douglass focuses on showing the horrors of slavery through making it more accessible from showing a story in a slave’s perspective. This change of lens and point of view makes it clear that the goal was always freedom, not rebellion. In this way, it shows that the rebellion is a part of the freedom. You can’t be freed without rebelling, and the point of view being from a slave’s perspective, allows the readers to feel allegiance and excitement for the slaves as they become free.
Herman Melville shows the black resistance through irony and indirection, which I think is what the editors were trying to state in their introduction. Because we saw the storyline through Delano’s point of view, there were a lot of pieces missing. Babo of course is not going to make it clear that he plans to take over the ship, so we see it through somebody else’s eyes and it’s a misdirection that is surprising to us. This misdirection shows something very different than it being clear.
On the opposing side, Frederick Douglass shows the struggles, the planning, the emotions that parallel the decision to leave your family for freedom. Douglass works harder in showing the themes throughout the novel, which then allows for the novel to be that much stronger. His themes are the letters, the poems/excerpts in the beginning of chapters, the character shifts, freedom, and friendship. These themes characterize it more similar to that of a novel. The literary themes throughout the novel make it easier to stay connected to the storyline, and connect with the characters. In this way, his themes move along the point of the story, therefore the readers still are left with an understanding of slavery on a deeper level. The character Mr. Listwell also allows the reader to feel connected to the novel, with being able to identify with him if they are of white descent. Because the novel is from the point of view of a slave – former slave – we see the whole thought process of what Madison Washington went through, and we are rooting for him to gain his freedom. He shows his true dedication to it very early on, when he shows it through his exclamations. “ Liberty I will have, or die in the attempt to gain it. This working that others may live in idleness! This cringing submission to insolence and curses! […] I will stand it no longer” (Douglass, 6). We become cheerleaders of sorts for Washington, which is a very different notion of how we react to Babo.
In a broader statement, I get the notion that black resistance was a heroic act to the slaves, and justifiably so. I learned that people perceive it in different ways, in regards to their race. If they are a slave, they see it as bettering the world, freedom, liberation. If they are a white person, however – aka, benefitting from the capitalist slave systematic oppression – then they’re going to see black resistance as a negative act, as cowardly, and a problem with the country. They’re going to see it as the majority of the characters in Melville’s novella, or as the majority of characters in Douglass’s novella. Both novellas, however, point the reader to the clear answer: black resistance was necessary, and it’s awesome that we have literature to show that more in depth.


4: Introduction to the Solar System and Its Formation
Surrounding the Sun is a complex system of worlds with a wide range of conditions: eight major planets, many dwarf planets, hundreds of moons, and countless smaller objects. Thanks largely to visits by spacecraft, we can now envision the members of the solar system as other worlds like our own, each with its own chemical and geological history, and unique sights that interplanetary tourists may someday visit. Some have called these past few decades the “golden age of planetary exploration,” comparable to the golden age of exploration in the fifteenth century, when great sailing ships plied Earth’s oceans and humanity became familiar with our own planet’s surface.
In this chapter, we discuss our planetary system and introduce the idea of comparative planetology—studying how the planets work by comparing them with one another. We want to get to know the planets not only for what we can learn about them, but also to see what they can tell us about the origin and evolution of the entire solar system. In the upcoming chapters, we describe the better-known members of the solar system and begin to compare them to the thousands of planets that have been discovered recently, orbiting other stars.
-
- 4.1: Overview of Our Planetary System
- Our solar system currently consists of the Sun, eight planets, five dwarf planets, nearly 200 known moons, and a host of smaller objects. The planets can be divided into two groups: the inner terrestrial planets and the outer giant planets. Pluto, Eris, Haumea, and Makemake do not fit into either category; as icy dwarf planets, they exist in an ice realm on the fringes of the main planetary system. The giant planets are composed mostly of liquids and gases.
-
- 4.2: Compostion and Structure of Planets
- The giant planets have dense cores roughly 10 times the mass of Earth, surrounded by layers of hydrogen and helium. The terrestrial planets consist mostly of rocks and metals. They were once molten, which allowed their structures to differentiate (that is, their denser materials sank to the center). The Moon resembles the terrestrial planets in composition, but most of the other moons—which orbit the giant planets—have larger quantities of frozen ice within them.
-
- 4.3: Dating Planetary Surfaces
- The ages of the surfaces of objects in the solar system can be estimated by counting craters: on a given world, a more heavily cratered region will generally be older than one that is less cratered. We can also use samples of rocks with radioactive elements in them to obtain the time since the layer in which the rock formed last solidified. The half-life of a radioactive element is the time it takes for half the sample to decay.
-
- 4.4: Planetary Evolution
- After their common beginning, each of the planets evolved on its own path. Different possible outcomes are illustrated by comparison of the terrestrial planets (Earth, Venus, Mars, Mercury, and the Moon). All are rocky, differentiated objects. The level of geological activity is proportional to mass: greatest for Earth and Venus, less for Mars, and absent for the Moon and Mercury. However, tides from another nearby world can also generate heat to drive geological activity.
-
- 4.5: Origin of the Solar System
- Regularities among the planets have led astronomers to hypothesize that the Sun and the planets formed together in a giant, spinning cloud of gas and dust called the solar nebula. Astronomical observations show tantalizingly similar circumstellar disks around other stars. Within the solar nebula, material first coalesced into planetesimals; many of these gathered together to make the planets and moons. The remainder can still be seen as comets and asteroids.
-
- 4.6: Formation of the Solar System
- Meteorites, comets, and asteroids are survivors of the solar nebula out of which the solar system formed. This nebula was the result of the collapse of an interstellar cloud of gas and dust, which contracted (conserving its angular momentum) to form our star, the Sun, surrounded by a thin, spinning disk of dust and vapor. Condensation in the disk led to the formation of planetesimals, which became the building blocks of the planets.
Thumbnail: This picture was taken by the Curiosity Rover on Mars in 2012. The image is reconstructed digitally from 55 different images taken by a camera on the rover’s extended mast, so that the many positions of the mast (which acted like a selfie stick) are edited out. (credit: modification of work by NASA/JPL-Caltech/MSSS).
Contributors and Attributions
-
Andrew Fraknoi (Foothill College), David Morrison (NASA Ames Research Center), Sidney C. Wolff (National Optical Astronomy Observatory) with many contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/astronomy ).


