30.2C: Bone Tissue and the Effects of Aging
As individuals age, bone resorption can outpace bone replacement, which can lead to osteoporosis and fractures.
- Analyze osteoclastic activity
Key Points
- Bone resorption is the process by which osteoclasts break down bone and release its minerals, resulting in a transfer of calcium from bone to blood.
- Bone resorption is highly regulated. It can be stimulated or inhibited by signals from other parts of the body depending on the demand for calcium.
- As people age, the rate of bone resorption far exceeds the rate of bone formation. Thus, bones can weaken, leading to conditions like osteoporosis.
- Osteoporosis can be prevented with lifestyle changes such as proper nutrition and exercise.
- The risks of osteoporosis can also be reduced through the prevention of falls. Fall-prevention advice includes exercise to tone deambulatory muscles, proprioception-improvement exercises, and equilibrium therapies.
Key Terms
- osteon : Any of the central canals and surrounding bony layers found in compact bone.
- periosteum : A membrane surrounding a bone.
EXAMPLES
Joints undergo substantial wear and tear. The longer one lives, the more a joint is used. Gardeners and flooring installers put a lot of stress on their knees. Use of knee pads can relieve some of that stress.
Bone resorption is the process by which osteoclasts break down bone and release the minerals, resulting in a transfer of calcium from bone to blood.
The Role of Osteoclasts
The osteoclasts are multi-nucleated cells that contain numerous mitochondria and lysosomes. These cells are responsible for the resorption of bone and are generally present on the outer layer of bone, just beneath the periosteum. Attachment of the osteoclast to the osteon begins the process. The osteoclast then induces an infolding of its cell membrane and secretes collagenase and other enzymes important in the resorption process. High levels of calcium, magnesium, phosphate, and collagen products are released into the extracellular fluid as the osteoclasts tunnel into the mineralized bone. Osteoclasts are also prominent in the tissue destruction commonly found in psoriatic arthritis and other rheumatology-related disorders.
Regulation of Bone Tissue
Osteoclast : Osteoclast, displaying many nuclei within its “foamy” cytoplasm above a bone’s surface.
Bone resorption is highly constructible, stimulated or inhibited by signals from other parts of the body depending on the demand for calcium.
Calcium-sensing membrane receptors in the parathyroid gland monitor calcium levels in the extracellular fluid. Low levels of calcium stimulate the release of parathyroid hormone (PTH) from chief cells of the parathyroid gland. In addition to its effects on the kidney and the intestine, PTH also increases the number and activity of osteoclasts to release calcium from bone, thus stimulating bone resorption. High levels of calcium in the blood, on the other hand, lead to decreased PTH release from the parathyroid gland. This decreases the number and activity of osteoclasts, resulting in less bone resorption.
Aging
As people get older, the rate of resorption tends to exceed the rate of replacement, leading to conditions like osteoporosis. Bone resorption can also be the result of disuse and the lack of stimulus for bone maintenance. For instance, astronauts undergo a certain amount of bone resorption due to the lack of gravity providing the proper stimulus for bone maintenance. In addition, certain medical conditions such as hormone imbalances can cause bone resorption to increase, leading to increased susceptibility to fractures.
Osteoporosis risks can be reduced with lifestyle changes and sometimes medication. Lifestyle change includes diet, exercise, and fall-prevention measures. Medication includes calcium, vitamin D, and bisphosphonates. Fall-prevention advice includes exercise to tone deambulatory muscles, proprioception-improvement exercises, and equilibrium therapies. With its anabolic effect, exercise may simultaneously stop or reverse osteoporosis, a component of frailty syndrome.
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- Bone remodeling period. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Bone_remodeling_period . License : CC BY-SA: Attribution-ShareAlike
- epiphyses. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/epiphyses . License : CC BY-SA: Attribution-ShareAlike
- diaphysis. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/diaphysis . License : CC BY-SA: Attribution-ShareAlike
- Bonemetabolism. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Bonemetabolism.svg . License : CC BY-SA: Attribution-ShareAlike
- The Bone-Muscle Relationship in Men and Women. Provided by : Hindawi Publishing Company. Located at : http://www.hindawi.com/journals/josteo/2011/702735/ . License : CC BY-SA: Attribution-ShareAlike
- Aerobic Exercise and Whole-Body Vibration in Offsetting Bone Loss in Older Adults. Provided by : Hindawi Publishing Company. Located at : http://www.hindawi.com/journals/jar/2011/379674/ . License : CC BY-SA: Attribution-ShareAlike
- Wolff's law. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Wolff's_law . License : CC BY-SA: Attribution-ShareAlike
- skeletal strength. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/skeletal%20strength . License : CC BY-SA: Attribution-ShareAlike
- muscle forces. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/muscle%20forces . License : CC BY-SA: Attribution-ShareAlike
- Wolff's law. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Wolff's%20law . License : CC BY-SA: Attribution-ShareAlike
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- Bone resorption. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Bone_resorption . License : CC BY-SA: Attribution-ShareAlike
- WikiArc/ The Wiki Archaeological Information Resource - Joint Disease and Bone Growth. Provided by : Wikiarc. Located at : http://www.wikiarc.org/jointdisease . License : CC BY-SA: Attribution-ShareAlike
- Osteoporosis. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Osteoporosis . License : CC BY-SA: Attribution-ShareAlike
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- periosteum. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/periosteum . License : CC BY-SA: Attribution-ShareAlike
- osteon. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/osteon . License : CC BY-SA: Attribution-ShareAlike
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- Osteoclast. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Osteoclast.jpg . License : CC BY-SA: Attribution-ShareAlike


What is a growth mindset, and how does it affect my learning?
What are performance goals versus learning goals?
In the previous sections of this chapter you have focused on a number of concepts and models about learning. One of the things they all have in common is that they utilize different approaches to education by presenting new ways to think about learning. In each of these, the common element has been a better understanding of yourself as a learner and how to apply what you know about yourself to your own learning experience. If you were to distill all that you have learned in this chapter so far down to a single factor, it would be about using your mindset to your best advantage. In this next section, you will examine how all of this works in a broader sense by learning about the significance of certain mindsets and how they can hinder or promote your own learning efforts.
Performance vs. Learning Goals
As you have discovered in this chapter, much of our ability to learn is governed by our motivations and goals. What has not yet been covered in detail has been how sometimes hidden goals or mindsets can impact the learning process. In truth, we all have goals that we might not be fully aware of, or if we are aware of them, we might not understand how they help or restrict our ability to learn. An illustration of this can be seen in a comparison of a student that has
performance
-based goals with a student that has
learning
-based goals.
If you are a student with strict performance goals, your primary psychological concern might be to appear intelligent to others. At first, this might not seem to be a bad thing for college, but it can truly limit your ability to move forward in your own learning. Instead, you would tend to play it safe without even realizing it. For example, a student who is strictly performance-goal-oriented will often only says things in a classroom discussion when they think it will make them look knowledgeable to the instructor or their classmates. For example, a performance-oriented student might ask a question that she knows is beyond the topic being covered (e.g., asking about the economics of Japanese whaling while discussing the book
Moby Dick
in an American literature course). Rarely will they ask a question in class because they actually do not understand a concept. Instead they will ask questions that make them look intelligent to others or in an effort to “stump the teacher.” When they do finally ask an honest question, it may be because they are more afraid that their lack of understanding will result in a poor performance on an exam rather than simply wanting to learn.
If you are a student who is driven by learning goals, your interactions in classroom discussions are usually quite different. You see the opportunity to share ideas and ask questions as a way to gain knowledge quickly. In a classroom discussion you can ask for clarification immediately if you don’t quite understand what is being discussed. If you are a person guided by learning goals, you are less worried about what others think since you are there to learn and you see that as the most important goal.
Another example where the difference between the two mindsets is clear can be found in assignments and other coursework. If you are a student who is more concerned about performance, you may avoid work that is challenging. You will take the “easy A” route by relying on what you already know. You will not step out of your comfort zone because your psychological goals are based on approval of your performance instead of being motivated by learning.
This is very different from a student with a learning-based psychology. If you are a student who is motivated by learning goals, you may actively seek challenging assignments, and you will put a great deal of effort into using the assignment to expand on what you already know. While getting a good grade is important to you, what is even more important is the learning itself.
If you find that you sometimes lean toward performance-based goals, do not feel discouraged. Many of the best students tend to initially focus on performance until they begin to see the ways it can restrict their learning. The key to switching to learning-based goals is often simply a matter of first recognizing the difference and seeing how making a change can positively impact your own learning.
What follows in this section is a more in-depth look at the difference between performance- and learning-based goals. This is followed by an exercise that will give you the opportunity to identify, analyze, and determine a positive course of action in a situation where you believe you could improve in this area.

No. Someone has always dictated how and what I learned.
Yes. I always look for ways to take control of what and how I learned.
I am uncertain. I never thought about it before.
Have you ever heard of learning styles or do you know your own learning style?
No. I have never heard of learning styles.
Yes. I have heard of learning styles and know my own.
Yes. I have heard of learning styles, but I don’t think they’re accurate or relate to me.
Which factors other than intelligence do you think have the greatest influence on learning?
