7.6.3: Drug-resistant Tuberculosis
Another very concerning example is that of drug-resistant tuberculosis (TB). The bacterium Mycobacterium tuberculosis causes disease most commonly in the lungs, although it can affect other organ systems. TB is spread through respiratory droplets, and develops slowly - only about 5-10% of those infected will actually develop the disease and become contagious - often due to a depressed immune system (often caused by HIV infection). Yet, the global disease burden is still high. It is estimated that 10 million cases of the disease happen every year worldwide, causing 1.5 million deaths (World Health Organization: WHO, 2019).
When the TB bacteria cause active disease, this can happen within 2 years or even decades after infection. Often other factors that depress the immune system are involved, such as co-infections (e.g. HIV), malnutrition, crowded housing, and lack of access to medical care. Mycobacterium tuberculosis has a thick, fatty surface that is difficult for the human immune system to break down, so white blood cells will typically envelope the bacteria instead - creating calcified “tubercles”. If the bacteria escapes these tubercles, then the bacteria can proliferate, causing active disease - mostly in the lungs but also in several other organ systems such as the central nervous system (CrashCourse, 2024).
Thanks to the development of antibiotics in the 1950s and 60s, we do have treatments and can cure this disease. Although TB has been eliminated from several countries and has the potential to be eradicated around the world, drug-resistant strains make this goal much more challenging. Certain factors increase the risk of drug-resistant TB, including:
- A full course of TB drug treatment is not completed, doses are missed or delayed
- High-quality drugs are not available
- The wrong drugs are prescribed (CDCTB, 2022)
In developing countries, other social determinants of health (SDOH) also influence these risk factors. For example, pharmaceutical companies are driven by profit incentives to develop and produce drugs, and since TB has been eliminated in many wealthy countries, there is less production of these drugs. Modern TB tests and drugs are (unnecessarily) expensive, and in order to ensure “compliance”, patients are often required to visit a clinic daily, or live in an inpatient facility for months. For someone already living in poverty, and with a severe illness, transportation and physical disability can be large barriers in getting treatment. Combine this with other SDOH such as access to nutrition, drinking water, and quality housing, and treating TB - especially drug-resistant TB - becomes much more challenging. Increases in the accessibility of testing, treatment, healthcare staff, and addressing barriers to treatment are all necessary if we want to eliminate TB (CrashCourse, 2024).


6: Quantum Mechanics
Quantum mechanics is a powerful framework for understanding the motions and interactions of particles at small scales, such as atoms and molecules. The ideas behind quantum mechanics often appear quite strange. In many ways, our everyday experience with the macroscopic physical world does not prepare us for the microscopic world of quantum mechanics. The purpose of this chapter is to introduce you to this exciting world.
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- 6.1: Prelude to Quantum Mechanics
- The quantum-computer processor is the “brain” of a quantum computer that operates at near-absolute zero temperatures. Unlike a digital computer, which encodes information in binary digits (definite states of either zero or one), a quantum computer encodes information in quantum bits or qubits (mixed states of zero and one). Quantum computers are discussed in the first section of this chapter.
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- 6.2: Wave functions
- In quantum mechanics, the state of a physical system is represented by a wave function. In Born’s interpretation, the square of the particle’s wave function represents the probability density of finding the particle around a specific location in space. Wave functions must first be normalized before using them to make predictions. The expectation value is the average value of a quantity that requires a wave function and an integration.
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- 6.3: The Heisenberg Uncertainty Principle
- The Heisenberg uncertainty principle states that it is impossible to simultaneously measure the x-components of position and of momentum of a particle with an arbitrarily high precision. The product of experimental uncertainties is always larger than or equal to \(\frac{\hbar}{2}\). The energy-time uncertainty principle expresses the experimental observation that a quantum state that exists only for a short time cannot have a definite energy.
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- 6.4: The Schrӧdinger Equation
- The Schrӧdinger equation is the fundamental equation of wave quantum mechanics. It allows us to make predictions about wave functions. When a particle moves in a time-independent potential, a solution of the time-dependent Schrӧdinger equation is a product of a time-independent wave function and a time-modulation factor. The Schrӧdinger equation can be applied to many physical situations.
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- 6.5: The Quantum Particle in a Box
- In this section, we apply Schrӧdinger’s equation to a particle bound to a one-dimensional box. This special case provides lessons for understanding quantum mechanics in more complex systems. The energy of the particle is quantized as a consequence of a standing wave condition inside the box.
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- 6.6: The Quantum Harmonic Oscillator
- The quantum harmonic oscillator is a model built in analogy with the model of a classical harmonic oscillator. It models the behavior of many physical systems, such as molecular vibrations or wave packets in quantum optics. The allowed energies of a quantum oscillator are discrete and evenly spaced. The energy spacing is equal to Planck’s energy quantum. The ground state energy is larger than zero. This means that, unlike a classical oscillator, a quantum oscillator is never at rest.
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- 6.7: Quantum Tunneling of Particles through Potential Barriers
- A quantum particle that is incident on a potential barrier of a finite width and height may cross the barrier and appear on its other side. This phenomenon is called ‘quantum tunneling.’ It does not have a classical analog. The tunneling probability is a ratio of squared amplitudes of the wave past the barrier to the incident wave.
Thumbnail: Schrödinger took the absurd implications of this thought experiment (a cat simultaneously dead and alive) as an argument against the Copenhagen interpretation. However, this interpretation remains the most commonly taught view of quantum mechanics.


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Full and truthful disclosure ________.
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Is often required under tort law
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Is a matter of ethics, not a legal requirement
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Is a risky business practice because it helps the competition
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Is a strictly personal choice
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Common IP law-mandated periods of protection are ________.
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Copyrights are protected for twenty years
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Patents are protected for fifty years
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Trademarks are protected for renewable ten-year periods
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Trade secrets are protected for fifteen years
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Acting ethically includes acting with which of the following?
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Integrity
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Disclosure
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Transparency
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All of these options are correct.
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Antitrust law:
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May help small entrepreneurs by ensuring big companies do not have monopolies
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May hurt small entrepreneurs by requiring them to disclose trade secrets
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Only applies to businesses with over 500 employees
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None of these options are correct.
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Patents ________.
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Are similar to trade secrets under IP law
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Allow a temporary monopoly for an invention
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Are protected under IP law for a period of ninety-nine years
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None of these options are correct.
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An example of a company(s) that have been fined millions of dollars for legal/ethical violations includes ________.
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Enron
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TOMS
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British Petroleum
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a and c
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The B-corporation moniker ________.
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Requires government certification
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Requires no certification
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Can only be used after an audit and certification by the B-corp organization
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Is a social media concept used primarily for marketing purposes
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What differentiates a social entrepreneurial enterprise from a traditional entrepreneur?
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Commitment to reduce pollution
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Pledge to undertake sustainable supply chain sourcing
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Effort to recycle water, waste, and other byproducts
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All of these options are correct.
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Core qualities for building a foundation of ethics-based behaviors include ________.
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Trust
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Fairness
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Excellence
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All of these options are correct.
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Millennial workers ________.
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Often do not place salary as their first priority
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Want work-life balance
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Usually expect their employer to have an ethical value system
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All of these options are correct.
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A diverse workforce ________.
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Is always optional
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Is usually required for government contracting
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Is always required under Title VII
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Is never required


1.2.2: How the World Was Made (Cherokee)
The earth is a great floating island in a sea of water. At each of the four corners there is a cord hanging down from the sky. The sky is of solid rock. When the world grows old and worn out, the cords will break, and then the earth will sink down into the ocean. Everything will be water again. All the people will be dead. The Indians are much afraid of this.
In the long time ago, when everything was all water, all the animals lived up above in Galun’lati, beyond the stone arch that made the sky. But it was very much crowded. All the animals wanted more room. The animals began to wonder what was below the water and at last Beaver’s grandchild, little Water Beetle, offered to go and find out. Water Beetle darted in every direction over the surface of the water, but it could find no place to rest. There was no land at all. Then Water Beetle dived to the bottom of the water and brought up some soft mud. This began to grow and to spread out on every side until it became the island which we call the earth. Afterwards this earth was fastened to the sky with four cords, but no one remembers who did this.
At first the earth was flat and soft and wet. The animals were anxious to get down, and they sent out different birds to see if it was yet dry, but there was no place to alight; so the birds came back to Galun’lati. Then at last it seemed to be time again, so they sent out Buzzard; they told him to go and make ready for them. This was the Great Buzzard, the father of all the buzzards we see now. He flew all over the earth, low down near the ground, and it was still soft. When he reached the Cherokee country, he was very tired; his wings began to flap and strike the ground. Wherever they struck the earth there was a valley; whenever the wings turned upwards again, there was a mountain. When the animals above saw this, they were afraid that the whole world would be mountains, so they called him back, but the Cherokee country remains full of mountains to this day. [This was the original home, in North Carolina.]
When the earth was dry and the animals came down, it was still dark. Therefore they got the sun and set it in a track to go every day across the island from east to west, just overhead. It was too hot this way. Red Crawfish had his shell scorched a bright red, so that his meat was spoiled. Therefore the Cherokees do not eat it.
Then the medicine men raised the sun a handsbreadth in the air, but it was still too hot. They raised it another time; and then another time; at last they had raised it seven handsbreadths so that it was just under the sky arch. Then it was right and they left it so. That is why the medicine men called the high place “the seventh height.” Every day the sun goes along under this arch on the under side; it returns at night on the upper side of the arch to its starting place.
