2.5: Cell Growth and Division
By the end of the section, you will be able to:
- Describe, in order, the stages of the cell cycle
- Discuss how the cell cycle is regulated
- Describe the implications of losing control over the cell cycle
- Describe the stages of mitosis and cytokinesis, in order
So far in this chapter, you have read numerous times of the importance and prevalence of cell division. While there are a few cells in the body that do not undergo cell division (such as gametes, red blood cells, most neurons, and some muscle cells), most somatic cells divide regularly. A somatic cell is a general term for a body cell, and all human cells, except for the cells that produce eggs and sperm (which are referred to as germ cells), are somatic cells. Somatic cells contain two copies of each of their chromosomes (one copy received from each parent). Each homologous pair of chromosomes includes the two copies of a single chromosome found in each somatic cell. The human is a diploid organism, having 23 homologous pairs of chromosomes in each of the somatic cells. The condition of having pairs of chromosomes is known as diploidy.
Cells in the body replace themselves over the lifetime of a person. For example, the cells lining the gastrointestinal tract must be frequently replaced when constantly “worn off” by the movement of food through the gut. But what triggers a cell to divide, and how does it prepare for and complete cell division? The cell cycle is the sequence of events in the life of the cell from the moment it is created at the end of a previous cycle of cell division until it then divides itself, generating two new cells.
The Cell Cycle
One “turn” or cycle of the cell cycle consists of two general phases: interphase, followed by mitosis and cytokinesis. Interphase is the period of the cell cycle during which the cell is not dividing. The majority of cells are in interphase most of the time. Mitosis is the division of genetic material, during which the cell nucleus breaks down and two new, fully functional, nuclei are formed. Cytokinesis divides the cytoplasm into two distinctive cells.
Interphase
A cell grows and carries out all normal metabolic functions and processes in a period called G 1 (Figure \(\PageIndex{1}\)). G 1 phase (gap 1 phase) is the first gap, or growth phase in the cell cycle. For cells that will divide again, G 1 is followed by replication of the DNA, during the S phase. The S phase (synthesis phase) is period during which a cell replicates its DNA.
After the synthesis phase, the cell proceeds through the G 2 phase. The G 2 phase is a second gap phase, during which the cell continues to grow and makes the necessary preparations for mitosis. Between G 1 , S, and G 2 phases, cells will vary the most in their duration of the G1 phase. It is here that a cell might spend a couple of hours, or many days. The S phase typically lasts between 8-10 hours and the G 2 phase approximately 5 hours. In contrast to these phases, the G 0 phase is a resting phase of the cell cycle. Cells that have temporarily stopped dividing and are resting (a common condition) and cells that have permanently ceased dividing (like nerve cells) are said to be in G 0 .
The Structure of Chromosomes
Billions of cells in the human body divide every day. During the synthesis phase (S, for DNA synthesis) of interphase, the amount of DNA within the cell precisely doubles. Therefore, after DNA replication but before cell division, each cell actually contains two copies of each chromosome. Each copy of the chromosome is referred to as a sister chromatid and is physically bound to the other copy. The centromere is the structure that attaches one sister chromatid to another. Because a human cell has 46 chromosomes, during this phase, there are 92 chromatids (46 × 2) in the cell. Make sure not to confuse the concept of a pair of chromatids (one chromosome and its exact copy attached during mitosis) and a homologous pair of chromosomes (two paired chromosomes which were inherited separately, one from each parent) (Figure \(\PageIndex{2}\)).
Mitosis and Cytokinesis
The mitotic phase of the cell typically takes between 1 and 2 hours. During this phase, a cell undergoes two major processes. First, it completes mitosis, during which the contents of the nucleus are equitably pulled apart and distributed between its two halves. Cytokinesis then occurs, dividing the cytoplasm and cell body into two new cells. Mitosis is divided into four major stages that take place after interphase (Figure \(\PageIndex{3}\)) and in the following order: prophase, metaphase, anaphase, and telophase. The process is then followed by cytokinesis.
Prophase is the first phase of mitosis, during which the loosely packed chromatin coils and condenses into visible chromosomes. During prophase, each chromosome becomes visible with its identical partner attached, forming the familiar X-shape of sister chromatids. The nucleolus disappears early during this phase, and the nuclear envelope also disintegrates.
A major occurrence during prophase concerns a very important structure that contains the origin site for microtubule growth. Recall the cellular structures called centrioles that serve as origin points from which microtubules extend. These tiny structures also play a very important role during mitosis. A centrosome is a pair of centrioles together. The cell contains two centrosomes side-by-side, which begin to move apart during prophase. As the centrosomes migrate to two different sides of the cell, microtubules begin to extend from each like long fingers from two hands extending toward each other. The mitotic spindle is the structure composed of the centrosomes and their emerging microtubules.