Motivation
Perseverance
Understanding how I learn
Good teachers and support
You can also take the anonymous
What Students Say
surveys to add your voice to this textbook. Your responses will be included in updates.
Students offered their views on these questions, and the results are displayed in the graphs below.
In the past, did you feel like you had control over your own learning?
Have you ever heard of learning styles or do you know your own learning style?
Which factors other than intelligence do you think have the greatest influence on learning?
Fixed vs. Growth Mindset
The research-based model of these two mindsets and their influence on learning was presented in 1988 by Carol Dweck.
7
In Dr. Dweck’s work, she determined that a student’s perception about their own learning accompanied by a broader goal of learning had a significant influence on their ability to overcome challenges and grow in knowledge and ability. This has become known as the Fixed vs. Growth Mindset model. In this model, the
performance
-goal-oriented student is represented by the
fixed
mindset, while the
learning
-goal-oriented student is represented by the
growth
mindset.
In the following graphic, based on Dr. Dweck’s research, you can see how many of the components associated with learning are impacted by these two mindsets.
The Growth Mindset and Lessons About Failing
Something you may have noticed is that a growth mindset would tend to give a learner grit and persistence. If you had learning as your major goal, you would normally keep trying to attain that goal even if it took you multiple attempts. Not only that, but if you learned a little bit more with each try you would see each attempt as a success, even if you had not achieved complete mastery of whatever it was you were working to learn.
With that in mind, it should come as no surprise that Dr. Dweck found that those people who believed their abilities could change through learning (growth vs. a fixed mindset) readily accepted learning challenges and persisted despite early failures.
Improving Your Ability to Learn
As strange as it may seem, research into fixed vs. growth mindsets has shown that if you believe you can learn something new, you greatly improve your ability to learn. At first, this may seem like the sort of feel-good advice we often encounter in social media posts or quotes that are intended to inspire or motivate us (e.g.,
believe in yourself!
), but in looking at the differences outlined between a fixed and a growth mindset, you can see how each part of the growth mindset path would increase your probability of success when it came to learning.
ACTIVITY
Very few people have a strict fixed or growth mindset all of the time. Often we tend to lean one way or another in certain situations. For example, a person trying to improve their ability in a sport they enjoy may exhibit all of the growth mindset traits and characteristics, but they find themselves blocked in a fixed mindset when they try to learn something in another area like computer programming or arithmetic.
In this exercise, do a little self-analysis and think of some areas where you may find yourself hindered by a fixed mindset. Using the outline presented below, in the far right column, write down how you can change your own behavior for each of the parts of the learning process. What will you do to move from a fixed to a growth mindset? For example, say you were trying to learn to play a musical instrument. In the
Challenges
row, you might pursue a growth path by trying to play increasingly more difficult songs rather than sticking to the easy ones you have already mastered. In the
Criticism
row, you might take someone’s comment about a weakness in timing as a motivation for you to practice with a metronome. For
Success of others
you could take inspiration from a famous musician that is considered a master and study their techniques.
Whatever it is that you decide you want to use for your analysis, apply each of the Growth characteristics to determine a course of action to improve.
Table 2.4
Parts of the learning process
Growth characteristic
What will you do to adopt a growth mindset?
Challenges
Embraces challenges
Obstacles
Persists despite setbacks
Effort
Sees effort as a path to success
Criticism
Learns from criticism
Success of Others
Finds learning and inspiration in the success of others
Footnotes
Dweck, C.S. & Leggett, E.L. (1988). A Social-Cognitive Approach to Motivation and Personality


3.1: ABC’s
Serves: 8
Cook Time: 5 min
Ingredients
- 8—10 Players
- Audience ‘Ask For’ suggestions:
- ‘Give us a suggestion for a heart-to-heart discussion between family members’
Instructions
- Players partner assigning ‘a’ or ‘b’
- Player ‘A’ must begin the scene with a sentence starting with the letter ‘A”
- Player ‘B” then must respond with the next sentence starting with the letter ‘B’.
- Players alternate letters of the alphabet until they reach ‘Z’.
Notes
To play this game players must create a rapid fire scene based on changing dialogue. Players must be in sentences with lets of the alphabet and alternate between each other with the next letter. Scene or relationships between the players are to be determined at random by the audience, or an ‘Ask For’.


5.5: Graphing Logarithmic Functions
- Identify the domain of logarithmic functions.
- Identify transformations of logarithmic functions.
- Graph logarithmic functions.
- Find equations of logarithmic functions that model graphical information.
Earlier in this chapter, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the cause for an effect .
To illustrate, suppose we invest \($2500\) in an account that offers an annual interest rate of 5%, compounded continuously. We already know that the balance in our account for any year \(t\) can be found with the equation \(A=2500e^{0.05t}\).
But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? Figure \(\PageIndex{1}\) shows this point on the logarithmic graph.
In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.
Finding the Domain of a Logarithmic Function
Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.
Recall that the exponential function is defined as \(y=b^x\) for any real number \(x\) and constant \(b>0\), \(b≠1\), where
- The domain of \(y\) is \((−\infty,\infty)\).
- The range of \(y\) is \((0,\infty)\).
In the last section, we learned that the logarithmic function \(y={\log}_b(x)\) is the inverse of the exponential function \(y=b^x\). So, as inverse functions:
- The domain of \(y={\log}_b(x)\) is the range of \(y=b^x\): \((0,\infty)\).
- The range of \(y={\log}_b(x)\) is the domain of \(y=b^x\): \((−\infty,\infty)\).
Transformations of the basic function \(y={\log}_b(x)\) behave similarly to those of other functions. Just as with other basic functions, we can apply the four types of transformations—shifts, stretches, compressions, and reflections—to the basic function without loss of shape.
We've seen that certain transformations can change the range of \(y=b^x\). Similarly, applying transformations to the basic function \(y={\log}_b(x)\) can change the domain . When finding the domain of a logarithmic function, it is important to remember that the domain consists only of positive real numbers . That is, the argument of the logarithmic function must be greater than zero.
For example, consider \(f(x)={\log}_4(2x−3)\). This function is defined for any values of \(x\) such that the argument, in this case \(2x−3\), is greater than zero. To find the domain, we set up an inequality and solve for \(x\):
\[\begin{align*} 2x-3&> 0 \qquad \text {Show the argument greater than zero}\\ 2x&> 3 \qquad \text{Add 3} \\ x&>\dfrac{3}{2} \qquad \text{Divide by 2} \\ \end{align*}\]
In interval notation, the domain of \(f(x)={\log}_4(2x−3)\) is \(\left(\dfrac{3}{2},\infty\right)\).
Find the domain of the following functions:
- \(f(x)={\log}_2(x+3)\)
- \(g(t)=4-{\log}(5−2t)\)
Solution
- The logarithmic function is defined only when the input is positive, so this function is defined when \(x+3>0\). Solving this inequality,
\[\begin{align*} x+3&> 0 \qquad \text{The input must be positive}\\ x&> -3 \qquad \text{Subtract 3} \end{align*}\]
The domain of \(f(x)={\log}_2(x+3)\) is \((−3,\infty)\).
-
This function is defined when \(5–2t>0\). Solving this inequality,
\[\begin{align*} 5-2t&> 0 \qquad \text{The input must be positive}\\ -2t&> -5 \qquad \text{Subtract 5}\\ t&< \dfrac{5}{2} \qquad \text{Divide by -2 and switch the inequality} \end{align*}\]
The domain of \(g(t)=4-{\log}(5−2t)\) is \(\left(–\infty,\dfrac{5}{2}\right)\).
What is the domain of \(h(x)=\log_4(x^2-4x+3)+2\)?
Solution
This function is defined when \(x^2-4x+3>0\). Solving this inequality using a sign chart,
\[\begin{align*} x^2-4x+3&> 0 \\[16pt] (x-3)(x-1)&>0 \end{align*} \]
The domain is \((−3,\infty)\)
What is the domain of \(f(x)=\log(3x+7)+2\)?
- Answer
-
\((-/dfrac{7}{3},\infty)\)
Graphing Logarithmic Functions
Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the basic function \(y={\log}_b(x)\) along with all its transformations: shifts, stretches, compressions, and reflections.
We begin with the basic function \(y={\log}_b(x)\). Because every logarithmic function of this form is the inverse of an exponential function with the form \(y=b^x\), their graphs will be reflections of each other across the line \(y=x\). To illustrate this, we can observe the relationship between the input and output values of \(y=2^x\) and its equivalent \(x={\log}_2(y)\) in Table \(\PageIndex{1}\).
| \(x\) | \(−3\) | \(−2\) | \(−1\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|---|---|---|
| \(2^x=y\) | \(\dfrac{1}{8}\) | \(\dfrac{1}{4}\) | \(\dfrac{1}{2}\) | \(1\) | \(2\) | \(4\) | \(8\) |
| \({\log}_2(y)=x\) | \(−3\) | \(−2\) | \(−1\) | \(0\) | \(1\) | \(2\) | \(3\) |
Using the inputs and outputs from Table \(\PageIndex{1}\), we can build another table to observe the relationship between points on the graphs of the inverse functions \(f(x)=2^x\) and \(g(x)={\log}_2(x)\). See Table \(\PageIndex{2}\).
| \(f(x)=2^x\) | \(\left(−3,\dfrac{1}{8}\right)\) | \(\left(−2,\dfrac{1}{4}\right)\) | \(\left(−1,\dfrac{1}{2}\right)\) | \((0,1)\) | \((1,2)\) | \((2,4)\) | \((3,8)\) |
|---|---|---|---|---|---|---|---|
| \(g(x)={\log}_2(x)\) | \(\left(\dfrac{1}{8},−3\right)\) | \(\left(\dfrac{1}{4},−2\right)\) | \(\left(\dfrac{1}{2},−1\right)\) | \((1,0)\) | \((2,1)\) | \((4,2)\) | \((8,3)\) |
As we’d expect, the \(x\)- and \(y\)-coordinates are reversed for the inverse functions. Figure \(\PageIndex{3}\) shows the graph of \(f\) and \(g\).