 You can teach the class how to do anything that interests you. Try to pick something you think other students would be interested in learning about.
-
Include the following:
- Give an introduction: Explain what you will teach the class to do.
- Explain why this skill is important, helpful or interesting for your classmates to learn.
- Your instruction video must include at least 3 steps.
- Make a video of yourself doing the task and explaining each of the steps.
- Conclusion: End with some final tips and/or summary of your presentation.
- Include a transcript: Write down what you're doing to say. Circle the content words in each sentence. Draw lines between words that are linked. Practice reading your transcript with the correct pronunciation.
Possible topics:
How to . . .
- make the perfect cup of coffee.
- change a bicycle tire.
- post on Instagram.
- make a TikTok video.
- make or do something from your country or culture.
Purpose: Why are we doing this?
- You will practice speech giving skills including volume, eye contact, organization and presenting information about a place you know well.
- You will review and practice using pronunciation features we have learned in class including stress and rhythm when saying content words and function words as well as linking in connected speech.
Rubric
50 Points Possible
| Criteria | Excellent | Good | Needs Improvement (resubmission required) | No Marks (resubmission required) |
|---|---|---|---|---|
| 10 | 7 | 5 | 0 | |
|
Content
|
The student meets all 4 requirements. | Student meets 3 of the 4 requirements. | Student meets 2 of 4 requirements. | Student meets 1 or fewer requirements. |
|
Speech organization and clarity (interesting and clear explanation given) |
The speech is well organized. The presentation is interesting and a clear explanation is given. | There are some errors in grammar, sentence structure and/or vocabulary that can occasionally cause confusion for the listener, but most of the presentation is clear. | There are some errors in grammar, sentence structure and/or vocabulary that can occasionally cause confusion for the listener, but most of the presentation is clear. | The presentation is poorly organized and difficult ot follow. |
| Delivery (volume, eye contact) | The student speaks loudly enough to be early comfortably and the student looks directly at the camera. The student has memorized most of the presentation and speaks confidently with little aid from notes. | The student looks at the video occasionally. The student is reading some of the presentation and has made some attempt to practice or memorize the presentation before recording. | The student is clearly reading most of the presentation and has made little attempt to practice or memorize the presentation before recording. The student is too reliant on notes. | The student's face is not seen during the video at any time and/or the student reads the entire presentation with little evidence that he/she understands what is being read. |
|
Pronunciation Focus
|
The student is very clear and easy to understand. The student uses rhythm and connected speech effectively. | The student is mostly clear and easy to understand. The student uses rhythm and connected speech effectively most of the time. | The student is understandable some of the time. The student attempts to use rhythm and connected speech some of the time but is not always successful. The student may sound robotic at times. | Most of the presentation is difficult to understand and/or the student does not attempt to use rhythm or connected speech. The student reads individual words, and sounds very robotic. |
| Visual (helps the presentation) | The student clearly models 3 or more steps while explaining each step. | No visual is provided. | ||
| Transcript | A transcript is included with content words and linking identified. | No transcript is provided. | ||
| Total |
Example
Analyze for Organization
Below are questions that will help you notice:
- the typical information usually included in a "How to" video
- the order in which information is usually presented
Watch the video again and complete the table.
| Questions | Your notes |
|---|---|
| What is the video about? |
This video is about how to make _________________. |
| What are the benefits of learning to do this? | |
| What tools or ingredients do you need? | |
| What are the steps ? | |
| What tips, helpful advice or important information were included? | |
| How does the video end? |
Brainstorm and Plan
Complete the table with information for your own video.
| Questions | Your notes |
|---|---|
| What is the video about ? |
This video is about how to _________________. |
| What are the benefits of learning to do this? | |
| What tools or ingredients do you need? | |
| What are the steps ? | |
| What tips, helpful advice or important information were included? | |
| How does the video end ? |
Story Board
Complete the storyboard before you make your video. Decide what you're going to say. Write your script in the first column. In the second column, make a sketch or take notes on what you will do in each part of the video.
| Script: What are you going to say? | Video: What will you see on the camera? | |
|---|---|---|
| 1 | ||
| 2 | ||
| 3 | ||
| 4 | ||
| 5 | ||
| 6 |
Post-Project Discussion
Your classmates' videos will be posted on an online forum. Watch two of your classmates' videos. Post a thoughtful comment or question to each classmate.


One of the most common misconceptions about accessible web design is that in order for a site to be accessible, it must have a simple, plain design with few or no images. Another myth is that an adequate, accessible site can be made by providing a “text-only” version of an existing website. This is a nuisance to maintain, as it requires you to keep not one but two versions of every single page.
Remember, not all disabled students are blind! People with mobility or hearing issues and even poor eyesight will certainly appreciate a well-thought-out, aesthetically pleasing website as much as anyone. As you’ll see, many of your accessibility changes will be tucked away in the code of your pages, where they will be a benefit to disabled users without detracting from your site in any way.
 Will you have a lot of material to read, or just a little? Will there be many pages or subpages?
The easier you can make it for students to find and read your course material, the easier it will be for them to learn.
Menus and Navigation
The way you plan your site’s navigation will affect your site’s usability for your entire audience. A good approach is to write down the different categories that apply to each of your pages, and then group the pages into these categories. The key is to find an intuitive balance between overwhelming the user with too many options, and burying important information too deep in the site.
For example, if your site is made up of these pages, you are running the risk of creating a very cluttered and busy navigation:
Course Content
Guidelines
Syllabus
Schedule
Messageboard
Chat
Mail
Submit Assignments
Assignment #1
Assignment #2
Assignment #3
Assignment #4
Grading
Help
You could try grouping your pages into these categories, and create subcategories within this structure:
About the Course—clicking reveals Guidelines, Syllabus, Schedule
Communicate—clicking reveals Messageboard, Chat, Mail
Help
Now your students only have to sort through five links instead of fourteen.
Use common sense when defining categories—there may be some links that a student might use several times a day, so you might want them to sit on the top level for quick and easy access.
When you are designing your site, and choosing where to place your navigation, keep these questions in mind:
Are the links grouped together in one place, where they can be easily found?
Are there so many links on the page that it becomes confusing?
Writing for The Web
Typically, users viewing websites do not read text as thoroughly as they do when reading printed text. Monitors have a lower resolution than printed material, which makes it less comfortable to stare at for long periods of time. Most online readers develop the habit of skimming the screen looking for key points rather than studying in detail. If it is necessary to read lengthy, wordy passages or papers, many users will print out the information to read it in comfort offline.
You can make it easier for readers to find what they need by:
Keeping your paragraphs short—one idea per paragraph
Using headers to announce and reinforce new themes
Using bulleted lists to group ideas into a simple, easy-to-read format
Writing for Learning-disabled Students
Being learning disabled doesn’t mean a student can’t learn—it may just mean that traditional learning methods are particularly difficult for that individual. Some students with difficulty reading may learn the same material just as well upon hearing it, or after seeing a graphic that explains the concept. For this reason, it can be helpful to explain key ideas in multiple different ways: text and a graphic or video that reinforces what is being taught.
The same principle applies to how you ask your students to express their understanding. For many students, the choice of whether to write a paper or give an oral presentation can make a huge difference in their ability to communicate what they have learned.
One of the biggest difficulties encountered by learning-disabled students is in interpreting academic demands and expectations. This can often be addressed by building checkpoints into assignments, such as “Submit a plan describing how you will approach this project.” This allows the instructor to assess whether the student has understood what is expected of them, before the student has invested too much time into a project that may be on the wrong track.
Clear, explicit instructions are of course vital, but they alone are not the solution—the student must actively engage and interpret the tasks and requirements themselves.
Additional Considerations
Some students with disabilities may require additional time to complete tasks such as self-tests and quizzes. A student using an alternative keyboard may not be able to type as fast as his classmates. Extend the allotted time for that student, or remove the time requirement.
Chat rooms are often inaccessible to users reading screen readers. Make sure that chat room participation is not a course requirement, or make arrangements for a disabled student to participate using other means such as a discussion room.
Using Correct Code: XHTML and CSS
HTML (Hypertext Markup Language) is the code used to describe web pages so they can be rendered in a browser. When HTML was created many years ago, no one could have predicted the sorts of dynamic, interactive pages that they would eventually be used to create. While HTML was easy to learn and fairly flexible, it had some significant limitations: for example, objects could not be placed anywhere on a page, but had to flow in a linear fashion, one item before the next. Creative designers found ways around these limitations: the TABLE tag was manipulated to allow precise placement of text and graphics.
But these clever fixes came with their own set of problems. Redesigning a website meant rewriting and rebuilding every single page of HTML on the site. Visually simple designs often required complex, bloated HTML. If code was written inaccurately, the web browser had to interpret the code as well as it could, slowing down the rendering of the page.
Note
Intermediate users: We recommend using Macromedia Dreamweaver to assist you in writing accessible code.
Novice users: If you’re not comfortable writing HTML code at all, we suggest Course Genie, a package from Horizon Wimba, which allows you to convert a Word document into a well-coded, accessible website that can be uploaded to WebCT.
To address these issues, HTML was given a fresh start by rewriting it using another language—XML, or eXtensible Markup Language. The result is called XHTML. Superficially, XHTML is not terribly different from HTML: the syntax is stricter, and some tags and attributes have been removed, but much of it is the same. The key is in the “extensible”. XHTML essentially lets you define new classes of objects.
What does this mean? Suppose you need all news-related images (but no others!) to be surrounded by a five-pixel blue border. Using old-style HTML, you would do this by wrapping every news image in a table tag.
Example \(\PageIndex{1}\)