Near the end of prophase there is an invasion of the nuclear area by microtubules from the mitotic spindle. The nuclear membrane has disintegrated, and the microtubules attach themselves to the centromeres that adjoin pairs of sister chromatids. The kinetochore is a protein structure on the centromere that is the point of attachment between the mitotic spindle and the sister chromatids. This stage is referred to as late prophase or “prometaphase” to indicate the transition between prophase and metaphase.
Metaphase is the second stage of mitosis. During this stage, the sister chromatids, with their attached microtubules, line up along a linear plane in the middle of the cell. A metaphase plate forms between the centrosomes that are now located at either end of the cell. The metaphase plate is the name for the plane through the center of the spindle on which the sister chromatids are positioned. The microtubules are now poised to pull apart the sister chromatids and bring one from each pair to each side of the cell.
Anaphase is the third stage of mitosis. Anaphase takes place over a few minutes, when the pairs of sister chromatids are separated from one another, forming individual chromosomes once again. These chromosomes are pulled to opposite ends of the cell by their kinetochores, as the microtubules shorten. Each end of the cell receives one partner from each pair of sister chromatids, ensuring that the two new daughter cells will contain identical genetic material.
Telophase is the final stage of mitosis. Telophase is characterized by the formation of two new daughter nuclei at either end of the dividing cell. These newly formed nuclei surround the genetic material, which uncoils such that the chromosomes return to loosely packed chromatin. Nucleoli also reappear within the new nuclei, and the mitotic spindle breaks apart, each new cell receiving its own complement of DNA, organelles, membranes, and centrioles. At this point, the cell is already beginning to split in half as cytokinesis begins.
The cleavage furrow is a contractile band made up of microfilaments that forms around the midline of the cell during cytokinesis. (Recall that microfilaments consist of actin.) This contractile band squeezes the two cells apart until they finally separate. Two new cells are now formed. One of these cells (the “stem cell”) enters its own cell cycle; able to grow and divide again at some future time. The other cell transforms into the functional cell of the tissue, typically replacing an “old” cell there.
Imagine a cell that completed mitosis but never underwent cytokinesis. In some cases, a cell may divide its genetic material and grow in size, but fail to undergo cytokinesis. This results in larger cells with more than one nucleus. Usually this is an unwanted aberration and can be a sign of cancerous cells.
Cell Cycle Control
A very elaborate and precise system of regulation controls direct the way cells proceed from one phase to the next in the cell cycle and begin mitosis. The control system involves molecules within the cell as well as external triggers. These internal and external control triggers provide “stop” and “advance” signals for the cell. Precise regulation of the cell cycle is critical for maintaining the health of an organism, and loss of cell cycle control can lead to cancer.
Mechanisms of Cell Cycle Control
As the cell proceeds through its cycle, each phase involves certain processes that must be completed before the cell should advance to the next phase. A checkpoint is a point in the cell cycle at which the cycle can be signaled to move forward or stopped. At each of these checkpoints, different varieties of molecules provide the stop or go signals, depending on certain conditions within the cell. A cyclin is one of the primary classes of cell cycle control molecules. A cyclin-dependent kinase (CDK) is one of a group of molecules that work together with cyclins to determine progression past cell checkpoints. Major cyclin-CDK complexes involved in regulating the cell cycle include Cyclin D - CDK4, Cyclin E - CDK2, Cyclin A - CDK2, and Cyclin B - CDK1 (Figure \(\PageIndex{4}\)). By interacting with many additional molecules, these triggers push the cell cycle forward unless prevented from doing so by “stop” signals, if for some reason the cell is not ready. At the G 1 checkpoint, the cell must be ready for DNA synthesis to occur. At the G 2 checkpoint the cell must be fully prepared for mitosis. Even during mitosis, a crucial stop and go checkpoint in metaphase ensures that the cell is fully prepared to complete cell division. The metaphase checkpoint ensures that all sister chromatids are properly attached to their respective microtubules and lined up at the metaphase plate before the signal is given to separate them during anaphase.
The Cell Cycle Out of Control: Implications
Most people understand that cancer or tumors are caused by abnormal cells that multiply continuously. If the abnormal cells continue to divide unstopped, they can damage the tissues around them, spread to other parts of the body, and eventually result in death. In healthy cells, the tight regulation mechanisms of the cell cycle prevent this from happening, while failures of cell cycle control can cause unwanted and excessive cell division. Failures of control may be caused by inherited genetic abnormalities that compromise the function of certain “stop” and “go” signals. Environmental insult that damages DNA can also cause dysfunction in those signals. Often, a combination of both genetic predisposition and environmental factors lead to cancer.
The process of a cell escaping its normal control system and becoming cancerous may actually happen throughout the body quite frequently. Fortunately, certain cells of the immune system are capable of recognizing cells that have become cancerous and destroying them. However, in certain cases the cancerous cells remain undetected and continue to proliferate. If the resulting tumor does not pose a threat to surrounding tissues, it is said to be benign and can usually be easily removed. If capable of damage, the tumor is considered malignant and the patient is diagnosed with cancer.