There is another world under this earth. It is like this one in every way. The animals, the plants, and the people are the same, but the seasons are different. The streams that come down from the mountains are the trails by which we reach this underworld. The springs at their head are the doorways by which we enter it. But in order to enter the other world, one must fast and then go to the water, and have one of the underground people for a guide. We know that the seasons in the underground world are different, because the water in the spring is always warmer in winter than the air in this world; and in summer the water is cooler.
We do not know who made the first plants and animals. But when they were first made, they were told to watch and keep awake for seven nights. This is the way young men do now when they fast and pray to their medicine. They tried to do this. The first night, nearly all the animals stayed awake. The next night several of them dropped asleep. The third night still more went to sleep. At last, on the seventh night, only the owl, the panther, and one or two more were still awake. Therefore, to these were given the power to see in the dark, to go about as if it were day, and to kill and eat the birds and animals which must sleep during the night.
Even some of the trees went to sleep. Only the cedar, the pine, the spruce, the holly, and the laurel were awake all seven nights. Therefore they are always green. They are also sacred trees. But to the other trees it was said, “Because you did not stay awake, therefore you shall lose your hair every winter.”
After the plants and the animals, men began to come to the earth. At first there was only one man and one woman. He hit her with a fish. In seven days a little child came down to the earth. So people came to the earth. They came so rapidly that for a time it seemed as though the earth could not hold them all.


4.5: Environmental Economics
It should be clear from reading this chapter that the well-being of people around the world is fundamentally linked to opportunities for biodiversity to survive and prosper. That means that when we destroy an ecosystem or let a species go extinct, we also put at risk our own ability to survive and prosper. To fully account for these risks, decisions that negatively affect biodiversity must account for all the costs and benefits (hidden or otherwise), including the impacts on ecosystem services, before the decision is implemented.
One of the most popular methods for accounting for potential harm to biodiversity, especially when weighing public policy and commercial decisions, is to attach market (or monetary) values to the ecosystem services. For some ecosystem services, it is rather straightforward to estimate a market value. For example, how much would it cost to replace a natural pollination service with hand pollination by farm workers? But for many services, estimating a monetary value is much more difficult. For example, how do we calculate the value of the Congo Basin’s carbon stocks? Where do we even start to estimate the value of breathing clean air, or knowing that dolphins exists?
Environmental economics strengthen arguments for biodiversity protection by examining the contribution of ecosystem services to global economies.
To examine these kinds of complex questions, conservation biologists look to a sub-discipline within economics called environmental economics. Environmental economics broadly examines the contribution of ecosystem services to global economies. An important component of this examination involves estimating the market value of all the different ecosystem services we benefit from, but it also includes studying the environmental costs of economic transactions, environmental policies, and other decisions that impact the environment.
4.5.1 Placing a price on the natural world
Approximating the market values of ecosystem services is no small feat, in part because nature’s contributions to people vary by location and perspective (Díaz et al., 2018). There also continue to be technical (e.g. Kling et al., 2012) and ethical (e.g. McCauley, 2006; Silvertown, 2015) disagreements about the need and methods used to translate nature’s services into monetary terms. Nevertheless, including such estimates has become a widely accepted norm in economics models and conservation activities (Guerry et al., 2015). To accomplish this task, economists rely on seven main methods to estimate the market values of ecosystem services (Farber et al., 2002):
- Market value: The price a person is willing to pay for a specific product or service. For example, how much is a person willing to pay for a bundle of firewood at a local market?
- Avoidance cost: The cost society avoids paying because a specific ecosystem service exists. For example, how much does society avoid paying for water filtration service otherwise provided by a region’s forests and wetlands?
- Replacement cost: The cost society would have incurred if a specific ecosystem service had to be replaced. For example, how much would society have to pay in extra healthcare costs and in clean-up costs for diseased carcasses to replace the sanitation services provided by vultures?
- Factor income: The additional income generated by the enhancement of an ecosystem service. For example, how much would a reduction in water pollution increase the income of fishermen through healthier fish populations?
- Travel cost: The additional travel cost a person is willing to pay to experience an ecosystem service otherwise not available to them. For example, how much extra is a person willing to pay for transport to participate in recreational activities at a cleaner lake?
- Hedonic pricing: The additional expense a person is willing to pay to experience an ecosystem service. For example, how much extra is someone willing to pay for a house with an ocean view, compared to an inland house?
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Contingent valuation: The additional expenses a person is willing to pay for an alternative hypothetical scenario. For example, how much is someone willing to pay for cleaner air, or the right to catch more fish?
The combined value of all of Earth’s ecosystem services may be double the current value of the global economy.
Using a combination of these methods, a range of ecosystem services have been valued in recent years. For example, the services offered by pollinating insects around the world have been valued at US $153 billion per year (Gallai et al., 2008). In just South Africa’s Western Cape Province, free pollination services provided by wild insects to the local fruit industry, valued at US $500 million, has been estimated at nearly US $360 million per year (Allsopp et al., 2008). The replacement cost of tropical forests is also increasingly appreciated in carbon sequestration markets, where heavy greenhouse gas emitters pay huge sums of money to conserve forests to become more carbon neutral (Section 10.4). For example, the United Nations Environmental Programme (UNEP) has estimated that their forests are worth 4.2 times more intact than the value that could be earned through logging; the value of just Kenya’s remaining Mau forest, if left intact, is estimated at US $1.3 billion per year (UNEP, 2012). One ambitious study estimated the value of all of Earth’s ecosystem services at US $145 trillion annually (Costanza et al., 2014), which is almost double the current US $78 trillion value of the global economy. By comparing the value of ecosystem services over time, Costanza et al. (2014) also estimated that we are losing US $4.3–20.2 trillion per year in ecosystem services through land degradation.
4.5.2 Environmental economics’ biggest contributions
Since its development, environmental economics has contributed to conservation biology in several very important ways. Perhaps the most important contribution is that it has enabled conservation biologists to better communicate the value of ecosystem goods and services to audiences like government officials and business leaders, who often base decisions on economic considerations. By doing this, environmental economics has also focussed our attention on the wide range of goods and services that biodiversity provides and has elevated these topics into corridors where they were not previously discussed. These efforts have already paid dividends; in 2012, several Africa countries signed the Gaborone Declaration, a pledge to integrate the value of ecosystem services into their economies.
Environmental economics enable us to better communicate the value of biodiversity to those who base decisions on economic considerations.
Environmental economics also enabled conservation biologists to better account for environmental impacts of environmental damaging activities. In doing so, the field highlighted how activities that appear profitable are running at an economic loss when properly accounting for otherwise ignored environmental (and social) damages. While such calculations have traditionally focussed on imbalances in overharvesting of material contributions (see negative externalities, Section 4.5.3), recent developments have also started accounting for damages inflicted on regulating services and nonmaterial contributions, such as the loss of nature’s contribution to climate regulation (Auffhammer et al., 2017; Hsiang et al., 2017).
4.5.3 Environmental economics’ biggest challenges
Despite all the direct and indirect contributions of environmental economics to biodiversity conservation, there are also several challenges facing the field. Some of these challenges relate to methodological complexities of valuing ecosystem services, but many challenges also have their roots in governance failures. Following is a discussion of the most important challenges facing environmental economics.
Accounting for negative externalities
Modern economics is built on the principle of voluntary transactions—that is, a transaction occurs only when it benefits all the stakeholders involved. However, environmental (and social) harm often arises when some hidden costs are passed on to people not directly involved in the transactions. The unregulated use of open-access resources—resources such as water, air, and fish populations that are freely used by many different groups of people—provide many opportunities for this kind of abuse. Consider a company that dumps chemical waste into a river instead of properly disposing of it. While the company may benefit from this cost-cutting measure, people further downstream bear the environmental and social costs of the company’s “free” waste disposal by having to contend with polluted drinking water, loss of swimming and other recreational opportunities, and loss of fish as a safe food source. Damage inflicted on rivers and other open-access resources also represent a classic example of the tragedy of the commons—while some people initially benefit from abusing the “free” ecosystem services, those values are gradually lost to all of society, including those who abused it (NRC, 2002).
Because negative externalities allow a small number of people to benefit at the expense of the rest of society, they often lead to market failures.
The hidden costs of economic transactions that are passed on to people not directly involved are generally known as negative externalities (Figure 4.10). Because negative externalities allow a small number of people to benefit at the expense of the rest of society, they often lead to market failures, characterized by transactions that do not lead to optimal outcomes for all stakeholders. Governments may correct for these kinds of market failures by imposing taxes on activities that are harmful to the environment. Carbon taxes imposed on greenhouse gas emitters (see climate change, Chapter 6) is a common example. But many times, governance structures fail, or even exacerbate, the impact of negative externalities, by artificially maintaining destructive activities with tax incentives, direct payments, and price regulations. For example, subsidies give foreign fishing fleets operating off Africa a competitive advantage over local fisherman and artificially inflate their profitability despite declining fish populations (Brashares et al., 2004; Sumaila and Pauly, 2006; Mallory, 2013). The financial incentives governments provide to maintain destructive activities are more often referred to as perverse subsidies (Myers and Kent, 2001). The size of perverse subsidies is often very large, regularly dwarfing conservation spending. For example, US $26 billion in subsidies were provided to the Africa’s fossil fuel industry just in 2015 (Whitley and van der Burg, 2015), compared to just US $381 million spent annually to secure Africa’s protected areas with lions (Lindsey et al., 2018).
There are many reasons why governance structures continue to fail nature and allow market failures to occur. For example, due to the prevailing mindset of pursuing economic growth at all costs, politicians, developers, and industries often skew their cost-benefit analyses by prioritising the short-term benefits gained from destructive sectors over long-term societal well-being and sustainability. Another factor is intense lobbying by industries benefitting from perverse subsidies, which leads to corruption and other questionable decisions. Solving these challenges will rely on a society that prioritises economic development (Section 15.1) and establishes structures (i.e. passing and enforcing environmental laws, Chapter 12) that fully account for negative externalities.