Every single image that needs a border would have to be treated this way.
Using XHTML saves you time and space. First define a class called “news” as having a five-pixel blue border.
Example \(\PageIndex{2}\)
.news {
border: 5px solid blue;
}
Then add an attribute to any image tag that needs to be in class “news”.
How does this work? The classes are defined within Cascading Stylesheets (CSS)—stylesheets, because they define the style of a page; cascading, because you can apply multiple stylesheets. You can define any style once and apply it throughout your entire site.
Tip A site that may help you visualize this process is CSS Zen Garden (
http://www.csszengarden.com
). Every design on the site uses the same XHTML code to define the different areas of the page. By swapping out only the stylesheet, the appearance of the site changes dramatically.
So with a single CSS file, you can now define the look and feel of an entire website consisting of hundreds of pages.
Why Can't I Do Things The Old Way?
Feel free to skip this section if you are new to building web pages or are already familiar with XHTML and CSS.
Tables Aren't Meant for Layout
If you ever built a website before CSS became widely accepted, chances are you built it using tables. You probably took a large image and chopped it up in an image editing program, then placed each chunk of the image into a borderless table to lay it out exactly where you wanted.
The first reason to avoid tables is that it’ll make redesigning your site much easier in the future. You won’t have to chop up new designs and recreate every page of your site any more—you can do it all with one change of your CSS sheet and maybe a few changes to the HTML.
But the main reason is that it simply isn’t all that accessible. Screen readers approach tables in a linear fashion; that is, they read out each column, left to right, and each row, top to bottom. If your table-based layout doesn’t correspond to this model, blind users may not receive the information in the order you intended it. They may hear the menu read out in pieces, in between parts of your main content, and as you can imagine, it is very confusing to navigate a page like this.
Many Old Tags Have Been Deprecated
XHTML no longer contains several tags that address the appearance of a site. The FONT tag, which used to be the only way to set the font appearance on a page, has been removed from HTML. This is because fonts can be much more efficiently defined and updated using CSS. Similarly, the CENTER tag has gone away, to be replaced by CSS formatting.
Note
There are many excellent resources, both online and offline, for learning XHTML and CSS. Here are some tutorials to get you started:
For the rest of this section, we will use XHTML and HTML interchangeably; the basic principles are the same, and most of the differences are in the accuracy and consistency of the code.
Text
Text makes the World Wide Web go ’round. The greatest amount of content on the Web is basic, plain text. Text is the most accessible media format there is—it is easy for all browsers and screen readers to handle.
There is one big thing you need to be most careful of, and that is the visibility of your text. Aging users, people with poor vision, or even people using a small monitor may not see your site’s text with the same clarity that you do. They may need to enlarge the size of the text to be able to read it better.
 A screen magnifier, such as ZoomText, will make a screen behave much as if a giant magnifying glass has been placed between the screen and the user. An even simpler way is to use the text size settings in the browser to increase the font size on the page.
When you define the appearance of your text in CSS, you have a choice between absolute or relative font sizes.
Absolute font sizes (pixels, points) should appear at the exact same size in every browser and every configuration. Text that is set to “12px” will appear as 12 pixels high. Designers often prefer absolute font sizes because they have greater control over the appearance of the text, and can dictate how much space a given block of text will occupy.
Relative font sizes (percentages, “em”) appear at a size relative to the user’s font settings. Text that is set to “90%” will appear at 90 percent of the current text size. If the user changes their font size to “larger”, the size of the text on the page will increase.
What is the implication here? Use relative font sizes at all times. Some browsers will allow absolute font sizes to scale up with the user settings, but not all. Your eyesight may be much better than that of some of your users, and what looks fine to you might cause problems for someone else. Make sure you give them the control of their screen.
Example
Example \(\PageIndex{4}\)
body, p {
font-family: Arial, Helvetica, sans-serif;
font-size: 0.9em;
color: #333333;
}
This will make the text for a page 0.9 em, or 90 percent, of its default size.
Be careful with the contrast and colours of your text. Whether your text is light on a dark background or dark on a light background, you need to make sure there is enough contrast between the text and the background for users with weaker vision to distinguish clearly. Additionally, if any information is conveyed by colour alone, reinforce the information with another method. In the example shown in Figure \(\PageIndex{2}\), the required fields are marked not only by a change in colour, but by bold text and an asterisk.
Images
Alt text
There is a very simple, built-in way to make sure your images are accessible: use ALT text. Figure 11.5 would be coded as follows:

When a screen reader encounters an alt attribute, it substitutes the text for the image, reading the text out loud. In order to make this as useful as possible for your users, you should choose text that is appropriately descriptive of the image. Include any details that are necessary to make the image make sense; don’t bother with trivial descriptions if they don’t add useful information.
Empty descriptions
There are some cases where an image does not require a description at all, or where a description would clutter the audio reading of the page.
Spacer (or transparent) images are typically 1x1 transparent images that are used to control the layout of a table-based website by pushing elements of the site into place. If your site is entirely CSS-based, you won’t really need these. If you are working on an older site, though, you may still be using them.
Decorative bullet graphics are often used in lists to illustrate a point.
Figure \(\PageIndex{4}\) shows three decorative bullets, which many people would mistakenly code as follows:
With code like this, a screen reader user will hear: “red bullet marketing plan blue bullet promotion plan yellow bullet licensing plan”.
Even though you don’t want screen readers to attempt to describe these images, you still need to define their alt text, or the screen reader will read out the filename instead. The alt text on a spacer image or a decorative graphic should be empty, i.e., alt=“ ”.
Tip Visually impaired users aren’t the only ones to benefit from ALT text—you will too! By describing your images, you’ll make it easier for search engines such as Google to index your content, and it’ll be easier for other users to find the content on your site.
Long descriptions
Alt text is good for a short sentence, but sometimes a complicated diagram or graph cannot be thoroughly described in one line of text. When this happens, use the ALT attribute for a quick summary, and the LONGDESC attribute:

The longdesc attribute is the URL for another web page, which should contain a complete description of the image in question.
Imagemaps
Imagemaps are just as easy to make accessible: add the alt text to the AREA tag for each clickable area within the map.
Links
We have already talked about menus and navigation and the importance of thinking about links.
Descriptive link text: If your link text is taken out of context, will it make sense? Many screen readers allow the user to pop up a list of only the links from the page. This is a useful way for a blind reader to navigate—unless your link text says “Click here”! Make sure your link includes enough text to clearly define the link, such as “Click here for the full schedule” or even “Full schedule”.
Unique link names: Similarly, if your link text is taken out of context, will a user see the same link text multiple times? Ten links that all say “Click here”, but point to different pages, would be frustrating.
Link separators: Link in a menu should be separated by more than just whitespace, for visually impaired users to better distinguish links from each other. Additionally, some older screen readers incorrectly read adjacent links as the same link.
Note
On the Web, links are usually underlined. Most web users are accustomed to clicking on underlined links. To this end, it is best not to underline anything that is not a link unless conventional style requires it.
Another alternative is to make each link into an item in an unordered list, and then use CSS to style the links. A screenreader will pause between list items, making the links more “listenable”.
To do this, you will need this CSS:
ul {
list-style: none;
}
ul li {
display: inline;
padding-right: 10px;
}
and this HTML:

About

Bio

Contact

Setting list-style to “none” will remove the bullets that are displayed by default before each list item, and setting display to “inline” will place all the list items on the same line. You can continue to style the list items with margin and padding settings as needed.
The Title Attribute
Similar to ALT text for images, the TITLE attribute can be used to make a link URL clearer. A person using a screen reader can set an option to read TITLE texts out loud instead of the link text. Most browsers display the TITLE text as a “tooltip”, or small popup, that appears for a few seconds when the link is moused over.
The TITLE attribute can actually be validly applied to most HTML elements, but is best supported in the A (hyperlink) tag.
Javascript and DHTML
Many people are fond of “drop-down” or rollout menus, which appear when the user moves the cursor over a top-level category. For many users, they are a quick way to jump straight to the page they need.
Many of these menus create accessibility issues. Some are very sensitive to mouse movement and will “roll up” the instant the mouse drifts outside the box—which can be a serious problem for users whose hands cannot control the mouse precisely. In addition, some of the Javascript and Dynamic HTML (DHTML) code needed to generate these menus is not understood by screen readers, and will be ignored. This can prevent many users from using the menus at all!
This doesn’t mean you can’t use Javascript or DHTML, but if you are using it for important functions like navigation, be sure that you have a fallback plan for browsers without Javascript. You can usually test this yourself by turning Javascript off in your browser.
Popup Windows
Popup windows have their purposes:
displaying extra information without making the user lose their place on the page
letting the user open a link to another site that they can look at later
advertising (often unwelcome)
Consider what happens when a screen reader encounters a new window. It will first announce that the new window has opened, and then shift focus to that window, reading out the new content. A blind user cannot quickly glance at the new window and put it aside for later; they must hear the content, decide whether or not it is relevant, and choose which window to continue reading.
Unexpected popups can also be a problem for users with learning disabilities, as the sudden appearance of a new window can be distracting and make them lose their place on the previous page.
As a general rule, warn the user if a link will open a new pop-up window. Additionally, consider whether the pop-up window is absolutely necessary. Traditionally, links to external sites were opened in new browser windows. This is preferred by many, but it is better to let the user choose: nearly all browsers let you right-click (or Control-click, if you are a Mac user) a link to open it in a new window.
Data Tables
We have established that you shouldn’t use tables for graphic layout, but that doesn’t mean you can’t use tables at all. Tables are indispensable for their original intended purpose: displaying tabular data in an organized and legible format.
Sighted users can easily glance at a data table, see where the row and column headers are, and find the piece of data they are seeking. But when a screen reader encounters a table, it reads it out in a linear fashion: row by row, each cell in order. If the table is very large, it is easy to lose track of which column you are listening to. And if the table is very complex, with merged cells that overlap multiple rows or columns, it may not make much sense when read out loud.
Figure \(\PageIndex{5}\) gives an example.
Table Headers
Every table should have clearly labelled table headers. Often developers have done this just by colouring the background of the header cells or making the text bold, but as we know, this visual information will be lost when run through a screen reader.
So how can we tell the browser itself where the table headers are? This can be done with the  tag, which works exactly like the  tag except it makes the distinction that the cell is a header. Plus, you can still style the  tag using CSS to make the headers look however you want.
Caption and Summary
The
 attribute gives all users a quick definition of the table. The  attribute provides more detail for screen readers.

Schedule for the 99 B-Line

 …
Scope
The <scope> attribute goes into a table header to tell the browser which header is associated with a given row or column. This helps remove ambiguity and allows the screen reader to provide the user more information about the given table. Two of the options are scope=“row” or scope=“col”.
Table \(\PageIndex{1}\): Student graduation data
Graduation year
GPA
Bob Smith
2002
3.4
Sara Miller
2004
3.8
This would be written as follows:

Tables with multiple layers of headers and categories can become quite complicated. XHTML does allow for further description of complex tables, including grouping sets of rows and associating cells with headings. These ideas may be of interest if you have many data tables. Here are some resources for complex tables:
Most of the changes we have talked about will improve your site’s accessibility without changing its functionality in any way. Now we are going to discuss a few things you can add to your site that will be of extra benefit to disabled users.
Skip to Content
While many experienced screen reader users listen to websites at very high speeds, there is still no audio equivalent to skimming the page. Sighted users can easily ignore any part of a website that is of no interest to them, or something they have seen before, such as the navigation.
One feature that will improve your website’s usability is a skip to content option. This is a link, coded to appear invisible to sighted users, that screen reader users can click to skip any navigation menus that they have already encountered and don’t need right now.
There are three steps to creating a skip navigation option.
Add an anchor link just before your main content starts:

Add a new class in your CSS:
.skiplink {display:none}
Now, anything that you assign to class “skiplink” will not be displayed in the browser.
Add this link right after the  declaration of your page:
Skip over navigation
Keyboard Shortcuts
The accesskey attribute allows you to predefine keyboard shortcuts to specific pages or form fields on your website.  Accesskeys are triggered by the user holding down ALT and pressing the specified key.
Simply define the key within an existing link to that page:
About This Site
Be careful not to override existing browser keyboard shortcuts that appear in the browser toolbar, such as F (File), E (Edit), V (View). To be certain, use only numbers as access keys; you are less likely to conflict with existing shortcut definitions. There is no automatic listing of what access keys are defined on a site, so you will have to list the keys that you have defined either on a separate page of your site or next to the appropriate links.
There are a few conventional shortcuts:
ALT-1: Home page
ALT-2: Skip to main content
ALT-9: Feedback
Not all browsers support accesskey yet, but those that don’t will simply ignore the attribute.