HOMEOSTATIC IMBALANCES
Cancer Arises from Homeostatic Imbalances
Cancer is an extremely complex condition, capable of arising from a wide variety of genetic and environmental causes. Typically, mutations or aberrations in a cell’s DNA that compromise normal cell cycle control systems lead to cancerous tumors. Cell cycle control is an example of a homeostatic mechanism that maintains proper cell function and health. While progressing through the phases of the cell cycle, a large variety of intracellular molecules provide stop and go signals to regulate movement forward to the next phase. These signals are maintained in an intricate balance so that the cell only proceeds to the next phase when it is ready. This homeostatic control of the cell cycle can be thought of like a car’s cruise control. Cruise control will continually apply just the right amount of acceleration to maintain a desired speed, unless the driver hits the brakes, in which case the car will slow down. Similarly, the cell includes molecular messengers, such as cyclins, that push the cell forward in its cycle.
In addition to cyclins, a class of proteins that are encoded by genes called proto-oncogenes provide important signals that regulate the cell cycle and move it forward. Examples of proto-oncogene products include cell-surface receptors for growth factors, or cell-signaling molecules, two classes of molecules that can promote DNA replication and cell division. In contrast, a second class of genes known as tumor suppressor genes sends stop signals during a cell cycle. For example, certain protein products of tumor suppressor genes signal potential problems with the DNA and thus stop the cell from dividing, while other proteins signal the cell to die if it is damaged beyond repair. Some tumor suppressor proteins also signal a sufficient surrounding cellular density, which indicates that the cell need not presently divide. The latter function is uniquely important in preventing tumor growth: normal cells exhibit a phenomenon called “contact inhibition;” thus, extensive cellular contact with neighboring cells causes a signal that stops further cell division.
These two contrasting classes of genes, proto-oncogenes and tumor suppressor genes, are like the accelerator and brake pedal of the cell’s own “cruise control system,” respectively. Under normal conditions, these stop and go signals are maintained in a homeostatic balance. Generally speaking, there are two ways that the cell’s cruise control can lose control: a malfunctioning (overactive) accelerator, or a malfunctioning (underactive) brake. When compromised through a mutation, or otherwise altered, proto-oncogenes can be converted to oncogenes, which produce oncoproteins that push a cell forward in its cycle and stimulate cell division even when it is undesirable to do so. For example, a cell that should be programmed to self-destruct (a process called apoptosis) due to extensive DNA damage might instead be triggered to proliferate by an oncoprotein. On the other hand, a dysfunctional tumor suppressor gene may fail to provide the cell with a necessary stop signal, also resulting in unwanted cell division and proliferation.
A delicate homeostatic balance between the many proto-oncogenes and tumor suppressor genes delicately controls the cell cycle and ensures that only healthy cells replicate. Therefore, a disruption of this homeostatic balance can cause aberrant cell division and cancerous growths.
Interactive Lin
Concept Review
The life of cell consists of stages that make up the cell cycle. After a cell is born, it passes through an interphase before it is ready to replicate itself and produce daughter cells. This interphase includes two gap phases (G 1 and G 2 ), as well as an S phase, during which its DNA is replicated in preparation for cell division. The cell cycle is under precise regulation by chemical messengers both inside and outside the cell that provide “stop” and “go” signals for movement from one phase to the next. Failures of these signals can result in cells that continue to divide uncontrollably, which can lead to cancer.
Once a cell has completed interphase and is ready for cell division, it proceeds through four separate stages of mitosis (prophase, metaphase, anaphase, and telophase). Telophase is followed by the division of the cytoplasm (cytokinesis), which generates two daughter cells. This process takes place in all normally dividing cells of the body except for the germ cells that produce eggs and sperm.
Review Questions
Q. Which of the following phases is characterized by preparation for DNA synthesis?
A. G 0
B. G 1
C. G 2
D. S
- Answer
-
Answer: B
Q. A mutation in the gene for a cyclin protein might result in which of the following?
A. a cell with additional genetic material than normal
B. cancer
C. a cell with less genetic material than normal
D. any of the above
- Answer
-
Answer: D
Q. What is a primary function of tumor suppressor genes?
A. stop all cells from dividing
B. stop certain cells from dividing
C. help oncogenes produce oncoproteins
D. allow the cell to skip certain phases of the cell cycle
- Answer
-
Answer: B
Critical Thinking Questions
Q. What would happen if anaphase proceeded even though the sister chromatids were not properly attached to their respective microtubules and lined up at the metaphase plate?
- Answer
-
Answer: One or both of the new daughter cells would accidently receive duplicate chromosomes and/or would be missing certain chromosomes.
Q. What are cyclins and cyclin-dependent kinases, and how do they interact?
- Answer
-
Answer: A cyclin is one of the primary classes of cell cycle control molecules, while a cyclin-dependent kinase is one of a group of molecules that work together with cyclins to determine progression past cell checkpoints. By interacting with many additional molecules, these triggers push the cell cycle forward unless prevented from doing so by “stop” signals, if for some reason the cell is not ready.