Determining ownership
Another problem that plagues environmental economists and other stakeholders is deciding who owns the commercial rights to biodiversity. Imagine a biochemist from a wealthy country traveling in a rural part of West Africa. The biochemist falls sick, but luckily local villagers help the chemist get better with the aid of a traditional healing plant. Once back home, the biochemist scientifically demonstrates that this plant can be used to synthesise a new effective medicine. Do the profits from this new medicine belong to the biochemist, the organization that sponsored his/her trip, or the local people in the area who helped the biochemist?
Scientists, economists, politicians, and others are currently debating who owns the commercial rights to the world’s biodiversity.
In the past, corporations and scientists (generally from wealthier countries) travelled extensively (often to poorer countries in the tropics) to collect species from which commercially valuable products might be obtained. These new products were then sold, but all profits were kept by the corporations while the people in the poorer source countries received little to no financial benefit. One such example is the production of palm oil, of which Malaysia and Indonesia currently contribute 85% of the global vegetable oil supply. This industry is entirely dependent on the oil palm (Elaeis guineensis, LC), and its specialist pollinator, the oil palm weevil (Elaeidobius kamerunicus), both imported from West Africa. Yet, West Africa have seen little benefit from the profits palm oil generated in Southeast Asia (Mbugua, 2017). (Note this exploitation goes multiple ways; for example, South America has also seen little benefit from profits generated from cacao production in West Africa.)
To combat this unfair exploitation, called biopiracy, many developing countries now require scientists and corporations to obtain permits before they can collect biological material for commercial or research purposes. Also, at the international level, nearly 100 countries have agreed to the fair sharing of benefits arising from the use of biological resources, through the Nagoya Protocol (see Section 12.2.1 for further discussion on international laws). Through these and similar laws and agreements, the hope is that a greater portion of the profits gained from biodiversity will be allocated to people who protect biodiversity and who live in the areas from where it is extracted.
A more inclusive approach
The valuation of ecosystem services has traditionally relied on generalised principles of economics and natural sciences. While this focus enabled scientists to develop broadly applicable themes and metrics in ecosystem evaluation, it also neglected the role of context and culture in understanding nature’s role in people’s lives. Many people have also remained uneasy about commodifying nature (i.e. giving it a market value), because some of the most important contributions of biodiversity are not easily converted into monetary metrics. Consequently, many feared that the transactional approach to ecosystem services would lead to social inequity concerns and alienate people offended by the idea that nature’s metaphysical properties must compete against commercial interests.
To address these concerns, the valuation and classification of ecosystem services are currently undergoing several major transformations. Prominently, the UN’s most recent classification scheme (Díaz et al., 2018) has given a more prominent voice to a wider range of stakeholders, including the social sciences, and recognizes the importance of culture and context in nature’s contributions to people. This exciting area of research is actively developing, and readers are encouraged to track developments and reactions associated with 2019 IPBES Global Assessment at https://www.ipbes.net/news/ipbes-global-assessment-preview .


15.21E: Rickettsial Diseases
- List the characteristics of Rickettsia species
Rickettsia is a genus of bacteria that can be transmitted by arthropod vectors to humans, causing disease. Rickettsia species are non-motile, Gram-negative, non-sporeforming, highly pleomorphic bacteria that can present as cocci (0.1 μm in diameter), rods (1–4 μm long), or thread-like (10 μm long). They are obligate intracellular parasites, and must replicate within the cytoplasm of eukaryotic host cells. Rickettsia are one of closest living relatives to bacteria that were the origin of the mitochondria organelle that exists inside most eukaryotic cells. Unlike viruses, Rickettsia possess true cell walls and are similar to other gram-negative bacteria. Despite a similar name, Rickettsia bacteria do not cause rickets, which is a result of vitamin D deficiency.
Rickettsia species are carried by many ticks, fleas, and lice, and cause diseases in humans such as typhus, rickettsialpox, Boutonneuse fever, African tick bite fever, Rocky Mountain spotted fever, Flinders Island spotted fever, and Queensland tick typhus (Australian Tick Typhus). They have also been associated with a range of plant diseases.
Rickettsia can be classified into three groups based on serology and DNA sequencing: spotted fever, typhus, and scrub typhus. All three of these groups contain human pathogens. Recent studies reclassify the scrub typhus group as a new genus – Orienta, and suggest that the spotted fever group should be divided into two clades. Rickettsia are widespread, and can be associated with arthropods, leeches, and protists. Rickettsia found in Arthropods are generally associated with reproductive manipulation (such as parthenogenesis) to persist in host lineage.
Unlike free-living bacteria, Rickettsia species contain no genes for anaerobic glycolysis or those involved in the biosynthesis and regulation of amino acids and nucleosides. In this regard, certain segments of Rickettsia genomes resemble that of mitochondria, and ATP production is the same as that in mitochondria. (With the exception of R. prowazekii , whose genome contains a complete set of genes encoding for the tricarboxylic acid cycle and the respiratory chain complex). The genomes of both Rickettsia and mitochondria are frequently said to be “small, highly derived products of several types of reductive evolution. ”
Key Points
- Rickettsia are obligate intracellular parasites, and must replicate within the cytoplasm of eukaryotic host cells.
- Rickettsia species are carried by many ticks, fleas, and lice, and cause diseases in humans such as typhus, rickettsialpox, Boutonneuse fever, African tick bite fever, Rocky Mountain spotted fever, Flinders Island spotted fever, and Queensland tick typhus (Australian Tick Typhus).
- Rickettsia are one of closest living relatives to bacteria that were the origin of the mitochondria organelle that exists inside most eukaryotic cells. Indeed, certain segments of Rickettsia genomes resemble that of mitochondria, and ATP production is the same as that in mitochondria.
Key Terms
- parthenogenesis : a form of asexual reproduction in which growth and development of embryos occurs without fertilization.
- pleomorphic : the ability to alter shape or size in response to environmental conditions.


14.4: Measuring Pressure
- Define gauge pressure and absolute pressure
- Explain various methods for measuring pressure
- Understand the working of open-tube barometers
- Describe in detail how manometers and barometers operate
In the preceding section, we derived a formula for calculating the variation in pressure for a fluid in hydrostatic equilibrium. As it turns out, this is a very useful calculation. Measurements of pressure are important in daily life as well as in science and engineering applications. In this section, we discuss different ways that pressure can be reported and measured.
Gauge Pressure vs. Absolute Pressure
Suppose the pressure gauge on a full scuba tank reads 3000 psi, which is approximately 207 atmospheres. When the valve is opened, air begins to escape because the pressure inside the tank is greater than the atmospheric pressure outside the tank. Air continues to escape from the tank until the pressure inside the tank equals the pressure of the atmosphere outside the tank. At this point, the pressure gauge on the tank reads zero, even though the pressure inside the tank is actually 1 atmosphere—the same as the air pressure outside the tank.
Most pressure gauges, like the one on the scuba tank, are calibrated to read zero at atmospheric pressure. Pressure readings from such gauges are called gauge pressure , which is the pressure relative to the atmospheric pressure. When the pressure inside the tank is greater than atmospheric pressure, the gauge reports a positive value. Some gauges are designed to measure negative pressure. For example, many physics experiments must take place in a vacuum chamber, a rigid chamber from which some of the air is pumped out. The pressure inside the vacuum chamber is less than atmospheric pressure, so the pressure gauge on the chamber reads a negative value. Unlike gauge pressure, absolute pressure accounts for atmospheric pressure, which in effect adds to the pressure in any fluid not enclosed in a rigid container.
The absolute pressure, or total pressure, is the sum of gauge pressure and atmospheric pressure:
\[p_{abs} = p_{g} + p_{atm} \label{14.11}\]
where p abs is absolute pressure, p g is gauge pressure, and p atm is atmospheric pressure.
For example, if a tire gauge reads 34 psi, then the absolute pressure is 34 psi plus 14.7 psi (p atm in psi), or 48.7 psi (equivalent to 336 kPa).
In most cases, the absolute pressure in fluids cannot be negative. Fluids push rather than pull, so the smallest absolute pressure in a fluid is zero (a negative absolute pressure is a pull). Thus, the smallest possible gauge pressure is p g = −p atm (which makes p abs zero). There is no theoretical limit to how large a gauge pressure can be.
Measuring Pressure
A host of devices are used for measuring pressure, ranging from tire gauges to blood pressure monitors. Many other types of pressure gauges are commonly used to test the pressure of fluids, such as mechanical pressure gauges. We will explore some of these in this section.
Any property that changes with pressure in a known way can be used to construct a pressure gauge. Some of the most common types include strain gauges, which use the change in the shape of a material with pressure; capacitance pressure gauges, which use the change in electric capacitance due to shape change with pressure; piezoelectric pressure gauges, which generate a voltage difference across a piezoelectric material under a pressure difference between the two sides; and ion gauges, which measure pressure by ionizing molecules in highly evacuated chambers. Different pressure gauges are useful in different pressure ranges and under different physical situations. Some examples are shown in Figure \(\PageIndex{1}\).
Manometers
One of the most important classes of pressure gauges applies the property that pressure due to the weight of a fluid of constant density is given by p = h\(\rho\)g. The U-shaped tube shown in Figure \(\PageIndex{2}\) is an example of a manometer ; in part (a), both sides of the tube are open to the atmosphere, allowing atmospheric pressure to push down on each side equally so that its effects cancel.