Observe the following from the graph:
- \(f(x)=2^x\) has a \(y\)-intercept at \((0,1)\) and \(g(x)={\log}_2(x)\) has an \(x\)- intercept at \((1,0)\).
- The domain of \(f(x)=2^x\), \((−\infty,\infty)\), is the same as the range of \(g(x)={\log}_2(x)\).
- The range of \(f(x)=2^x\), \((0,\infty)\), is the same as the domain of \(g(x)={\log}_2(x)\).
For any real number \(x\) and constant \(b>0\), \(b≠1\), we can see the following characteristics in the graph of \(f(x)={\log}_b(x)\):
- one-to-one function
- vertical asymptote: \(x=0\)
- domain: \((0,\infty)\)
- range: \((−\infty,\infty)\)
- \(x\) - intercept: \((1,0)\)
- \(y\)-intercept: none
- point at \((b,1)\)
- increasing if \(b>1\)
- decreasing if \(0<b<1\)
Figure \(\PageIndex{4}\) compares the graphs of logarithmic functions with \(b>1\) and \(0<b<1\)
Figure \(\PageIndex{5}\) shows how changing the base \(b\) in \(f(x)={\log}_b(x)\) can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the function \(\ln(x)\) has base \(e≈2.718\).)
Sketch the graph of \(f(x)={\log}_5(x)\). State the domain, range, and asymptote.
Solution
Before graphing, identify the behavior and key points for the graph.
- Since \(b=5\) is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote, \(x=0\), and the right tail will increase slowly without bound.
- The \(x\)-intercept is \((1,0)\).
- The point \((5,1)\) is on the graph.
- We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see Figure \(\PageIndex{6}\)).
The domain is \((0,\infty)\), the range is \((−\infty,\infty)\), and the vertical asymptote is \(x=0\).
Graphing Transformations of Logarithmic Functions
As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other basic functions. We can shift, stretch, compress, and reflect the basic function \(y={\log}_b(x)\) without loss of shape.
Graphing a Horizontal Shift
When a constant \(c\) is added to the input of the basic function \(f(x)={\log}_b(x)\), the result is a horizontal shift \(c\) units in the opposite direction of the sign on \(c\). To visualize horizontal shifts, we can observe the general graph of the basic function \(f(x)={\log}_b(x)\) and for \(c>0\) alongside the shift left, \(g(x)={\log}_b(x+c)\), and the shift right, \(h(x)={\log}_b(x−c)\). See Figure \(\PageIndex{7}\).
Sketch the horizontal shift \(f(x)={\log}_3(x−2)\) alongside its basic function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.
Solution
Since the function is \(f(x)={\log}_3(x−2)\), we notice \(x+(−2)=x–2\).
Thus \(c=−2\), so \(c<0\). This means we will shift the function \(f(x)={\log}_3(x)\) right 2 units.
The vertical asymptote is \(x=−(−2)\) or \(x=2\).
Consider the three key points from the basic function, \(\left(\dfrac{1}{3},−1\right)\), \((1,0)\), and \((3,1)\).
The new coordinates are found by adding \(2\) to the \(x\) coordinates.
Label the points \(\left(\dfrac{7}{3},−1\right)\), \((3,0)\),and \((5,1)\).
The domain is \((2,\infty)\),the range is \((−\infty,\infty)\),and the vertical asymptote is \(x=2\).
Sketch a graph of \(f(x)={\log}_3(x+4)\) alongside its basic function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.
- Answer
-
The domain is \((−4,\infty)\),the range \((−\infty,\infty)\),and the asymptote \(x=–4\).
Graphing a Vertical Shift
When a constant \(d\) is added to the basic function \(f(x)={\log}_b(x)\),the result is a vertical shift \(d\) units in the direction of the sign on \(d\). To visualize vertical shifts, we can observe the general graph of the basic function \(f(x)={\log}_b(x)\) alongside the shift up, \(g(x)={\log}_b(x)+d\) and the shift down, \(h(x)={\log}_b(x)−d\).See Figure \(\PageIndex{10}\).
Sketch a graph of \(f(x)={\log}_3(x)−2\) alongside its basic function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Solution
Since the function is \(f(x)={\log}_3(x)−2\),we will notice \(d={ }–2\). Thus \(d<0\).
This means we will shift the function \(f(x)={\log}_3(x)\) down \(2\) units.
The vertical asymptote is \(x=0\).
Consider the three key points from the basic function, \(\left(\dfrac{1}{3},−1\right)\), \((1,0)\), and \((3,1)\).
The new coordinates are found by subtracting \(2\) from the y coordinates.
Label the points \(\left(\dfrac{1}{3},−3\right)\), \((1,−2)\), and \((3,−1)\).
The domain is \((0,\infty)\),the range is \((−\infty,\infty)\), and the vertical asymptote is \(x=0\).
The domain is \((0,\infty)\),the range is \((−\infty,\infty)\),and the vertical asymptote is \(x=0\).
Identify the transformations and use them to sketch the graph of \(L(x)=2+\log_3(x+1)\). State the domain, range, and asymptote.
Solution
We can rewrite \(L(x)=2+\log_3(x+1)\) as \(L(x)=\log_3(x+1)+2\) which can be helpful for identifying transformations.
The basic function is \(y=\log_3(x)\). The graph will shift left 1 unit and up 2 units.
Shifting left 1 unit and up 2 units results in the x-intercept of the basic graph shifting to \((0,2)\). The point \((3,1)\) on the basic graph shifts to \((2,3)\). The horizontal asymptote \(x=−1\).
Using these key points, asymptote, and the shape of \(y=\log_3(x)\), shift the whole graph left \(1\) units and down \(3\) units.
The domain is \((-1,\infty)\), the range is \((−\infty,\infty)\), and the horizontal asymptote is \(x=-1\)
Identify the transformations and use them to sketch the graph of \(m(x)=\log_5(x-4)-3\). State the domain, range, and asymptote.
- Answer
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The transformations are shift right 4 units and shift down 3 units.
The domain is \((4,\infty)\), the range is \((−\infty,\infty)\), and the horizontal asymptote is \(x=4\)
Remember that the input for a logarithm is called the argument. When the argument is a single term, it is a common practice to not write parentheses. Example: \[f(x)={\log}_4(x)\text{ is equivalent to } f(x)={\log}_4x\nonumber\]
However, when the argument is more than one term, one must put parentheses around it. Example: \[y=\ln(x+3) \text{ is not equivalent to } y=\ln x+3\nonumber\]
The function on the left is a natural logarithmic function shifted left 3 units, whereas, the function on the right is a natural logarithmic function shifted up 3 units.
In graphing calculators, one needs to make sure functions are inputted properly also.
Graphing Reflections of \(f(x) = log_b(x)\)
When the basic function \(f(x)={\log}_b(x)\) is multiplied by \(−1\),the result is a reflection about the \(x\)-axis. When the input is multiplied by \(−1\),the result is a reflection about the \(y\)-axis. To visualize reflections, we restrict \(b>1\), and observe the general graph of the basic function \(f(x)={\log}_b(x)\) alongside the reflection about the \(x\)-axis, \(g(x)=−{\log}_b(x)\) and the reflection about the \(y\)-axis, \(h(x)={\log}_b(−x)\).
Sketch a graph of \(f(x)=\log(−x)\) alongside its basic function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Solution
Before graphing \(f(x)=log(−x)\), \(f(x)=log(−x)\),identify the behavior and key points for the graph.
- Since \(b=10\) is greater than one, we know that the basic function is increasing. Since the input value is multiplied by \(−1\), \(f(x)\) is a reflection of the basic graph about the \(y\) - axis. Thus, \(f(x)=\log(−x)\) will be decreasing as \(x\) moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote \(x=0\).
- The \(x\)-intercept is \((−1,0)\).
- We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.
The domain is \((−\infty,0)\), the range is \((−\infty,\infty)\), and the vertical asymptote is \(x=0\).
Graph \(f(x)=−\log(−x)\). State the domain, range, and asymptote.
- Answer
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The domain is \((−\infty,0)\),the range is \((−\infty,\infty)\),and the vertical asymptote is \(x=0\).