Glossary
- anaphase
- third stage of mitosis (and meiosis), during which sister chromatids separate into two new nuclear regions of a dividing cell
- cell cycle
- life cycle of a single cell, from its birth until its division into two new daughter cells
- centromere
- region of attachment for two sister chromatids
- centrosome
- cellular structure that organizes microtubules during cell division
- checkpoint
- progress point in the cell cycle during which certain conditions must be met in order for the cell to proceed to a subsequence phase
- cleavage furrow
- contractile ring that forms around a cell during cytokinesis that pinches the cell into two halves
- cyclin
- one of a group of proteins that function in the progression of the cell cycle
- cyclin-dependent kinase (CDK)
- one of a group of enzymes associated with cyclins that help them perform their functions
- cytokinesis
- final stage in cell division, where the cytoplasm divides to form two separate daughter cells
- diploid
- condition marked by the presence of a double complement of genetic material (two sets of chromosomes, one set inherited from each of two parents)
- G 0 phase
- phase of the cell cycle, usually entered from the G 1 phase; characterized by long or permanent periods where the cell does not move forward into the DNA synthesis phase
- G 1 phase
- first phase of the cell cycle, after a new cell is born
- G 2 phase
- third phase of the cell cycle, after the DNA synthesis phase
- homologous pair
- describes two copies of the same chromosome (not identical), one inherited from each parent
- interphase
- entire life cycle of a cell, excluding mitosis
- kinetochore
- region of a centromere where microtubules attach to a pair of sister chromatids
- metaphase
- second stage of mitosis (and meiosis), characterized by the linear alignment of sister chromatids in the center of the cell
- metaphase plate
- linear alignment of sister chromatids in the center of the cell, which takes place during metaphase
- mitosis
- division of genetic material, during which the cell nucleus breaks down and two new, fully functional, nuclei are formed
- mitotic phase
- phase of the cell cycle in which a cell undergoes mitosis
- mitotic spindle
- network of microtubules, originating from centrioles, that arranges and pulls apart chromosomes during mitosis
- prophase
- first stage of mitosis (and meiosis), characterized by breakdown of the nuclear envelope and condensing of the chromatin to form chromosomes
- S phase
- stage of the cell cycle during which DNA replication occurs
- sister chromatid
- one of a pair of identical chromosomes, formed during DNA replication
- somatic cell
- all cells of the body excluding gamete cells
- telophase
- final stage of mitosis (and meiosis), preceding cytokinesis, characterized by the formation of two new daughter nuclei
Contributors and Attributions
-
OpenStax Anatomy & Physiology (CC BY 4.0). Access for free at https://openstax.org/books/anatomy-and-physiology


2.1: Health Benefits of Physical Activity
Regular physical activity is one of the most important things you can do for your health. In many studies covering a wide range of issues, researchers have focused on exercise, as well as on the more broadly defined concept of physical activity. Exercise is a form of physical activity that is planned, structured, repetitive, and performed with the goal of improving health or fitness. So, although all exercise is physical activity, not all physical activity is exercise . Being more physically active and also doing more exercise can help to:
- Control your weight
- Reduce your risk of cardiovascular disease
- Reduce your risk for type 2 diabetes and metabolic syndrome
- Reduce your risk of some cancers
- Strengthen your bones and muscles
- Improve your mental health and mood
- Improve your ability to do daily activities and prevent falls
- Increase your chances of living longer
Begin Taking Steps to be More Physically Active Everyday
If you’re not sure about becoming active or boosting your level of physical activity because you’re afraid of getting hurt, the good news is that moderate-intensity aerobic activity, like brisk walking, is generally safe for most people.
Start slowly. Cardiac events, such as a heart attack, are rare during physical activity. But the risk does go up when you suddenly become much more active than usual. For example, you can put yourself at risk if you don’t usually get much physical activity and then all of a sudden do vigorous-intensity aerobic activity, like shoveling snow. That’s why it’s important to start slowly and gradually increase your level of activity.
If you have a chronic health condition such as arthritis, diabetes, or heart disease, talk with your doctor to find out if your condition limits, in any way, your ability to be active. Then, work with your doctor to come up with a physical activity plan that matches your abilities. If your condition stops you from meeting the minimum Guidelines, try to do as much as you can. What’s important is that you avoid being inactive. Even 60 minutes a week of moderate-intensity aerobic activity is good for you.
The bottom line is – the health benefits of physical activity far outweigh the risks of getting hurt. The following list explains how being more active can positively impact your health:
Control Your Weight
Looking to get to or stay at a healthy weight? Both diet and physical activity play a critical role in controlling your weight. You gain weight when the calories you burn, including those burned during physical activity, are less than the calories you eat or drink. When it comes to weight management, people vary greatly in how much physical activity they need. You may need to be more active than others to achieve or maintain a healthy weight.