A manometer with only one side open to the atmosphere is an ideal device for measuring gauge pressures. The gauge pressure is p g = h\(\rho\)g and is found by measuring h. For example, suppose one side of the U-tube is connected to some source of pressure p abs , such as the balloon in part (b) of the figure or the vacuum-packed peanut jar shown in part (c). Pressure is transmitted undiminished to the manometer, and the fluid levels are no longer equal. In part (b), p abs is greater than atmospheric pressure, whereas in part (c), pabs is less than atmospheric pressure. In both cases, p abs differs from atmospheric pressure by an amount h\(\rho\)g, where \(\rho\) is the density of the fluid in the manometer. In part (b), p abs can support a column of fluid of height h, so it must exert a pressure h\(\rho\)g greater than atmospheric pressure (the gauge pressure p g is positive). In part (c), atmospheric pressure can support a column of fluid of height h, so p abs is less than atmospheric pressure by an amount h\(\rho\)g (the gauge pressure p g is negative).
Barometers
Manometers typically use a U-shaped tube of a fluid (often mercury) to measure pressure. A barometer (Figure \(\PageIndex{3}\)) is a device that typically uses a single column of mercury to measure atmospheric pressure. The barometer, invented by the Italian mathematician and physicist Evangelista Torricelli (1608–1647) in 1643, is constructed from a glass tube closed at one end and filled with mercury. The tube is then inverted and placed in a pool of mercury. This device measures atmospheric pressure, rather than gauge pressure, because there is a nearly pure vacuum above the mercury in the tube. The height of the mercury is such that h\(\rho\)g = p atm . When atmospheric pressure varies, the mercury rises or falls.
Weather forecasters closely monitor changes in atmospheric pressure (often reported as barometric pressure), as rising mercury typically signals improving weather and falling mercury indicates deteriorating weather. The barometer can also be used as an altimeter, since average atmospheric pressure varies with altitude. Mercury barometers and manometers are so common that units of mm Hg are often quoted for atmospheric pressure and blood pressures.
A U-tube with both ends open is filled with a liquid of density \(\rho_{1}\) to a height h on both sides (Figure \(\PageIndex{1}\)). A liquid of density \(\rho_{2} < \rho_{1}\) is poured into one side and Liquid 2 settles on top of Liquid 1. The heights on the two sides are different. The height to the top of Liquid 2 from the interface is h 2 and the height to the top of Liquid 1 from the level of the interface is h 1 . Derive a formula for the height difference.
Strategy
The pressure at points at the same height on the two sides of a U-tube must be the same as long as the two points are in the same liquid. Therefore, we consider two points at the same level in the two arms of the tube: One point is the interface on the side of the Liquid 2 and the other is a point in the arm with Liquid 1 that is at the same level as the interface in the other arm. The pressure at each point is due to atmospheric pressure plus the weight of the liquid above it.
Pressure on the side with Liquid 1 = p 0 + \(\rho_{1}\) gh 1
Pressure on the side with Liquid 2 = p 0 + \(\rho_{2}\) gh 2
Solution
Since the two points are in Liquid 1 and are at the same height, the pressure at the two points must be the same. Therefore, we have
\[p_{0} + \rho_{1} gh_{1} = p_{0} + \rho_{2} gh_{2} \ldotp \nonumber\]
Hence,
\[\rho_{1} h_{1} = \rho_{2} h_{2} \ldotp \nonumber\]
This means that the difference in heights on the two sides of the U-tube is
\[h_{2} - h_{1} = \left(1 - \dfrac{p_{1}}{p_{2}}\right) h_{2} \ldotp \nonumber\]
The result makes sense if we set \(\rho_2 = \rho_1\), which gives h 2 = h 1 . If the two sides have the same density, they have the same height.
Mercury is a hazardous substance. Why do you suppose mercury is typically used in barometers instead of a safer fluid such as water?
Units of pressure
As stated earlier, the SI unit for pressure is the pascal (Pa), where
\[1\; Pa = 1\; N/m^{2} \ldotp\]
In addition to the pascal, many other units for pressure are in common use (Table \(\PageIndex{1}\)). In meteorology, atmospheric pressure is often described in the unit of millibars (mb), where
\[1000\; mb = 1 \times 10^{5}\; Pa \ldotp\]
The millibar is a convenient unit for meteorologists because the average atmospheric pressure at sea level on Earth is 1.013 x 10 5 Pa = 1013 mb = 1 atm. Using the equations derived when considering pressure at a depth in a fluid, pressure can also be measured as millimeters or inches of mercury. The pressure at the bottom of a 760-mm column of mercury at 0 °C in a container where the top part is evacuated is equal to the atmospheric pressure. Thus, 760 mm Hg is also used in place of 1 atmosphere of pressure. In vacuum physics labs, scientists often use another unit called the torr, named after Torricelli, who, as we have just seen, invented the mercury manometer for measuring pressure. One torr is equal to a pressure of 1 mm Hg.
| Unit | Definition |
|---|---|
| SI unit: the Pascal | $$1\; Pa = 1\; N/m^{2}$$ |
| English unit: pounds per square inch ( lb/in.2 or psi) | $$1\; psi = 7.015 \times 10^{3}\; Pa$$ |
| Other units of pressure | $$\begin{split} 1\; atm & = 760\; mm\; Hg \\ & = 1.013 \times 10^{5}\; Pa \\ & = 14.7\; psi \\ & = 29.9\; inches\; of\; Hg \\ & = 1013\; mb \end{split}$$ |
| $$1\; bar = 10^{5}\; Pa$$ | |
| $$1\; torr = 1\; mm\; Hg = 122.39\; Pa$$ |


1.5: Instrumentation and Scada
Student Learning Outcomes
After reading this chapter, you should be able to:
- Explain the differences between primary and secondary instrumentation
- List the various ways water utilities use SCADA to operate a distribution system
- Define the term telemetry and how it relates to a SCADA system
There are many different processes that need to be monitored by water utility operators. These include, but are not limited to flow rates, meter totalizers, chemical dosages, pressures, levels, and various water quality parameters.
Primary Instrumentation
Primary instrumentation is an instrument used to measure process variables. Some of the more common process variables measured in a water distribution system include but are not limited to, flows, pressures, levels, chemical dosages, and temperatures. This sort of process flow measurement provides water utility operators information on the efficiency and overall operation of the system. Pump stations, groundwater wells, storage tanks, and other facilities should be monitored to ensure they are operating correctly and to help maintain the quality and quantity of the drinking water supply.
Flow Sensors
Measure the flow of water is an important aspect of any distribution system. Flow measurements are used to monitor flows coming into a distribution system (wells, treatment plants, and purchased water sources), flows moving through a distribution system (pump stations), and flows delivered to customers (water service). Measuring flows is important for accounting for the amount of water being purchased, pumped, and sold. Flows can also be used to help track when a piece of equipment needs to be maintained and/or replaced. Meters for measuring flows are typically differential pressure and velocity. They either transmit a read directly to a register (similar to a car’s odometer) or have a pulse or electronic output for monitoring at remote locations.
Pressure Sensors
Pressures on the inlet or suction side of a pump and pressures on the outlet or discharge side of a pump are important parameters to track. Too high or too low of pressure can cause problems with pumping equipment and within a distribution system. Pressure sensors can provide direct-read outputs or provide electronic pulses, which can transmit readings to remote locations. There are four common pressure sensors:
- Strain gauge—This is the most widely used pressure gauge in modern instrumentation. It consists of a section of wire fastened to a diaphragm. The diaphragm moves changing the resistance of the wire. This changing resistance can be measured and transmitted by electrical circuits.
- Bellows sensor—A bellows sensor uses flexible copper that can expand and contract with varying pressures. This is a direct reading pressure gauge.
- Helical sensor—A spiral wound tubular element that coils and uncoils with changes in pressure is a helical sensor. This is a direct reading pressure gauge.
- Bourdon tube—This is a semicircular tube with an elliptical cross-section that tends to assume a circular cross-sectional shape with changes in pressure. This is a direct reading pressure gauge.
Level Sensors
Water levels in groundwater wells and water storage tanks are commonly measured. The depth to groundwater is important to ensure the groundwater table is not drawn down too low and it also indicates when the water level drops below the bowls within a well.  Water levels in storage tanks are also important to measure. Water storage tanks provide millions of gallons of water to consumers and it is critical that storage tanks do not overflow or run empty.
- Float mechanisms—A simple and inexpensive type of liquid level measuring device is a float that rides on the surface and drives a transducer through an arm or cable.
- Diaphragm element—This type of level sensor operates on the principle that the confined air in a tube compresses in relation to the head of water above the diaphragm. The change in pressure sensed is then related to a change in the head of water.
- Bubble tube—A bubble tube provides a constant flow of air in a tube, which is suspended in the water. The pressure required to discharge air from the tube is proportional to the head of water above the bottom of the tube. Bubbler tubes are not very common and are being replaced with newer electronic equipment.
Direct Electronic Sensors
Probes, variable resistance devices, and ultrasonic sensors are also being used to measure levels. A probe can be suspended in the water and has an electronic circuit that detects a change in capacitance between the probe and water. It then electronically converts this information into water depth. A wound resistor inside a semi-flexible envelope makes up a variable-resistance level sensor. As the water level rises, a portion of the resistor element temporarily shorts out and changes the resistance of the sensor. This resistance is converted to a level output signal. Transducers are common water level measuring devices, which translate the head of water over the unit into a signal (typically 4-20mA), which is then converted to feet of head or pressure.