Graphing Stretches and Compressions of \(y = log_b(x)\)
When the basic function \(f(x)={\log}_b(x)\) is multiplied by a constant \(a>0\), the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set \(a>1\) and observe the general graph of the basic function \(f(x)={\log}_b(x)\) alongside the vertical stretch, \(g(x)=a{\log}_b(x)\) and the vertical compression, \(h(x)=\dfrac{1}{a}{\log}_b(x)\). See Figure \(\PageIndex{17}\).
Sketch a graph of \(f(x)=2{\log}_4(x)\) alongside its basic function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Solution
Since the function is \(f(x)=2{\log}_4(x)\), we will notice \(a=2\).
This means we will stretch the function \(f(x)={\log}_4(x)\) by a factor of \(2\).
The vertical asymptote is \(x=0\).
Consider the three key points from the basic function, \(\left(\dfrac{1}{4},−1\right)\), \((1,0)\), and \((4,1)\).
The new coordinates are found by multiplying the \(y\) coordinates by \(2\).
Label the points \(\left(\dfrac{1}{4},−2\right)\), \((1,0)\), and \((4,2)\).
The domain is \((0, \infty)\), the range is \((−\infty,\infty)\), and the vertical asymptote is \(x=0\). See Figure \(\PageIndex{18}\).
The domain is \((0,\infty)\), the range is \((−\infty,\infty)\), and the vertical asymptote is \(x=0\).
Sketch a graph of \(f(x)=5{\log}(x+2)\). State the domain, range, and asymptote.
Solution
Remember: what happens inside basicheses happens first. First, we move the graph left \(2\) units, then stretch the function vertically by a factor of \(5\), as in Figure \(\PageIndex{19}\). The vertical asymptote will be shifted to \(x=−2\). The \(x\)-intercept will be \((−1,0)\). The domain will be \((−2,\infty)\). Two points will help give the shape of the graph: \((−1,0)\) and \((8,5)\). We chose \(x=8\) as the x -coordinate of one point to graph because when \(x=8\), \(x+2=10\), the base of the common logarithm.
The domain is \((−2,\infty)\), the range is \((−\infty,\infty)\),and the vertical asymptote is \(x=−2\).
Sketch a graph of the function \(f(x)=3{\log}(x−2)+1\). State the domain, range, and asymptote.
- Answer
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The domain is \((2,\infty)\), the range is \((−\infty,\infty)\), and the vertical asymptote is \(x=2\).
Summarizing Transformations of the Logarithmic Function
Now that we have worked with each type of transformation for the logarithmic function, we can summarize each to arrive at the general equation for translating exponential functions.
All transformations of the basic logarithmic function, \(y={\log}_b(x)\), have the form
\(f(x)=a{\log}_b(x+c)+d\)
where the basic function, \(y={\log}_b(x)\), \(b>1\),is
- shifted vertically up \(d\) units.
- shifted horizontally to the left \(c\) units.
- stretched vertically by a factor of \(|a|\) if \(|a|>0\).
- compressed vertically by a factor of \(|a|\) if \(0<|a|<1\).
- reflected about the x- axis when \(a<0\).
For \(f(x)=\log(−x)\), the graph of the basic function is reflected about the y-axis.
What is the vertical asymptote and domain of \(f(x)=−2{\log}_3(x+4)+5\)?
Solution
The vertical asymptote is at \(x=−4\).
Analysis
The coefficient, the base, and the upward transformation do not affect the asymptote. The shift of the curve \(4\) units to the left shifts the vertical asymptote to\(x=−4\). The domain is \((-4,\infty)\)
What is the vertical asymptote and domain of of \(f(x)=3+\ln(-x+1)\)?
- Answer
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The vertical asymptote is \(x=1\). The domain is \((-\infty, 1)\)
Finding Equations of Logarithmic Functions that Model Graphical Information
Write an equation of the natural logarithmic function that has no vertical stretch or compression that is graphed in Figure \(\PageIndex{21}\).
Solution
The vertical asymptote is at \(x=1\). There is a point at \((2,-5)\) that is one unit to the right of the asymptote which indicates this is the transformed x-intercept ( remember that for basic functions that the vertical asymptote is at \(x=0\) and the x-intercept is at \((1,0)\) ). This tells us the basic graph was shifted right 1 units and down 5 units. The equation for the graph is \(y=\ln(x-1)-5\)
Write an equation of the base 3 logarithmic function that is vertically stretched by 4, is reflected across the y-axis, and then shifted up 6 units.
- Answer
-
\(y=4\log_3(-x)+6\)


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13.16: Professional Associations for Sexual Health
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AASECT (American Association of Sexuality Educators Counselors & Therapists) www.aasect.org
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SSSS (Society for the Scientific Study of Sexuality) www.sexscience.org
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SSTAR (Society for Sex Therapy and Research) www.sstarnet.org
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AACAST (American Association of Couples and Sex Therapists) www.aacast.net
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WAS (World Association for Sexual Health) www.worldsexology.org
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WPATH (World Professional Association of Transgender Health) www. wpath.org
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WoCSHN (Women of Color Sexual Health Network)
www.wocshn.org
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IPSA (International Professional Surrogates Association) www.surrogatetherapy.org
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GLMA (Gay Lesbian Medical Association)
www.glma.org
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ISTI (Integrative Sex Therapy Institute)
www.integrativesextherapyinstitute.com
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TASHRA (The Alternative Sexualities Health Research Alliance)
www.tashra.org
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CARAS (Community Academic Consortium for Research on Alternative Sexualities)
www.carasresearch.org
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ISEE (Institute for Sexuality Education & Enlightenment) instituteforsexuality.com
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ISSWSH (International Society for the Study of Women’s Sexual Health) www.isswsh.org
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APA (American Psychological Association)
www.apa.org/topics/sexuality


9.6.5: Chapter Review
The Metric System
For the following exercises, determine the base unit of the metric system described: liter, gram, or meter.
amount of juice in a carton
weight of a puppy
height of a wall
For the following exercises, choose the smaller of the two units.
hectogram or centigram
decameter or kilometer
milliliter or deciliter
Convert 236 cm to meters (m).
Convert 137.5 mg to grams (g).
You purchase 15 kg of birdseed. You divide the birdseed into 10 bags. How many grams of birdseed are in each bag?
A track is 650 m long. If you run 8 laps around the track, how many kilometers did you run?
Measuring Area
For the following exercises, determine the most reasonable value for each area.
bedroom floor
12 km
2
12 m
2
12 cm
2
12 mm
2
nature preserve
800 km
2
800 m
2
800 cm
2
800 mm
2
desk surface
0.125 km
2
0.125 m
2
0.125 cm
2
0.125 mm
2
For the following exercises, convert the given area to the units shown.
650,000 cm 2 = __________ m 2
17 m 2 = __________ cm 2
759 cm 2 = __________ mm 2
A laptop is 30 cm by 20 cm. A sticker on the laptop cover measures 15 mm by 15 mm. How many square centimeters of the laptop cover is still visible?
A dining room wall is 2.75 m by 2.25 m. A window on the wall is 0.75 m by 1.5 m. How many square meters is the surface of the wall?
A floor is 6 m by 4 m. A rug on the floor measures 200 cm by 100 cm. How much of the floor, in m 2 , is not covered by the rug?
A ceiling is 4 m by 2.5 m. A skylight in the ceiling measures 75 cm by 80 cm. How much of the ceiling, in m 2 , would need to be painted if you decide to change the color of the ceiling?
Measuring Volume
For the following exercises, determine the most reasonable value for each volume.
aquarium
65,750 km
3
65,750 m
3
65,750 cm
3
65,750 mm
3
carton of soup
94,000 L
94 L
940,000 mL
940 mL
gift box
6,250 km
3
6,250 m
3
6,250 cm
3
6,250 mm
3
For the following exercises, convert the given volume to the units shown.
59,837 mm 3 = __________ cm 3
12.5 dm 3 = __________ mL
368 cm 3 = __________ mm 3
A box has dimensions of \(15\,{\text{cm}} \times 10\,{\text{cm}} \times 25\,{\text{cm}}\). The box currently holds 1,750 cm 3 of black beans. How many cubic centimeters of black beans are needed to completely fill the box?
The dimensions of a small storage unit are \(3\,{\text{m}} \times 3{\text{m}} \times 3\,{\text{m}}\). What is the volume of a medium storage area with dimensions twice the size of the small unit?
How much liquid, in liters, can a container with dimensions of \(35\,{\text{cm}} \times 15\,{\text{cm}} \times 20\,{\text{cm}}\) hold?
A box is 20 cm long and 15 cm wide. The volume of the box is 2,100 cm 3 . What is the height of the box?
Measuring Weight
What is the most reasonable estimate for the weight of a small stack of coins?
120 mg
12 g
1.2 kg
0.12 mt
For the following exercises, convert the given weight to the units shown.
7,900 g = __________ kg
81 g = __________ mg
0.12 kg = __________ g
For the following exercises, determine the total weight in the units shown.
four 450 mg tablets ________ g
six 45 g candy bars ________ kg
seven 1.5 g buttons ________ mg
A box of chocolate weighs 2.75 kg. If you share the candy bar equally with 5 friends, how many grams of chocolate does each person get?
You purchase 15 tomatoes that weigh 100 grams each. If tomatoes cost $1.18 per kilogram, how much did you pay?