To maintain your weight: Work your way up to 150 minutes of moderate-intensity aerobic activity, 75 minutes of vigorous-intensity aerobic activity, or an equivalent mix of the two each week. Strong scientific evidence shows that physical activity can help you maintain your weight over time. However, the exact amount of physical activity needed to do this is not clear since it varies greatly from person to person. It’s possible that you may need to do more than the equivalent of 150 minutes of moderate-intensity activity a week to maintain your weight.
To lose weight and keep it off: You will need a high amount of physical activity unless you also adjust your diet and reduce the amount of calories you’re eating and drinking. Getting to and staying at a healthy weight requires both regular physical activity and a healthy eating plan. The CDC has some great tools and information about nutrition, physical activity and weight loss. For more information, visit
Healthy Weight
.
Reduce Your Risk of Cardiovascular Disease
Heart disease and stroke are two of the leading causes of death in the United States. But following the Guidelines and getting at least 150 minutes a week (2 hours and 30 minutes) of moderate-intensity aerobic activity can put you at a lower risk for these diseases. You can reduce your risk even further with more physical activity. Regular physical activity can also lower your blood pressure and improve your cholesterol levels.
Reduce Your Risk of Type 2 Diabetes and Metabolic Syndrome
Regular physical activity can reduce your risk of developing type 2 diabetes and metabolic syndrome.  Research shows that lower rates of these conditions are seen with 120 to 150 minutes (2 hours to 2 hours and 30 minutes) a week of at least moderate-intensity aerobic activity. And the more physical activity you do, the lower your risk will be.
Already have type 2 diabetes? Regular physical activity can help control your blood glucose levels. To find out more, visit Managing Diabetes .
Reduce Your Risk of Some Cancers
Being physically active lowers your risk for two types of cancer: colon and breast. Research shows that:
- Physically active people have a lower risk of colon cancer than do people who are not active.
- Physically active women have a lower risk of breast cancer than do people who are not active.
Reduce your risk of endometrial and lung cancer. Although the research is not yet final, some findings suggest that your risk of endometrial cancer and lung cancer may be lower if you get regular physical activity compared to people who are not active.
Improve your quality of life. If you are a cancer survivor, research shows that getting regular physical activity not only helps give you a better quality of life, but also improves your physical fitness.
Strengthen Your Bones and Muscles
As you age, it’s important to protect your bones, joints and muscles. Not only do they support your body and help you move, but keeping bones, joints and muscles healthy can help ensure that you’re able to do your daily activities and be physically active. Research shows that doing aerobic, muscle-strengthening and bone-strengthening physical activity of at least a moderately-intense level can slow the loss of bone density that comes with age.
Hip fracture is a serious health condition that can have life-changing negative effects, especially if you’re an older adult. But research shows that people who do 120 to 300 minutes of at least moderate-intensity aerobic activity each week have a lower risk of hip fracture.
Regular physical activity helps with arthritis and other conditions affecting the joints. If you have arthritis, research shows that doing 130 to 150 (2 hours and 10 minutes to 2 hours and 30 minutes) a week of moderate-intensity, low-impact aerobic activity can not only improve your ability to manage pain and do everyday tasks, but it can also make your quality of life better.
Build strong, healthy muscles. Muscle-strengthening activities can help you increase or maintain your muscle mass and strength. Slowly increasing the amount of weight and number of repetitions you do will give you even more benefits, no matter your age
Improve Your Mental Health and Mood
Regular physical activity can help keep your thinking, learning, and judgment skills sharp as you age. It can also reduce your risk of depression and may help you sleep better. Research has shown that doing aerobic or a mix of aerobic and muscle-strengthening activities 3 to 5 times a week for 30 to 60 minutes can give you these mental health benefits. Some scientific evidence has also shown that even lower levels of physical activity can be beneficial.

How does this relate to physical activity? If you’re a physically active middle-aged or older adult, you have a lower risk of functional limitations than people who are inactive
Already have trouble doing some of your everyday activities? Aerobic and muscle-strengthening activities can help improve your ability to do these types of tasks.
Are you an older adult who is at risk for falls? Research shows that doing balance and muscle-strengthening activities each week along with moderate-intensity aerobic activity, like brisk walking, can help reduce your risk of falling.
Increase Your Chances of Living Longer
Science shows that physical activity can reduce your risk of dying early from the leading causes of death, like heart disease and some cancers. This is remarkable in two ways:
- Only a few lifestyle choices have as large an impact on your health as physical activity. People who are physically active for about 7 hours a week have a 40 percent lower risk of dying early than those who are active for less than 30 minutes a week.
- You don’t have to do high amounts of activity or vigorous-intensity activity to reduce your risk of premature death. You can put yourself at lower risk of dying early by doing at least 150 minutes a week of moderate-intensity aerobic activity.
Everyone can gain the health benefits of physical activity – age, ethnicity, shape or size do not matter.