9.4K: Evaluating Global Theories of Inequality
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Social theorists think differently about global inequality based on their sociological perspective.
Learning Objectives
- Differentiate between the positions on social inequality taken by functionalists, Marxists, modern liberalism, and social justice advocates
Key Points
- Functionalists are likely to believe that inequality is beneficial to societies and is naturally regulated by market forces to foster economic growth.
- Marxists, on the other hand, are likely to see inequality as detrimental to society and to advocate government regulation of the means of production and distribution of property.
- In modern liberal societies, individuals tend to value human rights according to the idea that all people are born with equal value.
- Social justice advocates generally argue that inequality is unfair, as it leaves some individuals with greater life chances and higher standards of living than others regardless of individual worth or merit.
Key Terms
- State-Oriented Approach : A strategy for reducing inequality in which governments instate policies to equally distribute opportunities and resources.
- Market-Oriented Approach : A perspective on inequality that asserts that a free market will result in prices that benefit the smooth functioning and growth of economies.
- Functionalist Approach : An approach that asserts that global inequality is not a problem at all, but rather benefits society as it produces an incentive structure to motivate highly capable individuals to pursue positions of power.
There is significant debate among sociologists, other social scientists, and policy makers over the best approach to global inequality. Some theorists who embrace a functionalist approach assert that global inequality is not a problem at all, but rather benefits society as it produces an incentive structure to motivate highly capable individuals to pursue positions of power. Functionalists are likely to embrace market-oriented approaches to inequality, on the basis that a free market will result in prices that benefit the smooth-functioning and growth of economies. Marxists, by contrast, see global inequality as indicative of exploitation and consider it a detriment to society. These thinkers are likely to support state -oriented approaches to regulating inequality, with governments instating policies to equally distribute opportunities and resources. Interactionists recognize global inequality, but consider it only in the context of individual relations and, therefore, see no role for state intervention.
Whatever sociological theory one adopts to explain the existence of inequality, not all theorists consider inequality to be a problem that needs correction. The idea that all members of a society should be equal is often associated with modern liberalism. In modern liberal societies, individuals tend to value human rights according to the idea that all people are born with equal value. The logic of human rights does not necessarily imply that all people should achieve equal status, but it does assume that all should have equal opportunities to advance, or Weberian life chances. Those who evaluate global inequality and consider it to violate human rights may advocate for solutions to inequality using the language of social justice. Social justice advocates generally argue that inequality is unfair, as it leaves some individuals with greater life chances and higher standards of living than others, regardless of individual worth or merit.
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12.10: Primary Sources
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Barry Goldwater, Republican Nomination Acceptance Speech (1964)
In 1964, Senator Barry Goldwater of Arizona accepted the Republican Party’s nomination for the presidency. In his speech, Goldwater refused to apologize for his strict conservative politics. “Extremism in the defense of liberty is no vice,” he said, and “moderation in the pursuit of justice is no virtue.”
Lyndon Johnson on Voting Rights and the American Promise (1965)
On March 15, 1965, Lyndon Baines Johnson addressed a joint session of Congress to push for the Voting Rights Act. In his speech, Johnson not only advocated policy, he borrowed the language of the civil rights movement and tied the movement to American history.
Lyndon Johnson, Howard University Commencement Address (1965)
On June 4, 1965, President Johnson delivered the commencement address at Howard University, the nation’s most prominent historically black university. In his address, Johnson explained why “opportunity” was not enough to ensure the civil rights of disadvantaged Americans.
National Organization for Women, “Statement of Purpose” (1966)
The National Organization for Women was founded in 1966 by prominent American feminists, including Betty Friedan, Shirley Chisolm, and others. The organization’s “statement of purpose” laid out the goals of the organization and the targets of its feminist vision.
George M. Garcia, Vietnam Veteran, Oral Interview (2012/1969)
In 2012, George Garcia sat down to be interviewed about his experiences as a corporal in the United States Marine Corps during the Vietnam War. Alternating between English and Spanish, Garcia told of early life in Brownsville, Texas, his time as a U.S. Marine in Vietnam, and his experience coming home from the war.
Civil rights activists protested against the injustice of segregation in a variety of ways. Here, in 1965, marchers, some carrying American flags, march from Selma to Montgomery, Alabama, to champion African American voting rights.
LBJ and Civil Rights Leaders (1964)
As civil rights demonstrations rocked the American South, civil rights legislation made its way through Washington D.C. Here, President Lyndon B. Johnson sits with civil rights leaders in the White House.
Women’s Liberation March (1970)
American popular feminism accelerated throughout the 1960s. The slogan “Women’s Liberation” accompanied a growing women’s movement but also alarmed conservative Americans. In this 1970 photograph, women march in Washington D.C. carrying signs reading, “Women Demand Equality,” “I’m a Second Class Citizen,” and “Women’s Liberation.”


3.4: Hidden connections
Problem 88 Two old women set out at sunrise and each walked with a constant speed. One went from A to B, and the other went from B to A . They met at noon, and continuing without a stop, they arrived respectively at B at 4 pm and at A at 9 pm. At what time was sunrise on that day?
Problem 89 A paddle-steamer takes five days to travel from St Louis to New Orleans, and takes seven days for the return journey. Assuming that the rate of flow of the current is constant, calculate how long it takes for a raft to drift from St Louis to New Orleans.
Problem 90 [From Paolo dell’Abbaco’s Trattato d’aritmetica ] “From here to Florence is 60 miles, and there is one who walks it in 8 days [in one direction], another in five days [in the opposite direction]. It is asked: Departing at the same time, in how many days will they meet?”
Problem 91 Notice that in Problem 88 sunrise occurs hours before noon, and that is the geometric mean of 4 and 9. Once this is pointed out, can you reformulate your solution to Problem 88 to solve a more general problem?


1.2: Rectangular Sleeve
For this example we want to build a template for a “chimney” or sleeve that will rest on a sloped roof surface. The roof surface will have a slope of 30° and the sleeve will have the dimensions of the drawing below.
Now that we have needed measurements we can begin to draw the stretch-out.
To help keep everything in order it helps to number points on stretch-out so they match up to each corner. Since this part will have a seam, we can use “0” to indicate the seam. Looking at the side & top view we can see that the points 1&4 have the same height as well as points 2&3 since they are in the same plane. The blue lines are there to show how the lines to extend to help develop the stretch-out.
Next we will transfer the measurements from the side view to the stretch-out to determine the shape of the template.
We know at points 0, 1 & 4, the height is the overall height measure at 9.46”, since all those point are in the same plane in the side view. For points 2 & 3 we can see that the overall height there is 6”. We need to transfer this to the stretch-out.  You can simply measure 6” down from each point on stretch-out at points 2 & 3 and mark it with a tic mark, you can also transfer the line 90° to the right and where the horizontal line intersects points 2 & 3 place a tic mark.
Now we can add a couple lines connecting the base of the stretch-out to vertical line 2 & 3 to show what the final shape of stretch-out will be. We can also remove any lines we used to transfer features from side view to stretch-out. The green line indicate the sheet size and shape the red lines indicate where the sheet will be formed.


6.1: The Workplace
The work world is always changing. Please click on the link below to see a YouTube video entitled “How the World of Work is Changing“ for an introduction to workplace changes in the last 20 years. Some of these changes are explained in this chapter including many changes that occurred only in the last few years due to the COVID-19 pandemic. Knowing about these workplace changes and trends will help you make informed decisions when preparing for your future career (Martinez & Shaker, 2021).
Video How the World of Work is Changing [youtu.be]
Trends are general directions in which elements are changing or moving. In the United States’ labor market and workplaces, some trends include certain types of jobs being in demand or decline and potential job growth or slow-downs and changes due to technology and corporate business structures. Knowing about these trends will help you prepare for your future career by being more informed about the current status of careers and potential changes expected in those careers. For example, you probably would not want to pursue a career in a field that is declining due to technological changes.
Several current workplace trends are listed later in this chapter, but these trends always are changing. To research the most current labor market information and trends in Pennsylvania, please visit the State of Pennsylvania’s Center for Workforce Information & Analysis. The State of Pennsylvania’s Center for Workforce Information & Analysis is the primary source of labor market information for the Commonwealth. Their mission is “to produce and provide timely, objective, and credible workforce and economic statistics and analysis to help our customers make informed policy, business and career decisions that strengthen Pennsylvania’s economic security” (Center for Workforce Information and Analysis, 2020).