A value-size package of ground chicken costs $23.00. The package weighs 2.5 kg. What is the cost per gram of the chicken?
Measuring Temperature
For the following exercises, convert the temperatures to the nearest degree.
365 °F = __________ °C
42 °C = __________ °F
575 °F = __________ °C
135 °C = __________ °F
–20 °F = __________ °C
73 °C = __________ °F
The temperature of a welder’s torch increases 1,112 ºF every 3 minutes. About how many minutes does it take for the torch to increase 1,200 ºC?
A substance is heated to 302 ºF. As the substance cools the temperature drops 4 ºC every 3 minutes. How many minutes until the temperature of the water reaches 130 ºC?
On November 1 the temperature was 40 ºF. On December 1 the temperature was 7 ºC. Which day was cooler?
The instructions to cook lasagna say to set the oven at 475 ºF. To the nearest degree, what is the temperature in degrees Celsius?


1.3: Fundamentals of Waves
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In this section, we formally introduce the concept of a wave and explain some basic characteristics.
To begin, let us consider not electromagnetic waves, but rather sound waves. To be clear, sound waves and electromagnetic waves are completely distinct phenomena. Sound waves are variations in pressure, whereas electromagnetic waves are variations in electric and magnetic fields. However, the mathematics that govern sound waves and electromagnetic waves are very similar, so the analogy provides useful insight. Furthermore, sound waves are intuitive for most people because they are readily observed. So, here we go:
Imagine standing in an open field and that it is completely quiet. In this case, the air pressure everywhere is about 101 kPa (101,000 N/m 2 ) at sealevel, and we refer to this as the quiescent air pressure . Sound can be described as the differential air pressure \(p ( x , y , z , t )\), which we define as the absolute air pressure at the spatial coordinates \((x, y, z)\) minus the quiescent air pressure. So, when there is no sound, \(p(x, y, z, t) = 0\). The function \(p\) as an example of a scalar field .
Let’s also say you are standing at \(x = y = z = 0\) and you have brought along a friend who is standing at \(x = d\); i.e., a distance \(d\) from you along the \(x\) axis. Also, for simplicity, let us consider only what is happening along the \(x\) axis; i.e., \(p(x, t)\).
At \(t = 0\), you clap your hands once. This forces the air between your hands to press outward, creating a region of increased pressure (i.e., \(p > 0\)) that travels outward. As the region of increased pressure moves outward, it leaves behind a region of low pressure where \(p < 0\). Air molecules immediately move toward this region of lower pressure, and so the air pressure quickly returns to the quiescent value, \(p = 0\). The traveling disturbance in \(p(x, t)\) is the sound of the clap. The disturbance continues to travel outward until it reaches your friend, who then hears the clap.
At each point in time, you can make a plot of \(p(x, t)\) versus \(x\) for the current value of \(t\). This is shown in Figure \(\PageIndex{1}\). At times \(t < 0\), we have simply \(p(x, t) = 0\). A short time after \(t = 0\), the peak pressure is located at slightly to the right of \(x = 0\). The pressure is not a simple impulse because interactions between air molecules constrain the pressure to be continuous over space. So instead, we see a rounded pulse representing the rapid build-up and similarly rapid decline in air pressure. A short time later \(p(x, t)\) looks very similar, except the pulse is now further away.
Figure \(\PageIndex{1}\): The differential pressure \(p(x, t)\) (top) a short time after the clap and (bottom) a slightly longer time after the clap. (CC BY 4.0; Y. Qin).Now: What precisely is \(p(x, t)\)? Completely skipping over the derivation, the answer is that \(p(x, t)\) is the solution to the acoustic wave equation (see “Additional References” at the end of this section):
\[\dfrac { \partial ^ { 2 } p } { \partial x ^ { 2 } } - \dfrac { 1 } { c _ { s } ^ { 2 } } \dfrac { \partial ^ { 2 } p } { \partial t ^ { 2 } } = 0 \label{1.1} \]
where \(c_s\) is the speed of sound, which is about 340 m/s at sea level. Just to emphasize the quality of the analogy between sound waves and electromagnetic waves, know that the acoustic wave equation is mathematically identical to equations that that govern electromagnetic waves
Although “transient” phenomena – analogous to a clap – are of interest in electromagnetics, an even more common case of interest is the wave resulting from a sinusoidally-varying source. We can demonstrate this kind of wave in the context of sound as well. Here we go:
In the previous scenario, you pick up a trumpet and blow a perfect A note. The A note is 440 Hz, meaning that the air pressure emerging from your trumpet is varying sinusoidally at a frequency of 440 Hz. Let’s say you can continue to blow this note long enough for the entire field to be filled with the sound of your trumpet. Now what does the pressure-versus-distance curve look like? Two simple observations will settle that question:
- \(p(x, t)\) at any constant position \(x\) is a sinusoid as a function of \(t\). This is because the acoustic wave equation is linear and time invariant, so a sinusoidal excitation (i.e., your trumpet) results in a sinusoidal response at the same frequency (i.e., the sound heard by your friend).
- \(p(x, t)\) at any constant time \(t\) is also a sinusoid as a function of \(x\). This is because the sound is propagating away from the trumpet and toward your friend, and anyone in between will also hear the A note, but with a phase shift determined by the difference in distances.
This is enough information to know that the solution must have the form:
\[p ( x , t ) = A _ { m } \cos ( \omega t - \beta x + \psi ) \label{1.2} \]
where \(ω = 2πf, f = 440 \; Hz\), and \(A_m, β,\) and \(ψ\) remain to be determined. You can readily verify that Equation \ref{1.2} satisfies the acoustic wave equation when
\[\beta = \dfrac { \omega } { c _ { s } } \nonumber \]
In this problem, we find \(\beta \cong\) 8.13 rad/m. This means that at any given time, the difference in phase measured between any two points separated by a distance of 1 m is 8.13 rad. The parameter \(β\) goes by at least three names: phase propagation constant , wavenumber , and spatial frequency . The term “spatial frequency” is particularly apt, since \(β\) plays precisely the same role for distance (\(x\)) as \(ω\) plays for time (\(t\)) – This is apparent from Equation \ref{1.2}. However, “wavenumber” is probably the more commonly-used term.
Definition: Wavenumber
The wavenumber \(β\) (rad/m) is the rate at which the phase of a sinusoidal wave progresses with distance.
Note that \(A_m\) and \(ψ\) are not determined by the wave equation, but instead are properties of the source. Specifically, \(A_m\) is determined by how hard we blow, and \(ψ\) is determined by the time at which we began to blow and the location of the trumpet. For simplicity, let us assume that we begin to blow at time \(t \ll 0\); i.e., in the distant past so that the sound pressure field has achieved steady state by \(t = 0\). Also, let us set \(ψ = 0\) and set \(A_m = 1\) in whatever units we choose to express \(p(x, t)\). We then have:
\[p(x, t) = cos (ωt − βx) \label{1.4} \]
Now we have everything we need to make plots of \(p(x)\) at various times. Figure \(\PageIndex{2a}\) shows \(p(x, t = 0)\). As expected, \(p(x, t = 0)\) is periodic in \(x\). The associated period is referred to as the wavelength λ. Since λ is the distance required for the phase of the wave to increase by \(2π\) rad, and because phase is increasing at a rate of β rad/m, we find: \[\lambda = \frac { 2 \pi } { \beta } \nonumber \] In the present example, we find \(λ \cong 77.3\) cm.
Wavelength \(λ = 2π/β\) is the distance required for the phase of a sinusoidal wave to increase by one complete cycle (i.e., \(2π\) rad) at any given time.
Now let us consider the situation at \(t = +1/4f\), which is \(t = 568 µs\) and \(ωt = π/2\). We see in Figure \(\PageIndex{2}\)(b) that the waveform has shifted a distance \(λ/4\) to the right. It is in this sense that we say the wave is propagating in the \(+x\) direction. Furthermore, we can now compute a phase velocity \(v_p\): We see that a point of constant phase has shifted a distance \(λ/4\) in time \(1/4f\), so
\[v_p = λf \nonumber \]
In the present example, we find \(v_p \cong 340\) m/s; i.e., we have found that the phase velocity is equal to the speed of sound \(c_s\). It is in this sense that we say that the phase velocity is the speed at which the wave propagates.
Figure \(\PageIndex{2}\): The differential pressure \(p(x, t)\) for (a) \(t = 0\), (b) \(t = 1/4f\) for “\(−βx\),” as indicated in Equation \ref{1.4} (wave traveling to right); and (c) \(t = 1/4f\) for “\(+βx\)” (wave traveling to left).Definition: Phase velocity
Phase velocity \(v_p = λf\) is the speed at which a point of constant phase in a sinusoidal waveform travels.