Contributors and Attributions
Public domain content
- Benefits of Physical Activity. Authored by: Centers for Disease Control and Prevention. Provided by: U.S. Department of Health and Human Services. Located at: https://www.cdc.gov/physicalactivity/basics/pa-health/index.htm . License: Public Domain: No Known Copyright


16.2: Galvanic cells and Electrodes
It is physically impossible to measure the potential difference between a piece of metal and the solution in which it is immersed. We can, however, measure the difference between the potentials of two electrodes that dip into the same solution, or more usefully, are in two different solutions. In the latter case, each electrode-solution pair constitutes an oxidation-reduction half cell , and we are measuring the sum of the two half-cell potentials .
This arrangement is called a galvanic cell . A typical cell might consist of two pieces of metal, one zinc and the other copper, each immersed each in a solution containing a dissolved salt of the corresponding metal. The two solutions are separated by a porous barrier that prevents them from rapidly mixing but allows ions to diffuse through.
If we connect the zinc and copper by means of a metallic conductor, the excess electrons that remain when Zn 2 + ions emerge from the zinc in the left cell would be able to flow through the external circuit and into the right electrode, where they could be delivered to the Cu 2 + ions which become "discharged", that is, converted into Cu atoms at the surface of the copper electrode. The net reaction is the oxidation of zinc by copper(II) ions:
\[Zn_{(s)} + Cu^{2+} \rightarrow Zn^{2+} + Cu_{(s)}\]
but this time, the oxidation and reduction steps (half reactions) take place in separate locations:
|
left electrode:
|
Zn (s) → Zn 2 + + 2 e – | oxidation |
|
right electrode:
|
Cu 2 + + 2 e – → Cu (s) | reduction |
Electrochemical cells allow measurement and control of a redox reaction
The reaction can be started and stopped by connecting or disconnecting the two electrodes. If we place a variable resistance in the circuit, we can even control the rate of the net cell reaction by simply turning a knob. By connecting a battery or other source of current to the two electrodes, we can force the reaction to proceed in its non-spontaneous, or reverse direction. By placing an ammeter in the external circuit, we can measure the amount of electric charge that passes through the electrodes, and thus the number of moles of reactants that get transformed into products in the cell reaction.
Electric charge q is measured in coulombs. The amount of charge carried by one mole of electrons is known as the Faraday , which we denote by F . Careful experiments have determined that 1 F = 96467 C. For most purposes, you can simply use 96,500 Coulombs as the value of the faraday. When we measure electric current, we are measuring the rate at which electric charge is transported through the circuit. A current of one ampere corresponds to the flow of one coulomb per second.
Charge Transport within the Cell
For the cell to operate, not only must there be an external electrical circuit between the two electrodes, but the two electrolytes (the solutions) must be in contact. The need for this can be understood by considering what would happen if the two solutions were physically separated. Positive charge (in the form of Zn 2 + ) is added to the electrolyte in the left compartment, and removed (as Cu 2 + ) from the right side, causing the solution in contact with the zinc to acquire a net positive charge, while a net negative charge would build up in the solution on the copper side of the cell. These violations of electroneutrality would make it more difficult (require more work) to introduce additional Zn 2 + ions into the positively-charged electrolyte or for electrons to flow into right compartment where they are needed to reduce the Cu 2 + ions, thus effectively stopping the reaction after only a chemically insignificant amount has taken place.
In order to sustain the cell reaction, the charge carried by the electrons through the external circuit must be accompanied by a compensating transport of ions between the two cells. This means that we must provide a path for ions to move directly from one cell to the other. This ionic transport involves not only the electroactive species Cu 2 + and Zn 2 + , but also the counterions , which in this example are nitrate, NO 3 - . Thus an excess of Cu 2 + in the left compartment could be alleviated by the drift of these ions into the right side, or equally well by diffusion of nitrate ions to the left. More detailed studies reveal that both processes occur, and that the relative amounts of charge carried through the solution by positive and negative ions depends on their relative mobilities , which express the velocity with which the ions are able to make their way through the solution. Since negative ions tend to be larger than positive ions, the latter tend to have higher mobilities and carry the larger fraction of charge.
In the simplest cells, the barrier between the two solutions can be a porous membrane, but for precise measurements, a more complicated arrangement, known as a salt bridge , is used. The salt bridge consists of an intermediate compartment filled with a concentrated solution of KCl and fitted with porous barriers at each end. The purpose of the salt bridge is to minimize the natural potential difference, known as the junction potential , that develops (as mentioned in the previous section) when any two phases (such as the two solutions) are in contact. This potential difference would combine with the two half-cell potentials so as introduce a degree of uncertainty into any measurement of the cell potential. With the salt bridge, we have two liquid junction potentials instead of one, but they tend to cancel each other out.
Cell description conventions
In order to make it easier to describe a given electrochemical cell, a special symbolic notation has been adopted. In this notation the cell we described above would be
Zn (s) | Zn 2 + (aq) || Cu 2 + (aq) | Cu (s)
There are several other conventions relating to cell notation and nomenclature that you are expected to know:
- The anode is where oxidation occurs, and the cathode is the site of reduction. In an actual cell, the identity of the electrodes depends on the direction in which the net cell reaction is occurring.