7.5: The Slope-Intercept Form of a Line
The General Form of a Line
We have seen that the general form of a linear equation in two variables is ax+by=c (Section 7.4). When this equation is solved for y, the resulting form is called the slope-intercept form. Let's generate this new form.
\(\begin{aligned}
ax + by &=c&\text{Subtract } ax \text{ from both sides. }\\
by &= -ax + c&\text{ Divide both sides by } b\\
\dfrac{by}{b} &= \dfrac{-ax}{b} + \dfrac{c}{b}\\
\dfrac{\not{b}y}{\not{b}}&= \dfrac{-ax}{b} + \dfrac{c}{b}\\
y&=\dfrac{-ax}{b} + \dfrac{c}{b}\\
\end{aligned}\)
This equation is of the form \(y = mx + b\) if we replace \(\dfrac{-a}{b}\) with \(m\) and constant \(\dfrac{c}{b}\) with \(b\). ( Note The fact that we let \(b = \dfrac{c}{b}\) is unfortunate and occurs because of the letters we have chosen to use in the general form. The letter \(b\) occurs on both sides of the equal sign and may not represent the same value at all. This problem is one of the historical convention and, fortunately, does not occur very often.)
The following examples illustrate this procedure.
Solve \(3x + 2y = 6\) for \(y\).
\(\begin{aligned}
3x + 2y&= 6& \text{ Subtract } 3x \text{ from both sides }\\
2y&= -3x + 6& \text{ Divide both sides by } 2\\
y &= -\dfrac{3}{2}x + 3
\end{aligned}\)
The equation is of the form \(y = mx + b\). In this case, \(m = -\dfrac{3}{2}\) and \(b = 3\).
Solve \(-15x + 5y = 20\) for \(y\).
\(\begin{aligned}
-15x + 5y&=20\\
5y&=15x + 20\\
y&=3x+4
\end{aligned}\)
This equation is of the form \(y = mx + b\). In this case, \(m = 3\) and \(b = 4\).
Solve \(4x-y = 0\) for \(y\).
\(\begin{aligned}
4x-y&=0\\
-y&=-4x\\
y&=4x
\end{aligned}\)
This equation is of the form \(y = mx + b\). In this case, \(m=4\) and \(b=0\). Notice that we can write \(y=4x\) as \(y = 4x + 0\)
The Slope-Intercept Form of a Line
The Slope-Intercept Form of a Line \(y=mx+b\)
A linear equation in two variables written in the form \(y=mx+b\) is said to be in slope-intercept form.
Sample Set A
The following equations are in slope-intercept form:
\(y = 6x - 7\). In this case \(m = 6\) and \(b = -7\)
\(y = -2x + 9\). In this case \(m = -2\) and \(b = 9\)
\(y = \dfrac{1}{5}x + 4.8\). In this case \(m = \dfrac{1}{5}\) and \(b = 4.8\)
\(y = 7x\). In this case \(m = 7\) and \(b = 0\) since we can write \(y = 7x\) as \(y = 7x + 0\).
The following equations are not in slope-intercept form.
\(2y = 4x - 1\). The coefficient of \(y\) is \(2\). To be in slope-intercept form, the coefficient of \(y\) must be \(1\).
\(y + 4x = 5\). The equation is not solved for \(y\). THe \(x\) and \(y\) appear on the same side of the equal sign.
\(y + 1 = 2x\). The equation is not solved for \(y\)
Practice Set A
The following equation are in slope-intercept form. In each case, specify the slope and \(y\)-intercept.
\(y=2x+7; m=?, b=?\)
- Answer
-
\(m=2, b=7\)
\(y=−4x+2; m=?, b=?\)
- Answer
-
\(m=−4,b=2\)
\(y=−5x−1; m=?, b=?\)
- Answer
-
\(m=−5,b=−1\)
\(y=\dfrac{2}{3}x−10; m=?, b=?\)
- Answer
-
\(m = \dfrac{2}{3}, b = -10\)
\(y = \dfrac{-5}{8}x + \dfrac{1}{2}; m=?, b=?\)
- Answer
-
\(m = \dfrac{-5}{8}, b=\dfrac{1}{2}\)
\(y=−3x; m=?, b=?\)
- Answer
-
\(m=−3, b=0\)
Slope and Intercept
When the equation of a line is written in slope-intercept form, two important properties of the line can be seen: the slope and the intercept . Let's look at these two properties by graphing several lines and observing them carefully.
Sample Set B
Graph the line \(y=x−3\).
| \(x\) | \(y\) | \((x, y)\) |
| \(0\) | \(−3\) | \((0, −3)\) |
| \(4\) | \(1\) | \((4, 1)\) |
| \(−2\) | \(−5\) | \((−2, −5)\) |
Looking carefully at this line, answer the following two questions.
At what number does this line cross the y-axis? Do you see this number in the equation?
The line crosses the y-axis at −3.
Place your pencil at any point on the line. Move your pencil exactly one unit horizontally to the right. Now, how many units straight up or down must you move your pencil to get back on the line? Do you see this number in the equation?
After moving horizontally one unit to the right, we must move exactly one vertical unit up. This number is the coefficient of \(x\).
Graph the line \(y = \dfrac{2}{3}x + 1\)
| \(x\) | \(y\) | \((x, y)\) |
| \(0\) | \(1\) | \((0, 1)\) |
| \(3\) | \(3\) | \((3, 3)\) |
| \(−3\) | \(−1\) | \((−3, −1)\) |
Looking carefully at this line, answer the following two questions.
At what number does this line cross the \(y\)-axis? Do you see this number in the equation?
The line crosses the \(y\)-axis at \(+1\).
Place your pencil at any point on the line. Move your pencil exactly one unit horizontally to the right. Now, how many units straight up or down must you move your pencil to get back on the line? Do you see this number in the equation?
After moving horizontally one unit to the right, we must move exactly \(\dfrac{2}{3}\) unit upward. This number is the coefficient of \(x\).
Practice Set B
Graph the line \(y = -3x + 4\)
| \(x\) | \(y\) | \((x, y)\) |
| \(0\) | - | - |
| \(3\) | - | - |
| \(2\) | - | - |
Looking carefully at this line, answer the following two questions.
At what number does the line cross the \(y\)-axis? Do you see this number in the equation?
- Answer
-
The line crosses the \(y\)-axis at \(+4\). After moving horizontally \(1\) unit to the right, we must move exactly \(3\) units downward.
Place your pencil at any point on the line. Move your pencil exactly one unit horizontally to the right. Now, how many units straight up or down must you move your pencil to get back on the line? Do you see this number in the equation?
- Answer
-
In the graphs constructed in Sample Set B and Practice Set B, each equation had the form \(y=mx+b\). We can answer the same questions by using this form of the equation (shown in the diagram).
\(y\) -Intercept
At what number does the line cross the \(y\)-axis? Do you see this number in the equation?
- Answer
-
In each case, the line crosses the \(y\)-axis at the constant \(b\). The number \(b\) is the number at which the line crosses the \(y\)-axis, and it is called the \(y\)-intercept. The ordered pair corresponding to the \(y\)-intercept is \((0, b)\).