Temperature Sensors
There are two main types of temperature measuring devices used in water, they are thermocouples and thermistors. A thermocouple uses two wires made of different materials, which are joined at two points. One wire is referred to as the sensing point and the other the reference junction. Temperature changes between the two points causes a voltage to be generated, which can then be read directly or transmitted. A thermistor uses a semiconductive material, such as cobalt oxide, which is compressed into a desired shape from the powder form and then it is heat-treated to form crystals to which the wires are attached. Temperature changes are reflected with a corresponding change in resistance through the wires.
Primary instrumentation provide real-time measurements of the condition of various pieces of equipment. Some of this information can be used for planning and scheduling routine maintenance and also can be used to indicate when something is out of the norm. These instruments are composed of a sensor, which responds to a physical condition being measured and an indicator, which converts the signal into a display on an indicator.
Equipment Sensors
Several parameters are monitored for different pieces of equipment in a distribution system. Electrical sensors are used to monitor voltage (volts), current (amps), resistance (ohms), and power (watts). A D’Arsonval meter is used to measure volts, amperes, and ohms on equipment. It is a current sensing device where an electromagnetic core is suspended between the poles of a permanent magnet.
While these types of sensors have been used over the years, digital sensors, which indicate values directly, are more commonly found in the industry nowadays.
The status of equipment is an important parameter to monitor. Common equipment status monitors include, but are not limited to vibration, position, speed, and torque sensors. Any time equipment turns on, off, or simply just runs, vibration occur. This is normal when the components within the equipment are in good condition and the flow of energy is smooth. However, as components age and begin to wear, vibration can increase. A vibration sensor, especially in locations where daily visual inspections are not possible, can be connected to the power circuit and shutdown the equipment if vibration exceeds a specified value.
Similar sensors measuring speed, torque, position, and various other parameters can also be used to monitor equipment and help protect against excessive damage by shutting down equipment at specified set points.
Process Analyzers
In addition to sensors used to monitor the status of equipment, various other processes are commonly monitored within the water utility industry. Measuring water quality is important and common. While measuring water quality parameters in drinking water treatment plants is routine, there are several water quality parameters monitored within distribution systems.
One of the most common water quality parameters measured in distribution systems is the disinfectant residual. Chlorine or chloramines are chemicals used to make sure the biological integrity of drinking water is maintained. Chlorine and chloramine residual analyzers are commonly used to measure the water quality on sources of supply such as groundwater wells and purchased water sources. The measurements can trigger alarms or can be automatically adjusted if the measured parameter falls outside predetermined set points.
Secondary Instrumentation
Secondary instrumentation converts signals from sensors and primary instrumentation. There are several ways instrumentation receives and transmits (indicates) parameters. There are a variety of ways including, but not limited to direct-reading indicators, which express values such as gallons per minute (gpm) for flows, volts from motors, and pressures expressed in pounds per square inch.
Some receivers and indicators collect and record data. Charts are sometimes used to express the values such as the strip chart shown below. Other recorders display total accumulated values or some combination of data collection and expression.
On analog devices, the values will range smoothly from the minimum and maximum values. They are generally easier to read the relative position of the value being displayed throughout the entire range. If values fall between the scaled values on the display they can easily be estimated. On digital devices, the accuracy tends to be better than analog systems and they are very easy to read. The values are typically decimal numbers on mechanical or electronic displays. However, estimating the exact value when the reading falls between divisions on the display is difficult.
When sensors and the indicators are not located in the same area, some type of equipment is needed to send the signal from the sensor to the indicator. In these instances, telemetry is often used. Early telemetry systems used audio tones or electrical pulses. Digital systems are common and se a binary code to transmit signals. A sensor signal feeds into a transmitter, which then generates a series of on-off pulses. The number of on-off pulses represents a number. For example, the pulse sequence of off-on-off-on represents the number five (5). The transmitting device in a digital system is referred to as a remote terminal unit (RTU) and the receiver is called a control terminal unit (CTU).
Whenever multiple signals need to be sent from more than one sensor over the same transmission line is needed, there are several employable methods. Tone-frequency sends signals over one wire or radio signal by having tone-frequency generators in the transmitter. Each parameter is sent at a different frequency. There are filters within the receiver, which sort out the signals and send them to the proper indicator. An example of this is a single voice grade telephone line. As many as twenty-one (21) frequencies can be sent over these types of systems.
Scanning equipment is used to transmit the value of several parameters one at a time in a specified sequence. The receiver decodes the signals and displays each one in a specific turn. Scanning equipment can also be combined with tone-frequency to allow even more signals over a single transmission line. Polling is a system used where each instrument has its own unique address. A system controller sends out a message requesting a specific piece of equipment to transmit its data.
Duplexing is the last process of transmitting signals we will discuss. There are three (3) types of duplexing systems: full-duplex, half-duplex, and simplex.
- In full-duplex systems, the signals can pass in both directions at the same time
- Half-duplex systems only allow signals to pass one direction at a time
- A simplex system only allows signals to pass in one direction
Control Systems
The idea of instrumentation and control systems is to obtain the ability to make changes or corrections in the parameters being measured. Yes, it is very valuable to know what a chlorine residual is at a groundwater well, but if you need to visit the location to make actual changes based on the signals being received from a sensor then valuable time can be lost. Therefore, control systems are extremely useful. A control system allows for adjustments to be made based on the data being transmitted and received. Control systems can be broken down into four (4) main types of systems.
- Direct Manual—A direct manual system, is the simplest and least complicated control system. Components are controlled by an operator that must physically visit each location to make a change. For example, if a signal is transmitted that requires a system to be turned off, in a direct manual control system, an operator must drive out to the location and manually turn the component off. This type of system has a low initial cost and has little complicated equipment to maintain. However, it does require labor and operator expertise and judgment.
- Remote Manual—In a remote manual control system, an operator can make adjustments to systems and components from a remote location. This type of system still requires operator expertise and judgment, but it requires less physical labor. An example of this type of control system could be when a pump needs to be turned on or off. Instead of requiring an operator to physically visit a location to perform this task, it can be controlled remotely from a control room or some remote location.
- Semiautomatic—This type of control system combines manual control from a remote location (control room) with automatic control of specific pieces of equipment. An example of this could be a circuit breaker. A breaker will disconnect automatically in response to an overload, but then it must be reset manually. This “resetting” can be remotely or at the facility.
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Automatic—Full automatic control is when equipment can turn on and off or adjust their operation in response to signals from sensors and analytical instruments. There are two general modes of automatic controls: on-off differential and proportional.
- On-Off Differential control systems turn equipment either full on or off in response to a signal. The rate of the equipment would need to be adjusted manually.
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Proportional control systems adjust variables automatically. Proportional control systems can be broken down into three (3) main types.
- Feedforward proportional control measures a variable such as chlorine dosage. The flow of water is being measured and the faster (or more) water flows through a meter, the chlorine feed system increases the amount of chlorine. This type of system is useful, but it cannot account for varying chlorine demand.
- Feedback proportional control measures the output of a process and will then react to adjust the operation of the piece of equipment. This type of system is also referred to as a “closed-loop” control system because it continuously self corrects. These systems can be troublesome if there are wide variations in the water flow rate.
- Combined control systems adjust in response to changes in the flow rate, but an analyzer monitoring the chlorine dosage makes minor adjustments in the feed rate of the chemical to maintain the selected residual being measured in the finished water.
Supervisory Control and Data Acquisition
The processes discussed in this chapter are wrapped up together into a Supervisory Control and Data Acquisition (SCADA) system. SCADA a system used in a variety of industries including drinking water treatment and distribution systems. There are field devices (primary instrumentation) such as sensors, which read various parameters, sending signals, which are received and transmitted (secondary instrumentation) through telemetry, to a centralized computer system. This allows an operator a complete view of a distribution system to see how things are operating.
A variety of components and processes are monitored in a drinking water distribution system, including but not limited to storage tank levels, pump station flow rates, pressures, groundwater well depths, and chemical feed such as disinfection systems.
Here is a simple example of a how a SCADA system works in a water distribution system.
A groundwater well provides water to the distribution system. So, how does this well turn on and start pumping water? Years ago, an operator would drive out to the well, open the gate to the facility, insert a key to unlock the control panel, and then turn a switch to power it on. Water would enter the distribution system through a network of pipes. If a customer turns on a faucet, water would come out. What is no one was using water when the well was turned on? The water would then flow to a water storage tank. This tank would begin to fill. What would happen if the well was not shut off? Obviously enough, the storage tank would overflow.  Without some sort of computer system, signal, or alarm, the operator would need to drive to the tank and visually look at the level of water.  This is not very complicated, but it is labor-intensive and time-consuming. Fast forward to the age of computers. A sensor could be installed in the water storage tank to monitor the level. With the help of a SCADA system, specific level set points can be programmed to tell another set of sensors when to turn on and off something like a groundwater well. This is a very simplistic example of how a SCADA system works, but it adequately explains the process of a sensor sending a signal, a computer system reading this signal, and responding with a function.
A control room is typically equipped with a human-machine interface (HMI). This interface is a window into the SCADA system. It graphically displays all the facilities within a system. The tanks, pumps, sources of supply are all interconnected to make sure water is continually distributed throughout the system. The system is usually equipped with various alarms to remotely notify operators through pagers, text messaging, or some other type of remote notification. This allows the system to continually function without someone sitting at a computer twenty-four (24) hours a day, seven (7) days a week. An alarm is received by an operator and then the fully trained operator would either verify the alarm was cleared after the process returned to normal operation or inform the operator that additional tasks might be needed to fix the system.
Sample Questions
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Which of the following is used to measure pressure?
- Volute
- Multi-jet
- Strain gauge
- All of the above
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Which of the following is used to measure low pressure?
- Bellows sensor
- Helical sensor
- Bourdon tube
- All of the above
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Bubble tubes must be installed ___________.