Recall that in Equation \ref{1.2} we declared that \(βx\) is subtracted from the argument of the sinusoidal function. To understand why, let’s change the sign of \(βx\) and see if it still satisfies the wave equation – one finds that it does. Next, we repeat the previous experiment and see what happens. The result is shown in Figure \(\PageIndex{2c}\). Note that points of constant phase have traveled an equal distance, but now in the \(−x\) direction. In other words, this alternative choice of sign for \(βx\) within the argument of the cosine function represents a wave that is propagating in the opposite direction. This leads us to the following realization:
If the phase of the wave is decreasing with \(βx\), then the wave is propagating in the \(+x\) direction. If the phase of the wave is increasing with \(βx\), then the wave is propagating in the \(−x\) direction
Since the prospect of sound traveling toward the trumpet is clearly nonsense in the present situation, we may neglect the latter possibility. However, what happens if there is a wall located in the distance, behind your friend? Then, we expect an echo from the wall, which would be a second wave propagating in the reverse direction and for which the argument of the cosine function would contain the term “\(+βx\).”
Finally, let us return to electromagnetics. Electromagnetic waves satisfy precisely the same wave equation (i.e., Equation \ref{1.1}) as do sound waves, except that the phase velocity is much greater. Interestingly, though, the frequencies of electromagnetic waves are also much greater than those of sound waves, so we can end up with wavelengths having similar orders of magnitude. In particular, an electromagnetic wave with \(λ = 77.3\) cm (the wavelength of the “A” note in the preceding example) lies in the radio portion of the electromagnetic spectrum.
An important difference between sound and electromagnetic waves is that electromagnetic waves are vectors; that is, they have direction as well as magnitude. Furthermore, we often need to consider multiple electromagnetic vector waves (in particular, both the electric field and the magnetic field ) in order to completely understand the situation. Nevertheless the concepts of wavenumber, wavelength, phase velocity, and direction of propagation apply in precisely the same manner to electromagnetic waves as they do to sound waves.


13.5: Structural and Procedural Algebra
When most people think about algebra from school, they think about “solving for x .” They imagine lots of equations with varying levels of complexity, but the goal is always the same: find the unknown quantity. This is a procedural view of algebra.
Even elementary students can be exposed to ideas in procedural algebra. This happens any time they think about unknown quantities and try to solve for them. For example, when first grade students learn to add and subtract numbers “within 10,’” they should frequently tackle problems like these:
- \(3 + \_\_\_ = 7 \ldotp\)
- Find several pairs of numbers that add up to 10.
Although procedural algebra is important, it’s not the most important skill, and it’s certainly not the whole story.
You also need to foster thinking about structural algebra in your students: using symbols to express meaning in a situation. If there is an x on your page, you should be able to answer, “what does the x mean? What does it represent?”
Most of what you’ve done so far in this chapter is structural algebra . You’ve used letters and symbols not to represent a single unknown quantity, but a varying quantity. For example, in Section 4 you used letters to represent the “figure number’” or “case number” in a growing pattern. The letters could take on different values, and the expressions gave you information: how many tiles or toothpicks or stars you needed to build that particular figure in that particular pattern.
- Consider the expression \(a + 3\). Give a real world situation that could be represented by this expression. Share your answer with your partner. Together, can you come up with even more ideas?
- Suppose the expression \(3c + 2\) represents the number of tiles used at any stage of a growing pattern.
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- Evaluate the expression at \(c = 1, 2,\; \text{and}\; 3\). What do the values tell you about the pattern?
- Can you describe in words how the pattern is growing?
- Can you design a pattern with tiles that grows according to this rule?
- Where do you see the “3” in your pattern? Where do you see the “2”’? Where do you see the “ ”?
Krystal was looking at this pattern, which may be familiar to you from the Problem Bank:
She wrote down the equation
\[y = 4x + 2 \ldotp \nonumber \]
In Krystal’s equation, what does represent? What does represent? How do you know?
Candice was thinking about this problem:
Today is Jennifer’s birthday, and she’s twice as old as her brother. When will she be twice as old as him again?
She wrote down the equation \(2n = m\). In Candice’s equation, what does represent? What does represent? How do you know?
Sarah and David collect old coins. Suppose the variable stands for the number of coins Sarah has in her collection, and stands for the number of coins David has in his collection. What would each of these equations say about their coin collections?
\[(a)\; k= \ell + 1 \qquad (b)\; k = \ell \qquad (c)\; 3k = 2 \ell \qquad (d)\; k = \ell - 11 \nonumber \]
The pictures below show balance scales containing bags and blocks. The bags are marked with a “?”’ because they contain some unknown number of blocks. In each picture:
- Each bag contains the same number of blocks.
- The scale is balanced.
For each picture, determine how many blocks are in each bag. Justify your answers.
(a)
(b)
(c)
When he was working on Problem 24, Kyle wrote down these three equations.
(i) \[2m = 6 \ldotp \nonumber \]
(ii) \[2x = x + 3 \ldotp \nonumber \]
(iii) \[z + 5 = 2z + 3 \ldotp \nonumber \]
Match each equation to a picture, and justify your choices. Then solve the equations, and say (in a sentence) what the solution represents.
Draw a balance puzzle that represents the equation
\[2h + 3 = h + 8 \ldotp \nonumber \]
Now solve the balance puzzle. Where is the “ ” in your puzzle? What does it represent?
Draw a balance puzzle that represents the equation
\[3b + 7 = 3b + 2 \ldotp \nonumber \]
Now solve the equation. Explain what happens.
Which equation below is most like the one in Problem 27 above? Justify your choice.
- $5+3=8 \ldotp$
- $\frac{2}{3} + \frac{1}{2} = \frac{3}{5} \ldotp$
- $5+3 = y \ldotp$
- $\frac{a}{5} = \frac{5}{a} \ldotp$
- $n + 3 = m \ldotp$
- $3x = 2x+ x \ldotp$
- $5k = 5k + 1 \ldotp$
Draw a balance puzzle that represents the equation
\[4 \ell + 7 = 4 \ell + 7 \ldotp \nonumber \]
Now solve the equation. Explain what happens.
Which equation below is most like the one in Problem 29 above? Justify your choice.
- $5+3 = 8 \ldotp$
- $\frac{2}{3} + \frac{1}{2} = \frac{3}{5} \ldotp$
- $5+3 = y \ldotp$
- $\frac{a}{5} = \frac{5}{a} \ldotp$
- $n + 3 = m \ldotp$
- $3x = 2x + x \ldotp$
- $5k = 5k + 1 \ldotp$
Create a balance puzzle where the solution is not a whole number of blocks. Can you solve it? Explain your answer.
There are three piles of rocks: pile A, pile B, and pile C. Pile B has two more rocks than pile A. Pile C has four times as many rocks as pile A. The total number of rocks in all three piles is 14.
- Use x to represent the number of rocks in pile A, and write equations that describe the rules above. Then find the number of rocks in each pile.
- Use x to represent the number of rocks in pile B, and write equations that describe the rules above. Then find the number of rocks in each pile.
- Use x to represent the number of rocks in pile C, and write equations that describe the rules above. Then find the number of rocks in each pile.
Look back at Problems 21–32. Which of them felt like structural algebraic thinking ? Which felt like procedural algebraic thinking ? Did any of the problems feel like they involved both kinds of thinking?
Variables and Equations
You have seen that in algebra, letters and symbols can have different meanings depending on the context.
- A symbol could stand for some unknown quantity .
- A symbol could stand for some quantity that varies . (Hence the term “variable” to describe these symbols.)
In much the same way, equations can represent different things.
- They can represent a problem to be solved. This is the traditional procedural algebra type of question.
- They can represent a relationship between two or more quantities. For example, \(A = s^{2}\) represents the relationship between the area of a square and its side length.
- They can represent identities : mathematical truths. For example, $x^{2} - 1 = (x + 1 )(x - 1)$is always true, for every value of . There is nothing to solve for, and no relationship between varying quantities. (If you do try to “solve for ,” you will get the equation \(0 = 0\), much like you saw in Problem 29. Not very satisfying!)
Give an example of each type of equation. Be sure to say what the symbols in the equations represent.
Answer the following questions about the equation
\[x^{2} - 1 = (x + 1)(x - 1) \ldotp \nonumber \]
- Evaluate both sides of the above equation for \(x = 1, 2, 3, 4,\; \text{and}\; 5\). What happens?
- Use the distributive property of multiplication over addition to expand the right side of the equation and simplify it.
- Use the equation to compute \(99^{2}\) quickly, without using a calculator. Explain how you did it.


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20.7: The Black Vampyre is Gothic? Big Surprise, Right?