- If electrons flow from the left electrode to the right electrode (as depicted in the above cell notation) when the cell operates in its spontaneous direction, the potential of the right electrode will be higher than that of the left, and the cell potential will be positive.
- "Conventional current flow" is from positive to negative, which is opposite to the direction of the electron flow. This means that if the electrons are flowing from the left electrode to the right, a galvanometer placed in the external circuit would indicate a current flow from right to left.
Electrodes and Electrode Reactions
An electrode reaction refers to the net oxidation or reduction process that takes place at an electrode. This reaction may take place in a single electron-transfer step, or as a succession of two or more steps. The substances that receive and lose electrons are called the electroactive species.
This process takes place within the very thin interfacial region at the electrode surface, and involves quantum-mechanical tunneling of electrons between the electrode and the electroactive species. The work required to displace the H 2 O molecules in the hydration spheres of the ions constitutes part of the activation energy of the process.
In the example of the Zn/Cu cell we have been using, the electrode reaction involves a metal and its hydrated cation; we call such electrodes metal-metal ion electrodes. There are a number of other kinds of electrodes which are widely encountered in electrochemistry and analytical chemistry.
Ion-ion Electrodes
Many electrode reactions involve only ionic species, such as \(Fe^{2+}\) and \(Fe^{3+}\). If neither of the electroactive species is a metal, some other metal must serve as a conduit for the supply or removal of electrons from the system. In order to avoid complications that would arise from electrode reactions involving this metal, a relatively inert substance such as platinum is commonly used. Such a half cell would be represented as
Pt(s) | Fe 3 + (aq) , Fe 2 + (aq) || ...
and the half-cell reaction would be
\[Fe^{2+}(aq) \rightarrow Fe^{3+} (aq) + e^-\]
The reaction occurs at the surface of the electrode (Fig 4 above). The electroactive ion diffuses to the electrode surface and adsorbs (attaches) to it by van der Waals and Coulombic forces . In doing so, the waters of hydration that are normally attached to any ionic species must be displaced. This process is always endothermic, sometimes to such an extent that only a small fraction of the ions be able to contact the surface closely enough to undergo electron transfer, and the reaction will be slow. The actual electron-transfer occurs by quantum-mechanical tunnelling .
Gas Electrodes
Some electrode reactions involve a gaseous species such as \(H_2\), \(O_2\), or \(Cl_2\). Such reactions must also be carried out on the surface of an electrochemically inert conductor such as platinum. A typical reaction of considerable commercial importance is
\[Cl^-(aq) \rightarrow ½ Cl_2(g) + e^- \]
Similar reactions involving the oxidation of \(Br_2\) or \(I_2\) also take place at platinum surfaces.
Insoluble–salt Electrodes
A typical electrode of this kind consists of a silver wire covered with a thin coating of silver chloride, which is insoluble in water. The electrode reaction consists in the oxidation and reduction of the silver:
\[AgCl(s) + e^– → Ag(s) + Cl^–(aq)\]
The half cell would be represented as
\[ ... || Cl^– (aq) | AgCl (s) | Ag (s)\]
Although the usefulness of such an electrode may not be immediately apparent, this kind of electrode finds very wide application in electrochemical measurements, as we shall see later.
Reference Electrodes
In most electrochemical experiments our interest is concentrated on only one of the electrode reactions. Since all measurements must be on a complete cell involving two electrode systems, it is common practice to employ a reference electrode as the other half of the cell. The major requirements of a reference electrode are that it be easy to prepare and maintain, and that its potential be stable. The last requirement essentially means that the concentration of any ionic species involved in the electrode reaction must be held at a fixed value. The most common way of accomplishing this is to use an electrode reaction involving a saturated solution of an insoluble salt of the ion. One such system, the silver-silver chloride electrode has already been mentioned:
\[Ag | AgCl(s) | Cl^–(aq) || ...\]
\[Ag(s) + Cl^–(aq) →AgCl(s) + e^–\]
This electrode usually takes the form of a piece of silver wire coated with AgCl. The coating is done by making the silver the anode in an electrolytic cell containing HCl; the Ag + ions combine with Cl – ions as fast as they are formed at the silver surface.
The other common reference electrode is the calomel electrode ; calomel is the common name for mercury(I) chloride. Such a half cell would be represented as
\[Hg | Hg^{2+}(aq) | KCl || ...\]
and the half-cell reaction would be
\[Hg(l) + Cl^– → ½ HgCl2(s) + e^–\]
The potentials of both of these electrodes have been very accurately determined against the hydrogen electrode. The latter is seldom used in routine electrochemical measurements because it is more difficult to prepare; the platinum surface has to be specially treated by preliminary electrolysis. Also, there is need for a supply of hydrogen gas which makes it somewhat cumbersome and hazardous.
Summary and additional notes
Make sure you thoroughly understand the following essential ideas which have been presented above. It is especially important that you know the precise meanings of all the highlighted terms in the context of this topic.