- At the top of the tank
- In the middle of the tank
- At the bottom of the tank
- Anywhere in the tank
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Which of the following temperature measuring devices uses two wires?
- Thermometer
- Thermister
- Thermostat
- Thermocouple
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Which of the following temperature measuring devices uses cobalt oxide as a semiconductive material?
- Thermometer
- Thermister
- Thermostat
- Thermocouple
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Which type of control would not account for varying chlorine demand?
- Feedback proportional
- Feedforward proportional
- Combined
- All of the above
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Which of the following is the smallest measurement of power?
- Voltage
- Amperes
- Ohms
- Watts
-
Amperes is a measurement of ___________.
- Current
- Resistance
- Power
- Voltage
-
SCADA stands for ___________.
- Self Contained Analog Digital Assembly
- Superior Computer And Data Acquisition
- Supervisory Computer And Digital Assembly
- Supervisory Control And Data Acquisition
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Which of the following allows signals to only pass in one direction?
- Half-duplex
- Full-duplex
- Simplex
- All of the above


4.1: Derivatives and Graphs
As we’ve seen, one of the most important connections between a function and its derivative is that a positive derivative means the quantity is increasing, and a negative derivative means the quantity is decreasing.
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Outside temperature has a positive derivative from 3am to 3pm, and a negative derivative from 3pm to 3am. Draw a graph of this, and label each part of the graph as “increasing” or “decreasing”.
With the positive derivative from 3am to 3pm, this should go up and be labeled “increasing”. From 3pm to 3am, the graph is going down and labeled “decreasing”.
At the interface between increasing and decreasing, at 3pm, is when the temperature is the highest. This is the key to one of the most useful applications of calculus: optimization! Optimization is either finding out when a quantity is maximized, or as high as possible, or finding out when a quantity is minimized, or as low as possible. Often this is at the interface of increasing and decreasing, and thus at the where a function goes from positive derivative to negative derivative. Hence, one of the most important maxims of calculus: optimization happens when the derivative is zero! We will come back to this in a future section.
For now, using this idea of when the derivative is positive, negative, or zero, we can draw a rough sketch of the derivative based on the graph of a function. Let’s see an example
When sketching the derivative, keep this idea in mind: slopes become \(y\)-values . First, let’s mark where the derivative is zero:
These are the places that, in the derivative graph, have zero for the \(y\)-value. That means these are the \(x\)-intercepts!
Now let’s mark where the derivative is positive, and where it is negative.
Finally, we can use this as a rough guide for a sketch, again keeping in mind slope becomes \(y\)-values. Here, the derivative is in black, while the original function is in grey.
Let’s see another example.
In this case, the graph is always going down, so the derivative is always negative. That doesn’t really tell you a lot about what the derivative graph looks like. However, there is a special point called an inflection point right here:
It’s not where the slope is zero, but it is where the slope gets the closest to zero. Everywhere else the slope is more negative than at the inflection point. So in the derivative graph this becomes a maximum or highest point, like this:
As long as that inflection point is the highest point (but still a negative \(y\)-value due to the negative slope of the original graph), that’s about the best you can do on that one.


11.2: Plaque Assay (Day 1)
Unlike bacteria, many of which can be grown on an artificial nutrient medium, viruses require a living host cell for replication. Infected host cells (eukaryotic or prokaryotic) can be cultured and grown, and then the growth medium can be harvested as a source of virus. In this lab, we will be quantifying bacteriophages (viruses that attack bacteria) using a plaque assay. This means that we must grow both bacteria and virus.
As the virus replicates and grows, it will lyse bacteria resulting in an area of clearing that can be observed on a bacterial lawn. These clearings are called plaques. A plaque assay, therefore, is a method by which the number of plaques observed can be used to estimate the amount of virus present in the original culture. The technique required to do so is called the pour-plate technique because you will mix the virus with the bacteria with melted agar and then pour the mixture onto a plate.
The unit of viral titer (concentration) is plaque-forming units per milliliter (pfu/ml).
As we do not know the starting concentration of virus, we will need to do a serial dilution (similar to the standard plate count). After incubation, the number of plaques formed will be used to calculate the original viral (phage) titer.
Plague Assay:
Complete the Following Lab with Your Partner:
1. Obtain 14 microcentrifuge tubes and label either #1-7 or E. coli #1-7.
2. Obtain 7 nutrient agar plates. Label them A-G and place them in the 37°C incubator to warm up until needed.
3. Aseptically add 990 μL of sterile saline to tubes #1.
4. Aseptically add 900 μL of sterile saline to tubes #2-7.
5. Finger vortex the E. coli broth. Then, aseptically add 300 μL to all seven E. coli microcentrifuge tubes.
6. Vortex the T4 bacteriophage suspension and then aseptically transfer 10 μL to tube #1. Close the lid and vortex. This is your 10-2 dilution.
Enterobacteria phage T4 is a bacteriophage that infects Escherichia coli bacteria.
7. Aseptically transfer 100 μL from tube #1 to tube #2, close the lid and vortex. This is your 10-3 dilution.
8. Aseptically transfer 100 μL from tube #2 to tube #3, close the lid and vortex. This is your 10-4 dilution.
9. Aseptically transfer 100 μL from tube #3 to tube #4, close the lid and vortex. This is your 10-5 dilution.
10. Aseptically transfer 100 μL from tube #4 to tube #5, close the lid and vortex. This is your 10-6 dilution.
11. Aseptically transfer 100 μL from tube #5 to tube #6, close the lid and vortex. This is your 10-7 dilution.
12. Aseptically transfer 100 μL from tube #6 to tube #7, close the lid and vortex. This is your 10-8 dilution.
13. Aseptically transfer 100 μL of SALINE into tube E. coli #1, close the lid and vortex. This will become your negative control.
14. Aseptically transfer 100 μL of dilution tube #2 to E. coli #2, close the lid and vortex.
15. Aseptically transfer 100 μL of dilution tube #3 to E. coli #3, close the lid and vortex.
16. Aseptically transfer 100 μL of dilution tube #4 to E. coli #4, close the lid and vortex.
17. Aseptically transfer 100 μL of dilution tube #5 to E. coli #5, close the lid and vortex.
18. Aseptically transfer 100 μL of dilution tube #6 to E. coli #6, close the lid and vortex.
19. Aseptically transfer 100 μL of dilution tube #7 to E. coli #7, close the lid and vortex.
20. Let all seven tubes stand for 15 minutes. We are allowing time for the virus to infect the bacteria.
21. Remove one soft agar tube (containing 2.5 ml of agar) from the hot water bath and add the entire contents of E. coli #1 to it. Using a sterile plastic transfer pipet, mix the contents by pipetting up and down twice.
22. Immediately pour the mixture onto plate A. Then, gently tilt the plate back and forth until the soft agar mixture is spread evenly across the solid medium.
Remember that you need to take the soft agar plates and nutrient agar plates out one at a time. Otherwise, the agar will solidify before you have a chance to pour.
23. Allow the agar to solidify completely with the lid on at your table (15-20 minutes). Then, invert and place in the 37°C incubator to grow overnight.


4: States of Consciousness
This chapter will discuss states of consciousness with a particular emphasis on sleep. The different stages of sleep will be identified, and sleep disorders will be described. The chapter will close with discussions of altered states of consciousness produced by psychoactive drugs, hypnosis, and meditation.
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- 4.1: Prelude to States of Consciousness
- Our lives involve regular, dramatic changes in the degree to which we are aware of our surroundings and our internal states. While awake, we feel alert and aware of the many important things going on around us. Our experiences change dramatically while we are in deep sleep and once again when we are dreaming.
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- 4.2: What Is Consciousness?
- Consciousness describes our awareness of internal and external stimuli. Awareness of internal stimuli includes feeling pain, hunger, thirst, sleepiness, and being aware of our thoughts and emotions. Awareness of external stimuli includes seeing the light from the sun, feeling the warmth of a room, and hearing the voice of a friend. We experience different states of consciousness and different levels of awareness on a regular basis.
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- 4.3: Sleep and Why We Sleep
- We spend approximately one-third of our lives sleeping.. Some animals never sleep (e.g., several fish and amphibian species); other animals can go extended periods of time without sleep and without apparent negative consequences (e.g., dolphins); yet some animals (e.g., rats) die after two weeks of sleep deprivation. Why do we devote so much time to sleeping? Is it absolutely essential that we sleep? This section will consider these questions and explore various explanations for why we sleep.
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- 4.4: Stages of Sleep
- Sleep is not a uniform state of being. Instead, sleep is composed of several different stages that can be differentiated from one another by the patterns of brain wave activity that occur during each stage. These changes in brain wave activity can be visualized using EEG and are distinguished from one another by both the frequency and amplitude of brain waves. Sleep can be divided into two different general phases: REM sleep and non-REM (NREM) sleep.
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- 4.5: Sleep Problems and Disorders
- Many people experience disturbances in their sleep at some point in their lives. Depending on the population and sleep disorder being studied, between 30% and 50% of the population suffers from a sleep disorder at some point in their lives. This section will describe several sleep disorders as well as some of their treatment options.
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- 4.6: Substance Use and Abuse
- While we all experience altered states of consciousness in the form of sleep on a regular basis, some people use drugs and other substances that result in altered states of consciousness as well. This section will present information relating to the use of various psychoactive drugs and problems associated with such use. This will be followed by brief descriptions of the effects of some of the more well-known drugs commonly used today.
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- 4.7: Other States of Consciousness
- Our states of consciousness change as we move from wakefulness to sleep. We also alter our consciousness through the use of various psychoactive drugs. This final section will consider hypnotic and meditative states as additional examples of altered states of consciousness experienced by some individuals.