Place your pencil at any point on the line. Move your pencil exactly one unit horizontally to the right. Now, how many units straight up or down must you move your pencil to get back on the line? Do you see this number in the equation?
- Answer
-
To get back on the line, we must move our pencil exactly \(m\) vertical units.
Slope
The number \(m\) is the coefficient of the variable \(x\). The number \(m\) is called the
slope
of the line and it is the number of units that \(y\) changes when \(x\) is increased by \(1\) unit. Thus, if \(x\) changes by \(1\) unit, \(y\) changes by \(m\) units.
Since the equation \(y=mx+b\) contains both the slope of the line and the \(y\)-intercept, we call the form \(y=mx+b\) the
slope-intercept
form.
The slope-intercept form of a straight line is \(y=mx+b\)
The slope of the line is \(m\), and the \(y\)-intercept is the point \((0 , b)\).
The Slope is a Measure of the Steepness of a Line
The word slope is really quite appropriate. It gives us a measure of the steepness of the line. Consider two lines, one with slope \(\dfrac{1}{2}\) and the other with slope \(3\). The line with slope \(3\) is steeper than is the line with slope \(\dfrac{1}{2}\). Imagine your pencil being placed at any point on the lines. We make a \(1\)-unit increase in the \(x\)-value by moving the pencil one unit to the right. To get back to one line we need only move vertically \(\dfrac{1}{2}\) unit, whereas to get back onto the other line we need to move vertically \(3\) units.
Sample Set C
Find the slope and the y-intercept of the following lines.
\(y = 2x + 7\)
The line is in the slope-intercept form \(y=mx+b\). The slope is \(m\), the coefficient of \(x\). Therefore, \(m=2\). The \(y\)-intercept is the point \((0, b)\). Since \(b=7\), the \(y\)-intercept is \((0, 7)\).
Slope:\(2\)
\(y\)-intercept: \((0, 7)\)
\(y = -4x + 1\)
The line is in the slope-intercept form \(y=mx+b\). The slope is \(m\), the coefficient of \(x\). Therefore, \(m=-4\). The \(y\)-intercept is the point \((0, b)\). Since \(b=1\), the \(y\)-intercept is \((0, 1)\).
Slope:\(-4\)
\(y\)-intercept: \((0, 1)\)
\(3x + 2y = 5\).
This equation is written in general form. We can put the equation in slope-intercept form by solving for \(y\).
\(\begin{aligned}
3x+2y&=5\\
2y&=-3x+5\\
y&=-\dfrac{3}{2}x + \dfrac{5}{2}
\end{aligned}\)
Now the equation is in slope-intercept form.
Slope:\(-\dfrac{3}{2}\)
\(y\)-intercept: \((0, \dfrac{5}{2})\)
Practice Set C
Find the slope and \(y\)-intercept of the line \(2x+5y=15\).
- Answer
-
Solving for \(y\) we get \(y = \dfrac{-2}{5}x + 3\). Now \(m = \dfrac{-2}{5}\) and \(b = 3\).
The Formula for the Slope of a Line
We have observed that the slope is a measure of the steepness of a line. We wish to develop a formula for measuring this steepness.
It seems reasonable to develop a slope formula that produces the following results:
Steepness of line \(1>\) steepness of line 2.
Consider a line on which we select any two points. We’ll denote these points with the ordered pairs \((x_1, y_1 )\) and \((x_2, y_2 )\). The subscripts help us to identify the points.
\((x_1, y_1)\) is the first point. Subscript \(1\) indicates the first point.
\((x_2, y_2)\) is the second point. Subscript \(2\) indicates the second point.
The difference in \(x\) values \((x_2−x_1)\) gives us the horizontal change, and the difference in \(y\) values \((y_2−y_1)\) gives us the vertical change. If the line is very steep, then when going from the first point to the second point, we would expect a large vertical change compared to the horizontal change. If the line is not very steep, then when going from the first point to the second point, we would expect a small vertical change compared to the horizontal change.
We are comparing changes. We see that we are comparing:
\(\begin{aligned}
\text{The vertical change}&\text{ to }&\text{ the horizontal change}\\
\text{The change in }y&\text{ to }&\text{ the change in }x\\
y_1-y_1&\text{ to }&x_2-x_1
\end{aligned}\)
This is a comparison and is therefore a ratio . Ratios can be expressed as fractions. Thus, a measure of the steepness of a line can be expressed as a ratio.
The slope of a line is defined as the ratio
\(\text{Slope } = \dfrac{\text{ change in } y}{\text{ change in } x}\)
Mathematically, we can write these changes as
\(\text{Slope } = \dfrac{y_2-y_1}{x_2-x_1}\)
The slope of a nonvertical line passing through the points \((x_1, y_1)\) and \((x_2, y_2)\) is found by the formula:
\(m = \dfrac{y_2 - y_1}{x_2 - x_1}\)
Sample Set D
For the two given points, find the slope of the line that passes through them.
\((0,1)\) and \((1,3)\).
Looking left to right on the line we can choose \((x_1, y_1)\) to be \((0,1)\), and \((x_2, y_2)\) to be \((1, 3)\). Then,
\(m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{3-1}{1-0} = \dfrac{2}{1} = 2\)
This line has slope \(2\). It appears fairly steep. When the slope is written in fraction form, \(2 = \dfrac{2}{1}\), we can see, by recalling the slope formula, that as \(x\) changes \(1\) unit to the right (because of the \(+1\)) \(y\) changes \(2\) units upward (because of the \(+2\).
\(m = \dfrac{\text{ change in } y}{\text{ change in } x} = \dfrac{2}{1}\).
Notice that as we look left to right, the line rises.
\((2,2)\) and \((4,3)\).
Looking left to right on the line we can choose \((x_1, y_1)\) to be \((2,2)\), and \((x_2, y_2)\) to be \((4, 3)\). Then,
\(m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{3-2}{4-2} = \dfrac{1}{2}\)
This line has slope \(\dfrac{1}{2}\). Thus, as \(x\) changes \(2\) units to the right (because of the \(+2\), \(y\) changes \(1\) unit upward (because of the \(+1\)).
\(m = \dfrac{\text{ change in } y}{\text{ change in } x} = \dfrac{1}{2}\).
Notice that in examples 16 and 17, both lines have positive slopes, \(+2\) and \(+\dfrac{1}{2}\), and both lines rise as we look left to right.
\((-2, 4)\) and \((1,1)\).
Looking left to right on the line we can choose \((x_1, y_1)\) to be \((-2,4)\), and \((x_2, y_2)\) to be \((1,1)\). Then,
\(m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{1-4}{1-(-2)} = \dfrac{-3}{1 + 2} = \dfrac{-3}{3} = -1\)