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Before we delve into why
The Black Vampire
is considered Gothic, let’s first define the term Gothic. We all have a general idea on what the term Gothic means in the sense of literature, but there’s more to the term/style then just supernatural stuff. According to Study.com, the definition of Gothic literature is as follows. “The term
Gothic fiction
refers to a style of writing that is characterized by elements of fear, horror, death, and gloom, as well as romantic elements, such as nature, individuality, and very high emotion. These emotions can include fear and suspense.” (Study.com) So when we think of Gothic literature, we normally think of the “horror” aspect of the definition, but we should also be paying attention to the second part of the definition that includes the romantic elements. If you look back at every Gothic texts you may have read, these types of elements are present and
The Black Vampire
is no exception. So, instead of focusing on the obvious horror elements found in the text, I will be observing the romantic elements and will be showing examples of when these elements are most prominent. “The element seemed very familiar to him; and he swam back with much grace and agility; parting the sparkling waves with his jet black members, polished like ebony, but reflecting no single beam of light” (16). If we observe the referenced text above, we see some references to nature, the added detail to the waves, but more importantly, the part that is comparing skin to polished ebony. This use of nature to further develop detail in a certain aspect of the story pervades the rest of the narrative. For example: “…and her interesting infant, the first pledge of her pure and perfect love, had been precociously sucked, like an unripe orange, and nothing left but its beautiful and tender skin” (19). The final aspect of Gothic that I will cover is the portion of romantic elements that cover extreme emotions. The emotion that I will be covering is fear, not the fear within the narrative, but rather the real-world fear found in the “moral.” “Yet in the figurative sense, and in the moral world, our climate is perhaps more prolific than any other, in enchanters,-Vampyres,-and the whole infernal brood of sorcery and witchcraft.” (41). The moral then goes onto give examples of the real monsters in society: the fraudulent trafficker, the corrupted and senseless Clerk, Brokers, Country Bank Directors, etc; all Vampyres, and there is one thing that they are after to please their thirst: your money (BUM BUM BUM!).


7.4: COMMERCE
Buildings for commerce have appeared over time. Early systems of trade and barter in some places eventually became formalized in ways that required marketplaces and commercial establishments with temporary or permanent housing. While open-air markets with vendor stalls continue to be used in many places, in others shops or full buildings evolved for commercial and service transactions.
An early example appeared in ancient Athens, Greece, in the area where the open market or agora , was also located.The Stoa of Attalos, built by King Attalos II of Pergamon (r. 159-133 BCE), was comprised of a two story covered walkway made of marble and limestone with columns on one side and a closed wall on the other. (Figure 7.33) Along the closed wall, there were twenty-one rooms on each level with each room providing space for a shop. These rooms were similar in character and purpose to those we noted on the ground floors of Roman villas and apartment buildings, but they provided for a more concentrated shopping area.
Our modern provisions for shopping centers and department stores were designed with different ideas about merchandising, sales, and consumerism but, as we have seen with the rapid rise of on-line shopping for durable and perishable goods, this scenario will likely be ever evolving. Indeed, grocery and department stores may become completely passé. But their development in the nineteenth and twentieth centuries presented new possibilities for architectural design.
An example is the Carson, Pirie, Scott and Company Store in Chicago, designed by Louis Sullivan (1856-1924, USA) and built in 1904. (Figure 7.34) One of the early applications for steel frame, or “skeleton frame,” construction that made the development of skyscrapers possible, this sort of building also opened new possibilities for retail and office space. Here, the large ground-floor windows and corner entrance could provide a great deal of display space for attracting pedestrians while the expansive multi- story interior offered shoppers a wide array of goods, especially compared to the sorts of small shops and markets that had been its predecessors.
Not only the structure but also the decorative approach was innovative, as Sullivan combined Beaux Arts ideas with Art Nouveau motifs in the building’s surface design. (Figure 7.35) The elaborate, curvilinear, plant-based motifs central to the Art Nouveau movement, c. 1890-1910, in cast metal relief panels above the doors and ground floor windows added to visual appeal for potential customers.
New designs emerged for other commercial firms in this era as well. The Austrian Postal Savings Bank in Vienna, Austria, designed by architect Otto Wagner (1841-1918, Austria) has a huge multi story façade covering a broad open interior space on the ground level; its sleek and modern aesthetic was startlingly new and different when it was completed in 1905. (Figure 7.36) One of Wagner’s aims in the design was to create a sense of strength and solidity that engendered trust and a feeling of financial security in customers. The main banking customer area is filled with natural light. Wagner used marble, steel, and polished glass for the simplified decoration of the reinforced concrete building, turning away from the Art Nouveau aesthetic and replacing it with his sense of modernism.
The use of steel and reinforced concrete that facilitated the advent of the skyscraper truly revolutionized architecture and began a contest for height that continues today. Wealthy entrepreneurs and ambitious developers from around the world have joined in the competition for buildings of modern distinction. One example is the Chrysler Building in New York City, designed by William van Alen (1883-1954, USA). (Figure 7.37) Its décor in the Art Deco style (c. 1920-1940), including the ribbed, sun- burst pattern made of stainless steel in the building’s terraced crown, celebrates American industrialism and the automobile. At 1,046 feet, the Chrysler Building was for eleven months after its completion in 1930 the tallest in the world. (It was surpassed in 1931 by the Empire State Building at 1,454 feet.)
A more recent example is the Petronas Twin Towers in Kuala Lampur, Malaysia, designed by César Pelli (b. 1926, Argentina, lives USA). (Figure 7.38) Inaugurated in 1999, they were the tallest buildings for several years and remain the tallest twin towers to this day. The buildings’ design motifs are inspired by Islamic art and culture; for example, the shape of each tower is the Muslim symbol of Rub el Hizb, or two overlapping squares that form an eight-pointed star. Both structures house commercial and business concerns and symbolize the architecture of modern business.
In the late twentieth century, architectural ingenuity, new materials, and the potential of computer design led some architects to develop radically innovative approach- es to structures that might house any number of different types of needs. Among the most innovative in this regard is Frank Gehry (b. 1929, Canada, lives USA), who has designed buildings all over the world including museums, business towers, residences, and theaters.
In Los Angeles, he created the Walt Disney Concert Hall, completed in 2003. (Figure 7.39) Using titanium sheathing for multiform, swooping curvilinear forms and volumes, his buildings are sculptural in effect from a visual standpoint. Yet in each case, his buildings have proven effective and dynamic in creating spaces for the activities they house. The acoustics of the concert hall are widely praised as is the beauty of the architectural form in capturing the whimsical spirit of Walt Disney, the creator of so many American comics, cartoons, and movies.


4.14: Putting It Together- Infancy
Contributors and Attributions
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We have explored the dramatic story of the first two years of life. Rapid physical growth, neurological development, language acquisition, the movement from hands-on to mental learning, an expanding emotional repertoire, and the initial conceptions of self and others make this period of life very exciting. These abilities are shaped into more sophisticated mental processes, self-concepts, and social relationships during the years of early childhood.
Babies begin to learn about the world around them from a very early age. Children’s early experiences, meaning the bonds they form with their parents and their first learning experiences, affect their future physical, cognitive, emotional and social development. Various organizations and agencies are dedicated to helping parents (and other caregivers), educators, and health care providers understand the importance of early healthy development. Healthy development means that children of all abilities, including those with special health care needs, are able to grow up where their social, emotional, and educational needs are met. Having a safe and loving home and spending time with family―playing, singing, reading, and talking―are very important. Proper nutrition, exercise, and sleep can also make a big difference; and effective parenting practices are key to supporting healthy development (CDC, 2019). The need to invest in very young children is important to maximize their future well-being.


2.4: A Preview of Some Modern Physics
“Mommy, why do you and Daddy have to go to work?” “To make money, sweetie-pie.” “Why do we need money?” “To buy food.” “Why does food cost money?” When small children ask a chain of “why” questions like this, it usually isn't too long before their parents end up saying something like, “Because that's just the way it is,” or, more honestly, “I don't know the answer.”
The same happens in physics. We may gradually learn to explain things more and more deeply, but there's always the possibility that a certain observed fact, such as conservation of mass, will never be understood on any deeper level. Science, after all, uses limited methods to achieve limited goals, so the ultimate reason for all existence will always be the province of religion. There is, however, an appealing explanation for conservation of mass, which is atomism, the theory that matter is made of tiny, unchanging particles. The atomic hypothesis dates back to ancient Greece, but the first solid evidence to support it didn't come until around the eighteenth century, and individual atoms were never detected until about 1900. The atomic theory implies not only conservation of mass, but a couple of other things as well.
First, it implies that the total mass of one particular element is conserved. For instance, lead and gold are both elements, and if we assume that lead atoms can't be turned into gold atoms, then the total mass of lead and the total mass of gold are separately conserved. It's as though there was not just a law against pickpocketing, but also a law against surreptitiously moving money from one of the victim's pockets to the other. It turns out, however, that although chemical reactions never change one type of atom into another, transmutation can happen in nuclear reactions, such as the ones that created most of the elements in your body out of the primordial hydrogen and helium that condensed out of the aftermath of the Big Bang.
Second, atomism implies that mass is quantized , meaning that only certain values of mass are possible and the ones in between can't exist. We can have three atoms of gold or four atoms of gold, but not three an a half. Although quantization of mass is a natural consequence of any theory in which matter is made up of tiny particles, it was discovered in the twentieth century that other quantities, such as energy, are quantized as well, which had previously not been suspected.
self-check:
Is money quantized?