Contributors and Attributions
-
Rose M. Spielman with many significant contributors. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the creative commons license and may not be reproduced without the prior and express written consent of Rice University. For questions regarding this license, please contact<EMAIL_ADDRESS>.Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.


6.5: Sinusoidal Graphs
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The London Eye (London Eye photo by authors, 2010, CC-BY) is a huge Ferris wheel 135 meters (394 feet) tall in London, Eng land, which completes one rotation every 30 minutes. When we look at the behavior of this Ferris wheel it is clear that it completes 1 cycle, or 1 revolution, and then repeats this revolution over and over again.
This is an example of a periodic function, because the Ferris wheel repeats its revolution or one cycle every 30 minutes, and so we say it has a period of 30 minutes.
In this section, we will work to sketch a graph of a rider’s height above the ground over time and express this height as a function of time.
Definition: PERIODIC FUNCTIONS
A periodic function is a function for which a specific horizontal shift, \(P\), results in the original function: \(f(x+P)=f(x)\) for all values of \(x\). When this occurs we call the smallest such horizontal shift with \(P > 0\) the period of the function.
You might immediately guess that there is a connection here to finding points on a circle, since the height above ground would correspond to the \(y\) value of a point on the circle. We can determine the \(y\) value by using the sine function. To get a better sense of this function’s behavior, we can create a table of values we know, and use them to sketch a graph of the sine and cosine functions.
Listing some of the values for sine and cosine on a unit circle,
| \(\theta\) | 0 | \(\dfrac{\pi }{6}\) | \(\dfrac{\pi }{4}\) | \(\dfrac{\pi }{3}\) | \(\dfrac{\pi }{2}\) | \(\dfrac{2\pi }{3}\) | \(\dfrac{3\pi }{4}\) | \(\dfrac{5\pi }{6}\) | \(\pi\) |
| cos | 1 | \(\dfrac{\sqrt{3} }{2}\) | \(\dfrac{\sqrt{2} }{2}\) | \(\dfrac{1}{2}\) | 0 | \(-\dfrac{1}{2}\) | \(-\dfrac{\sqrt{2} }{2}\) | \(-\dfrac{\sqrt{3} }{2}\) | -1 |
| sin | 0 | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{2} }{2}\) | \(\dfrac{\sqrt{3} }{2}\) | 1 | \(\dfrac{\sqrt{3} }{2}\) | \(\dfrac{\sqrt{2} }{2}\) | \(\dfrac{1}{2}\) | 0 |
Here you can see how for each angle, we use the \(y\) value of the point on the circle to determine the output value of the sine function.
Plotting more points gives the full shape of the sine and cosine functions.
Notice how the sine values are positive between 0 and \(\pi \), which correspond to the values of sine in quadrants 1 and 2 on the unit circle, and the sine values are negative between \(\pi \) and 2\(\pi \), corresponding to quadrants 3 and 4.
Like the sine function we can track the value of the cosine function through the 4 quadrants of the unit circle as we place it on a graph.
Both of these functions are defined for all real numbers, since we can evaluate the sine and cosine of any angle. By thinking of sine and cosine as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval [-1, 1].
DOMAIN AND RANGE OF SINE AND COSINE
The domain of sine and cosine is all real numbers, \((-\infty ,\infty )\).
The range of sine and cosine is the interval [-1, 1].
Both these graphs are called sinusoidal graphs.
In both graphs, the shape of the graph begins repeating after 2\(\pi \). Indeed, since any coterminal angles will have the same sine and cosine values, we could conclude that \[\sin (\theta +2\pi )=\sin (\theta )\nonumber\] and \[\cos (\theta +2\pi )=\cos (\theta )\nonumber\]
In other words, if you were to shift either graph horizontally by 2\(\pi \), the resulting shape would be identical to the original function. Sinusoidal functions are a specific type of periodic function.
PERIOD OF SINE AND COSINE
The periods of the sine and cosine functions are both 2\(\pi \).
Looking at these functions on a domain centered at the vertical axis helps reveal symmetries.
sine cosine
The sine function is symmetric about the origin, the same symmetry the cubic function has, making it an odd function. The cosine function is clearly symmetric about the y axis, the same symmetry as the quadratic function, making it an even function.
NEGATIVE ANGLE IDENTITIES
The sine is an odd function, symmetric about the origin , so \[\sin (-\theta )=-\sin (\theta )\]
The cosine is an even function, symmetric about the y -axis, so \[\cos (-\theta )=\cos (\theta )\]
These identities can be used, among other purposes, for helping with simplification and proving identities.
You may recall the cofunction identity from last chapter, \[\sin (\theta )=\cos \left(\dfrac{\pi }{2} -\theta \right)\nonumber\]
Graphically, this tells us that the sine and cosine graphs are horizontal transformations of each other. We can prove this by using the cofunction identity and the negative angle identity for cosine.
\[\sin (\theta )=\cos \left(\dfrac{\pi }{2} -\theta \right)=\cos \left(-\theta +\dfrac{\pi }{2} \right)=\cos \left(-\left(\theta -\dfrac{\pi }{2} \right)\right)=\cos \left(\theta -\dfrac{\pi }{2} \right)\]
Now we can clearly see that if we horizontally shift the cosine function to the right by \(\pi \)/2 we get the sine function.
Remember this shift is not representing the period of the function. It only shows that the cosine and sine function are transformations of each other.
Example \(\PageIndex{1}\)
Simplify \(\dfrac{\sin (-\theta )}{\tan (\theta )}\).
Solution
We start by using the negative angle identity for sine.
\[\dfrac{-\sin (\theta )}{\tan (\theta )}\nonumber\] Rewriting the tangent
\[\dfrac{-\sin (\theta )}{\sin (\theta )/\cos(\theta)}\nonumber\] Inverting and multiplying
\[-\sin (\theta )\cdot \dfrac{\cos (\theta )}{\sin (\theta )}\nonumber\] Simplifying we get
\[-\cos (\theta )\nonumber\]
Transforming Sine and Cosine
Example \(\PageIndex{2}\)
A point rotates around a circle of radius 3. Sketch a graph of the \(y\) coordinate of the point.
Solution
Recall that for a point on a circle of radius \(r\), the \(y\) coordinate of the point is \(y=r\sin (\theta )\), so in this case, we get the equation\[y(\theta )=3\sin (\theta )\nonumber\]
The constant 3 causes a vertical stretch of the \(y\) values of the function by a factor of 3.
Notice that the period of the function does not change.
Since the outputs of the graph will now oscillate between -3 and 3, we say that the amplitude of the sine wave is 3.
Exercise \(\PageIndex{1}\)
What is the amplitude of the function \(f(\theta )=7\cos (\theta )\)? Sketch a graph of this function.
- Answer
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7
Example \(\PageIndex{3}\)
A circle with radius 3 feet is mounted with its center 4 feet off the ground. The point closest to the ground is labeled \(P\). Sketch a graph of the height above ground of the point \(P\) as the circle is rotated, then find a function that gives the height in terms of the angle of rotation.
Solution
Sketching the height, we note that it will start 1 foot above the ground, then increase up to 7 feet above the ground, and continue to oscillate 3 feet above and below the center value of 4 feet.
Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other.
We decide to use a cosine function because it starts at the highest or lowest value, while a sine function starts at the middle value. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection.
Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of one, so this graph has been vertically stretched by 3, as in the last example.
Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4. Putting these transformations together,
\[h(\theta )=-3\cos (\theta )+4\nonumber\]
Definition: MIDLINE
The center value of a sinusoidal function, the value that the function oscillates above and below, is called the midline of the function, corresponding to a vertical shift.
The function \(f(\theta )=\cos (\theta )+k\) has midline at \(y = k\).
Exercise \(\PageIndex{2}\)
What is the midline of the function \(f(\theta )=3\cos (\theta )-4\)? Sketch a graph of the function.
- Answer
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-4
To answer the Ferris wheel problem at the beginning of the section, we need to be able to express our sine and cosine functions at inputs of time. To do so, we will utilize composition. Since the sine function takes an input of an angle, we will look for a function that takes time as an input and outputs an angle. If we can find a suitable \(\theta (t)\) function, then we can compose this with our \(f(\theta )=\cos (\theta )\) function to obtain a sinusoidal function of time: \(f(t)=\cos (\theta (t))\).
Example \(\PageIndex{4}\)
A point completes 1 revolution every 2 minutes around a circle of radius 5. Find the \(x\) coordinate of the point as a function of time, if it starts at (5, 0).
Solution
Normally, we would express the \(x\) coordinate of a point on a unit circle using \(x=r\cos (\theta )\), here we write the function \(x(\theta )=5\cos (\theta )\).
The rotation rate of 1 revolution every 2 minutes is an angular velocity. We can use this rate to find a for mula for the angle as a function of time. The point begins at an angle of 0. Since the point rotates 1 revolution = 2\(\pi \) radians every 2 minutes, it rotates \(\pi \) radians every minute. After t minutes, it will have rotated:
\[\theta (t)=\pi {\kern 1pt} t\text{ radians}\nonumber\]
Composing this with the cosine function, we obtain a function of time.
\[x(t)=5\cos (\theta (t))=5\cos (\pi {\kern 1pt} t)\nonumber\]
Notice that this composition has the effect of a horizontal compression, changing the period of the function.