This line has slope \(-1\).
When the slope is written in fraction form, \(m = -1 = \dfrac{-1}{+1}\), we can see that as \(x\) changes \(1\) unit to the right (because of the \(+1\), \(y\) changes \(1\) unit downward (because of the \(-1\)).
Notice also that this line has a negative slope and declines as we look left to right.
\((1, 3)\) and \((5, 3)\).
\(m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{3-3}{5-1} = \dfrac{0}{4} = 0\)
This line has \(0\) slope. This means it has no rise and, therefore, is a horizontal line. This does not mean that the line has no slope, however.
\((4,4)\) and \((4,0)\).
\(m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{0-4}{4-4} = \dfrac{-4}{0}\)
Since division by \(0\) is undefined, we say that vertical lines have undefined slope. Since there is no real number to represent the slope of this line, we sometimes say that vertical lines have undefined slope , or no slope .
Practice Set D
Find the slope of the line passing through \((2,1)\) and \((6,3)\). Graph this line on the graph of problem 2 below.
- Answer
-
\(m = \dfrac{3-1}{6-2} = \dfrac{2}{4} = \dfrac{1}{2}\)
Find the slope of the line passing through \((3,4)\) and \((5,5)\). Graph this line.
- Answer
-
The line has slope \(\dfrac{1}{2}\)
Compare the lines of the following problems. Do the lines appear to cross? What is it called when lines do not meet (parallel or intersecting)? Compare their slopes. Make a statement about the condition of these lines and their slopes.
- Answer
-
The lines appear to be parallel. Parallel lines have the same slope, and lines that have the same slope are parallel
Before trying some problems, let’s summarize what we have observed.
- The equation \(y=mx+b\) is called the slope-intercept form of the equation of a line. The number \(m\) is the slope of the line and the point \((0,b)\) is the \(y\)-intercept.
- The slope, \(m\), of a line is defined as the steepness of the line, and it is the number of units that \(y\) changes when \(x\) changes \(1\) unit.
-
The formula for finding the slope of a line through any two given points \((x_1,y_1)\) and \((x_2,y_2)\) is:
- \(m = \dfrac{y_2-y_1}{x_2-x_1}\)
- The fraction \(\dfrac{y_2-y_1}{x_2-x_1}\) represents the \dfrac{\text{ change in } y}{\text{ change in } x}
- As we look at a graph from left to right, lines with positive slope rise and lines with negative slope decline.
- Parallel lines have the same slope.
- Horizontal lines have 0 slope.
- Vertical lines have undefined slope (or no slope).
Exercises
For the following problems, determine the slope and y-intercept of the lines.
\(y=3x+4\)
- Answer
-
slope = \(3\); \(y\)-intercept= \((0,4)\)
\(y=2x+9\)
\(y=9x+1\)
- Answer
-
slope = \(9\); \(y\)-intercept= \((0,1)\)
\(y=7x+10\)
\(y=−4x+5\)
- Answer
-
slope = \(-4\); \(y\)-intercept= \((0,5)\)
\(y=−2x+8\)
\(y=−6x−1\)
- Answer
-
slope = \(-6\); \(y\)-intercept= \((0,-1)\)
\(y=−x−6\)
\(y=−x+2\)
- Answer
-
slope = \(-1\); \(y\)-intercept= \((0,2)\)
\(2y=4x+8\)
\(4y=16x+20\)
- Answer
-
slope = \(4\); \(y\)-intercept= \((0,5)\)
\(−5y=15x+55\)
\(−3y=12x−27\)
- Answer
-
slope = \(-4\); \(y\)-intercept= \((0,9)\)
\(y = \dfrac{3}{5} - 8\)
\(y = \dfrac{2}{7} - 12\)
- Answer
-
slope = \(\dfrac{2}{7}\); \(y\)-intercept= \((0,-12)\)
\(y = \dfrac{-1}{8}x + \dfrac{2}{3}\)
\(y = \dfrac{-4}{5} - \dfrac{4}{7}\)
- Answer
-
slope = \(-\dfrac{4}{5}\); \(y\)-intercept= \((0,-\dfrac{4}{7})\)
\(−3y=5x+8\)
\(−10y=−12x+1\)
- Answer
-
slope = \(\dfrac{6}{5}\); \(y\)-intercept= \((0,-\dfrac{1}{10})\)
\(−y=x+1\)
\(−y=−x+3\)
- Answer
-
slope = \(1\); \(y\)-intercept= \((0,-3)\)
\(3x−y=7\)
\(5x+3y=6\)
- Answer
-
slope = \(-\dfrac{5}{3}\); \(y\)-intercept= \((0,2)\)
\(−6x−7y=−12\)
\(−x+4y=−1\)
- Answer
-
slope = \(\dfrac{1}{4}\); \(y\)-intercept= \((0,-\dfrac{1}{4})\)
For the following problems, find the slope of the line through the pairs of points.
\((1, 6), (4,9)\)
\((1, 3), (4,7)\)
- Answer
-
\(m = \dfrac{4}{3}\)
\((3, 5), (4,7)\)
\((6, 1), (2,8)\)
- Answer
-
\(m = -\dfrac{7}{4}\)
\((0, 5), (2,−6)\)
\((−2, 1), (0,5)\)
- Answer
-
\(m=2\)
\((3, −9), (5,1)\)
\((4, −6), (−2,1)\)
- Answer
-
\(m = -\dfrac{7}{6}\)
\((−5, 4), (−1,0)\)
\((−3, 2), (−4,6)\)
- Answer
-
\(m=−4\)
\((9, 12), (6,0)\)
\((0, 0), (6,6)\)
- Answer
-
\(m=1\)
\((−2, −6), (−4,−1)\)
\((−1, −7), (−2,−9)\)
- Answer
-
\(m=2\)
\((−6, −6), (−5,−4)\)
\((−1, 0), (−2,−2)\)
- Answer
-
\(m=2\)
\((−4, −2), (0,0)\)
\((2, 3), (10,3)\)
- Answer
-
\(m=0\) (horizontal line \(y=3\))
Make a statement about the slopes of parallel lines.
\((4, −2), (4,7)\)
\((8, −1), (8,3)\)
- Answer
-
No slope (vertical line at \(x=8\))
\((4, 2), (6,2)\)
\((5, −6), (9,−6)\)
- Answer
-
\(m=0\) (horizontal line at \(y=−6\))
Do lines with a positive slope rise or decline as we look left to right?
Do lines with a negative slope rise or decline as we look left to right?
- Answer
-
decline
For the following problems, determine the slope and y-intercept of the lines. Round to two decimal places.
\(3.8x+12.1y=4.26\)
- Answer
-
slope=\(−0.31\)
\(y\)−intercept=\((0,0.35)\)
\(8.09x+5.57y=−1.42\)
\(10.813x−17.0y=−45.99\)
- Answer
-
slope=\(0.64\)
\(y\)−intercept=\((0,2.71)\)
\(−6.003x−92.388y=0.008\)
For the following problems, find the slope of the line through the pairs of points. Round to two decimal places.
\((5.56, 9.37), (2.16, 4.90)\)
- Answer
-
\(m=1.31\)
\((33.1, 8.9), (42.7, −1.06)\)
\((155.89, 227.61), (157.04,227.61)\)
- Answer
-
\(m=0\) (horizontal line at \(y=227.61\))
\((0.00426, −0.00404), (−0.00191, −0.00404)\)
\((88.81, −23.19), (88.81, −26.87)\)
- Answer
-
No slope (vertical line \(x=88.81\))
\((−0.0000567, −0.0000567), (−0.00765, 0.00764)\)
Exercises for Review
Simplify \((x^2y^3w^4)^0\).
- Answer
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\(1\) if \(xyw≠0\)
Solve the equation \(3x−4(2−x)−3(x−2)+4=0\).
When four times a number is divided by five, and that result is decreased by eight, the result is zero. What is the original number?
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\(10\)
Solve \(−3y+10=x+2\) if \(x=−4\).
Graph the linear equation \(x+y=3\).
- Answer
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