(answer in the back of the PDF version of the book)
If atomism is starting to make conservation of mass seem inevitable to you, then it may disturb you to know that Einstein discovered it isn't really conserved. If you put a 50-gram iron nail in some water, seal the whole thing up, and let it sit on a fantastically precise balance while the nail rusts, you'll find that the system loses about \(6\times10^{-12}\) kg of mass by the time the nail has turned completely to rust. This has to do with Einstein's famous equation \(E=mc^2\). Rusting releases heat energy, which then escapes out into the room. Einstein's equation states that this amount of heat, \(E\), is equivalent to a certain amount of mass, \(m\). The \(c\) in the \(c^2\) is the speed of light, which is a large number, and a large amount of energy is therefore equivalent to a very small amount of mass, so you don't notice nonconservation of mass under ordinary conditions. What is really conserved is not the mass, \(m\), but the mass-plus-energy, \(E+mc^2\). The point of this discussion is not to get you to do numerical exercises with \(E=mc^2\) (at this point you don't even know what units are used to measure energy), but simply to point out to you the empirical nature of the laws of physics. If a previously accepted theory is contradicted by an experiment, then the theory needs to be changed. This is also a good example of something called the correspondence principle , which is a historical observation about how scientific theories change: when a new scientific theory replaces an old one, the old theory is always contained within the new one as an approximation that works within a certain restricted range of situations. Conservation of mass is an extremely good approximation for all chemical reactions, since chemical reactions never release or consume enough energy to change the total mass by a large percentage. Conservation of mass would not have been accepted for 110 years as a fundamental principle of physics if it hadn't been verified over and over again by a huge number of accurate experiments.
Contributors and Attributions
Benjamin Crowell (Fullerton College). Conceptual Physics is copyrighted with a CC-BY-SA license.


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Summary
1.1
Computing from Inception to Today
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Early computers were used predominantly by engineers and scientists to handle large amounts of data. Key companies in the evolution of computing include IBM, Hewlett Packard, Xerox, Apple, and Microsoft.
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With the invention of the microprocessor, computers became available to the average consumer.
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Computing technology has increased efficiencies, decreased errors, opened new opportunities, and enhanced business–customer relationships.
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Key technologies in computing include mobile devices, digital imaging, and machine learning.
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Advances in technology are applicable to nearly every industry. The rapid pace of technological change has distinct career implications.
1.2
Computer Hardware and Networks
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Hardware components that make up a computer are the motherboard, CPU, microprocessor, and memory, as well as the keyboard, mouse, and other peripheral devices.
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Computers process and store data through the information processing cycle.
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Networks are connections of two or more computer systems, such as LANs and WANs. Routers, switches, and firewalls are basic components of networks.
1.3
The Internet, Cloud Computing, and the Internet of Things
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The internet was initially rooted in government and military applications.
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Advances in technology such as HTML, URL, and DNS made the internet more accessible to the average user.
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Programs such as Zoom, Google Docs, and X (Twitter) foster new types of interactions between businesses and their customers.
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Through sites such as eBay and Amazon, e-commerce has become an efficient way to purchase products and services, not just from companies but also from individuals.
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Cloud computing refers to storing IT resources on a virtual server rather than on the actual computers where the resources are used.
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The use of cloud computing in the workplace has increased productivity and saved money for many businesses.
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Cloud computing enables individuals to access a wide variety of resources such as media files, documents, and photos without the need for computers with massive storage capacities.
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The Internet of Things (IoT) connects the physical to the virtual; with simple changes to product designs, many products can be connected to the internet.
1.4
Safety, Security, Privacy, and the Ethical Use of Technology
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Computer safety and security are paramount considerations for a company’s effective operations.
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Privacy in the digital world has become a major focus of corporate efforts to maintain the trust of employees, the public, and stakeholders.
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Common security issues include data and identity theft, cybercrime, phishing, and hacking.
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Measures that prevent computer security breaches include encryption, firewalls, password management, data erasure, and data masking.


30.3B: Aging and the Skeleton
In osteoporosis, bone mineral density (BMD) is reduced and the integrity of bone proteins is altered, increasing the risk of fracture.
- Construct a diagnosis and treatment plan for osteoporosis
Key Points
- The risk of osteoporosis can be reduced with lifestyle changes and sometimes medication, both aimed at restoring the amounts and types of necessary bone-associated proteins.
- Osteoporosis itself has no symptoms; its main consequence is the increased risk of bone fractures.
- Treatments for osteoporosis include changes in diet and exercise, increased consumption of calcium and vitamin D, and prescriptions for bisphosphonates.
Key Terms
- bisphosphonates : A class of drugs that prevents the loss of bone mass, used to treat osteoporosis and similar diseases. They are the most commonly prescribed drugs used to treat osteoporosis.
- bone mineral density : Abbreviated BMD, the amount of mineral matter per square centimeter of bone. BMD is used in medicine as an indirect indicator of osteoporosis and fracture risk.
- osteoporosis : A disease in which the bones become extremely porous and are subject to fracture most common in post-menopausal women.
Osteoporosis is a bone disease that leads to an increased risk of fracture. In osteoporosis, the bone mineral density (BMD) is reduced, bone microarchitecture deteriorates, and the amount and variety of proteins in bone is altered.
Types of Osteoporosis
(a) Osteoporotic Fractures and (b) Woman with Osteoporosis: Image shows compression fractures of the lumbar spine. Hyperkyphosis, colloquially called a “dowager’s hump,” refers to an anteriorly concave curvature of the thoracic region of the spine.
The disease may be classified as primary type one, primary type two, or secondary. The form of osteoporosis most common in women after menopause is referred to as primary type one or postmenopausal osteoporosis.
Primary type two osteoporosis or senile osteoporosis occurs after age 75 and is seen in both females and males at a ratio of two to one. Finally, secondary osteoporosis may arise at any age and affect men and women equally. This form of osteoporosis results from chronic illness or prolonged use of medications such as glucocorticoids. Osteoporosis affects 55% of Americans aged 50 and above. Of these, approximately 80% are women.
Symptoms
Osteoporosis itself has no symptoms; its main consequence is the increased risk of fractures caused by weakened bones. Osteoporotic fractures, called fragility fractures, occur in situations where healthy people would not normally break a bone. These often affect the vertebral column, rib, hip, and wrist. The underlying mechanism in all cases of osteoporosis is an imbalance between bone resorption and bone formation.
Physical deformation can occur secondary to compromised bone densities. One example is hyperkyphosis, colloquially called a “dowager’s hump,” which refers to an anteriorly concave curvature of the thoracic region of the spine.
Treatment
Osteoporosis risks can be reduced with lifestyle changes and sometimes medication. Lifestyle changes include diet, exercise, and fall prevention. Exercises to tone deambulatory muscles and improve proprioception are recommended to decrease risk for falls. In addition, the anabolic effect of exercise can sometimes prevent progression of osteoporosis. Medication includes calcium, vitamin D, and bisphosphonates.
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- Curation and Revision. Authored by : Boundless.com. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
CC LICENSED CONTENT, SPECIFIC ATTRIBUTION
- Embryogenesis. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Skeleta.../Embryogenesis . License : CC BY-SA: Attribution-ShareAlike
- II. Osteology. Provided by : Saylor. Located at : http://www.saylor.org/site/wp-conten...h2with6c6d.pdf . License : Public Domain: No Known Copyright
- Ossification. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Ossification . License : CC BY-SA: Attribution-ShareAlike
- endochondral ossification. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/endocho...20ossification . License : CC BY-SA: Attribution-ShareAlike
- trabecula. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/trabecula . License : CC BY-SA: Attribution-ShareAlike
- mesenchyme. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/mesenchyme . License : CC BY-SA: Attribution-ShareAlike
- intramembranous ossification. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/intrame...20ossification . License : CC BY-SA: Attribution-ShareAlike
- Bone Growth. Provided by : Wikimedia. Located at : upload.wikimedia.org/wikipedi...one_growth.png . License : Public Domain: No Known Copyright
- Osteoporosis. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Osteoporosis . License : CC BY-SA: Attribution-ShareAlike
- bone mineral density. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/bone%20mineral%20density . License : CC BY-SA: Attribution-ShareAlike
- bisphosphonates. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/bisphosphonates . License : CC BY-SA: Attribution-ShareAlike
- osteoporosis. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/osteoporosis . License : CC BY-SA: Attribution-ShareAlike
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16.0: Prelude to Oscillatory Motion and Waves
What do an ocean buoy, a child in a swing, the cone inside a speaker, a guitar, atoms in a crystal, the motion of chest cavities, and the beating of hearts all have in common? They all oscillate —-that is, they move back and forth between two points. Many systems oscillate, and they have certain characteristics in common. All oscillations involve force and energy. You push a child in a swing to get the motion started. The energy of atoms vibrating in a crystal can be increased with heat. You put energy into a guitar string when you pluck it.
Some oscillations create waves . A guitar creates sound waves. You can make water waves in a swimming pool by slapping the water with your hand. You can no doubt think of other types of waves. Some, such as water waves, are visible. Some, such as sound waves, are not. But every wave is a disturbance that moves from its source and carries energy . Other examples of waves include earthquakes and visible light. Even subatomic particles, such as electrons, can behave like waves.
By studying oscillatory motion and waves, we shall find that a small number of underlying principles describe all of them and that wave phenomena are more common than you have ever imagined. We begin by studying the type of force that underlies the simplest oscillations and waves. We will then expand our exploration of oscillatory motion and waves to include concepts such as simple harmonic motion, uniform circular motion, and damped harmonic motion. Finally, we will explore what happens when two or more waves share the same space, in the phenomena known as superposition and interference.
Glossary
- oscillate
- moving back and forth regularly between two points
- wave
- a disturbance that moves from its source and carries energy