To see how the period relates to the stretch or compression coefficient \(B\) in the equation \(f(t)=\sin \left(Bt\right)\), note that the period will be the time it takes to complete one full revolution of a circle. If a point takes \(P\) minutes to complete 1 revolution, then the angular velocity is \(\dfrac{2\pi \; {\rm radians}}{P\; {\rm minutes}}\). Then \(\theta (t)=\dfrac{2\pi }{P} t\). Composing with a sine function,
\[f(t)=\sin (\theta (t))=\sin \left(\dfrac{2\pi }{P} t\right)\nonumber\]
From this, we can determine the relationship between the coefficient \(B\) and the period:
\[B=\dfrac{2\pi }{P}\nonumber\]
Notice that the stretch or compression coefficient \(B\) is a ratio of the “normal period of a sinusoidal function” to the “new period.” If we know the stretch or compression coefficient \(B\), we can solve for the “new period”: \[P=\dfrac{2\pi }{B}\nonumber\]
Summarizing our transformations so far:
TRANSFORMATIONS OF SINE AND COSINE
Given an equation in the form \[f(t)=A\sin \left(Bt\right)+k\text{ or }f(t)=A\cos \left(Bt\right)+k\]
\(A\) is the vertical stretch, and is the amplitude of the function.
\(B\) is the horizontal stretch/compression, and is related to the period , \(P\) , by \(P=\dfrac{2\pi }{B}\).
\(k\) is the vertical shift and determines the midline of the function.
Example \(\PageIndex{5}\)
What is the period of the function \(f(t)=\sin \left(\dfrac{\pi }{6} t\right)\)?
Solution
Using the relationship above, the stretch/compression factor is \(B=\dfrac{\pi }{6}\), so the period will be \[P=\dfrac{2\pi }{B} =\dfrac{2\pi }{\dfrac{\pi }{6} } =2\pi \cdot \dfrac{6}{\pi } =12\nonumber\]
While it is common to compose sine or cosine with functions involving time, the composition can be done so that the input represents any reasonable quantity.
Example \(\PageIndex{6}\)
A bicycle wheel with radius 14 inches has the bottom-most point on the wheel marked in red. The wheel then begins rolling down the street. Write a formula for the height above ground of the red point after the bicycle has travelled \(x\) inches.
Solution
The height of the point begins at the lowest value, 0, increases to the highest value of 28 inches, and continues to oscillate above and below a center height of 14 inches. In terms of the angle of rotation, \(\theta\):
\[h(\theta )=-14\cos (\theta )+14\nonumber\]
In this case, \(x\) is representing a linear distance the wheel has travelled, corresponding to an arclength along the circle . Since arclength and angle can be related by \(s=r\theta\), in this case we can write \(x=14\theta\), which allows us to express the angle in terms of \(x\):
\[\theta (x)=\dfrac{x}{14}\nonumber\]
Composing this with our cosine-based function from above,
\[h(x)=h(\theta (x))=-14\cos \left(\dfrac{x}{14} \right)+14=-14\cos \left(\dfrac{1}{14} x\right)+14\nonumber\]
The period of this function would be \(P=\dfrac{2\pi }{B} =\dfrac{2\pi }{\dfrac{1}{14} } =2\pi \cdot 14=28\pi\), the circumference of the circle. This makes sense – the wheel completes one full revolution after the bicycle has travelled a distance equivalent to the circumference of the wheel.
Example \(\PageIndex{7}\)
Determine the midline, amplitude, and period of the function \(f(t)=3\sin \left(2t\right)+1\).
Solution
The amplitude is 3
The period is \(P=\dfrac{2\pi }{B} =\dfrac{2\pi }{2} =\pi\)
The midline is at \(y=1\)
Amplitude, midline, and period, when combined with vertical flips, allow us to write equations for a variety of sinusoidal situations.
Exercise \(\PageIndex{3}\)
If a sinusoidal function starts on the midline at point (0,3), has an amplitude of 2, and a period of 4, write a formula for the function.
- Answer
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\[f(x)=2\sin \left(\dfrac{\pi }{2} x\right)+3\nonumber\]
Example \(\PageIndex{8}\)
Find a formula for the sinusoidal function graphed here.
Solution
The graph oscillates from a low of -1 to a high of 3, putting the midline at \(y = 1\), halfway between.
The amplitude will be 2, the distance from the midline to the highest value (or lowest value) of the graph.
The period of the graph is 8. We can measure this from the first peak at \(x = -2\) to the second at \(x = 6\). Since the period is 8, the stretch/compression factor we will use will be
\[B=\dfrac{2\pi }{P} =\dfrac{2\pi }{8} =\dfrac{\pi }{4}\nonumber\]
At \(x = 0\), the graph is at the midline value, which tells us the graph can most easily be represented as a sine function. Since the graph then decreases, this must be a vertical reflection of the sine function. Putting this all together,
\[f(t)=-2\sin \left(\dfrac{\pi }{4} t\right)+1\nonumber\]
With these transformations, we are ready to answer the Ferris wheel problem from the beginning of the section.
Example \(\PageIndex{9}\)
The London Eye is a huge Ferris wheel in London, England, which completes one rotation every 30 minutes. The diameter of the wheel is 120 meters, but the passenger capsules sit outside the wheel. Suppose the diameter at the capsules is 130 meters, and riders board from a platform 5 meters above the ground. Express a rider’s height above ground as a function of time in minutes.
Solution
It can often help to sketch a graph of the situation before trying to find the equation.
With a diameter of 130 meters, the wheel has a radius of 65 meters. The height will oscillate with amplitude of 65 meters above and below the center.
Passengers board 5 meters above ground level, so the center of the wheel must be located 65 + 5 = 70 meters above ground level. The midline of the oscillation will be at 70 meters.
The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with period of 30 minutes.
Lastly, since the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a flipped cosine curve.
Putting these together:
Amplitude: 65
Midline: 70
Period: 30, so \[B=\dfrac{2\pi }{30} =\dfrac{\pi }{15}\nonumber\]
Shape: negative cosine
An equation for the rider’s height would be
\[h(t)=-65\cos \left(\dfrac{\pi }{15} t\right)+70\nonumber\]
Exercise \(\PageIndex{4}\)
The Ferris wheel at the Puyallup Fair (Photo by photogirl7.1, www.flickr.com/photos/kitkaph...86205/sizes/z/, CC-BY) ha s a diameter of about 70 feet and takes 3 minutes to complete a full rotation. Passengers board from a platform 10 feet above the ground. Write an equation for a rider’s height above ground over time.
- Answer
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\[h(t)=-35\cos \left(\dfrac{2\pi }{3} t\right)+45\nonumber\]
While these transformations are sufficient to represent many situations, occasionally we encounter a sinusoidal function that does not have a vertical intercept at the lowest point, highest point, or midline. In these cases, we need to use horizontal shifts. Since we are combining horizontal shifts with horizontal stretches, we need to be careful. Recall that when the inside of the function is factored, it reveals the horizontal shift.
HORIZONTAL SHIFTS OF SINE AND COSINE
Given an equation in the form \[f(t)=A\sin \left(B(t-h)\right)+k\text{ or }f(t)=A\cos \left(B(t-h)\right)+k\]
\(h\) is the horizontal shift of the function
Example \(\PageIndex{10}\)
Sketch a graph of \(f(t)=3\sin \left(\dfrac{\pi }{4} t-\dfrac{\pi }{4} \right)\).
Solution
To reveal the horizontal shift, we first need to factor inside the function:
\[f(t)=3\sin \left(\dfrac{\pi }{4} (t-1)\right)\nonumber\]
This graph will have the shape of a sine function, starting at the midline and increasing, with an amplitude of 3. The period of the graph will be \[P=\dfrac{2\pi }{B} =\dfrac{2\pi }{\dfrac{\pi }{4} } =2\pi \cdot \dfrac{4}{\pi } =8\nonumber\]
Finally, the graph will be shifted to the right by 1.
In some physics and mathematics books, you will hear the horizontal shift referred to as phase shift . In other physics and mathematics books, they would say the phase shift of the equation above is \(\dfrac{\pi }{4}\), the value in the unfactored form. Because of this ambiguity, we will not use the term phase shift any further, and will only talk about the horizontal shift.
Example \(\PageIndex{11}\)
Find a formula for the function graphed here.
Solution
With highest value at 1 and lowest value at -5, the midline will be halfway between at -2.
The distance from the midline to the highest or lowest value gives an amplitude of 3.
The period of the graph is 6, which can be measured from the peak at \(x = 1\) to the next peak at \(x = 7\), or from the distance between the lowest points. This gives \[B=\dfrac{2\pi }{P} =\dfrac{2\pi }{6} =\dfrac{\pi }{3}\nonumber\]
For the shape and shift, we have more than one option. We could either write this as:
A cosine shifted 1 to the right
A negative cosine shifted 2 to the left
A sine shifted 1/2 to the left
A negative sine shifted 2.5 to the right
While any of these would be fine, the cosine shifts are easier to work with than the sine shifts in this case, because they involve integer values. Writing these:
\[y(x)=3\cos \left(\dfrac{\pi }{3} (x-1)\right)-2\nonumber\]or
\[y(x)=-3\cos \left(\dfrac{\pi }{3} (x+2)\right)-2\nonumber\]
Again, these functions are equivalent, so both yield the same graph.
Exercise \(\PageIndex{5}\)
Write a formula for the function graphed here.
- Answer
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Two possibilities: \[f(x)=4\cos \left(\dfrac{\pi }{5} (x-3.5)\right)+4\text{ or }f(x)=4\sin \left(\dfrac{\pi }{5} (x-1)\right)+4\nonumber\]
Important Topics of This Section
- Periodic functions
- Sine and cosine function from the unit circle
- Domain and range of sine and cosine functions
- Sinusoidal functions
- Negative angle identity
- Simplifying expressions
- Transformations
- Amplitude
- Midline
- Period
- Horizontal shifts