- A galvanic cell (sometimes more appropriately called a voltaic cell ) consists of two half-cells joined by a salt bridge or some other path that allows ions to pass between the two sides in order to maintain electroneutrality .
- The conventional way of representing an electrochemical cell of any kind is to write the oxidation half reaction on the left and the reduction on the right. Thus for the reaction
Zn (s) + Cu 2 + → Zn 2 + + Cu (s)
we write
Zn (s) | Zn 2 + (aq) || Cu 2 + (aq) | Cu (s)
in which the single vertical bars represent phase boundaries . The double bar denotes a liquid-liquid boundary which in laboratory cells consists of a salt bridge or in ion-permeable barrier. If the net cell reaction were written in reverse, the cell notation would become
Cu (s) | Cu 2 + (aq) || Zn 2+ (aq) | Zn (s)
Remember: the R eduction process is always shown on the R ight .
-
at the electrode surface. The energy required to displace water molecules from the hydration shell of an ion as it approaches the electrode surface constitutes an
activation energy
which can slow down the process. Even larger activation energies (and slower reactions) occur when a molecule such as O
2
is formed or consumed.


21.10: Clear and Effective Sentences
This section will help you write strong sentences that convey your meaning clearly and concisely. See Editing Focus: Sentence Structure for a related discussion and practice on effective sentences.
Emphasis
The most emphatic place in a sentence is the end. To achieve the strongest emphasis, end with the idea you want readers to remember. Place introductory, less important, or contextual information earlier in the sentence. Consider the differences in these two sentences.
Less Emphatic Angel needs to start now if he wants to have an impact on his sister’s life.
More Emphatic If Angel wants to have an impact on his sister’s life, he needs to start now
Concrete Nouns
General nouns name broad classes or categories of things ( man , dog , city ); concrete nouns refer to particular things ( Michael , collie , Chicago ). Concrete nouns provide a more vivid and lively reading experience because they create stronger images that activate readers’ senses. The examples below show how concrete nouns, combined with specific details, can make writing more engaging.
All General Nouns Approaching the library, I see people and dogs milling about outside , but no subjects to write about. I’m tired from my walk and go inside.
Revised with Concrete Nouns Approaching Brandon Library , I see skateboarders and bikers weaving through students who talk in clusters on the library steps . A friendly collie waits for its owner to return. Subjects to write about? Nothing strikes me as especially interesting. Besides, my heart is still pounding from the walk up the hill. I wipe my sweaty forehead and go inside.
Active Voice
Active voice refers to the way a writer uses verbs in a sentence. Verbs have two “voices”: active and passive. In the active voice , the subject of the sentence acts—the subject performs the action of the verb. In the passive voice , the subject receives the action, and the object actually becomes the subject. Although some passive sentences are necessary and clear, a paper full of passive-voice constructions lacks vitality and becomes wordy.
Active-voice verbs make something happen. By using active verbs wherever possible, you will create stronger, clearer, and more concise sentences.
Passive Voice On the post-training survey, the anti-harassment tutorial was rated highly informative by employees.
Revised in Active Voice On the post-training survey, employees rated the anti-harassment tutorial highly informative.
Conciseness
Concise writing considers the importance of every word. Editing sentences for emphasis, concrete nouns, and active voice will help you write clearly and precisely, as will the following strategies. To be concise, eliminate wasted words and filler— not ideas, information, description, or details that will interest readers or help them follow your thoughts. (For more on conciseness, see Editing Focus: Sentence Structure .)
Use Action Verbs
Using action verbs is one of the most direct ways to cut unneeded words. Whenever you find a phrase like the ones below, consider substituting an action verb.
| Instead of the phrase . . . | Use an action verb |
| reach a decision, come to a decision | decide |
| made a choice | chose |
| hold a meeting | meet |
| arrive at a conclusion | conclude |
| have a discussion | discuss |
Cut Unnecessary Words and Phrases
Eliminate words and phrases that do not add meaning. Consider the following sentences, which say essentially the same thing.
Wordy In almost every situation that I can think of, with few exceptions, it will make good sense for you to look for as many places as possible to cut out needless, redundant, and repetitive words and phrases from the papers, reports, paragraphs, and sentences you write for college assignments. (49 words)
Concise Whenever possible, cut needless words and phrases from your college writing. (11 words)
The wordy sentence is full of early-draft language in three chunks. The first chunk comes at the beginning of the sentence. Notice how In almost every situation that I can think of, with few exceptions, it will make good sense for you to look for as many places as possible is reduced to Whenever possible in the concise sentence.
The second chunk of the wordy sentence is needless , redundant , and repetitive . The concise version reduces those four words to needless because the words have the same meaning. The third chunk of the wordy sentence comes at the end. Notice how papers , reports , paragraphs , and sentences you write for college assignments is reduced to your college writing . The meaning, although expanded to all writing, remains the same.
The following phrases are common fillers that add nothing to meaning. They should be avoided.
- a person by the name of
- for all intents and purposes
- in a manner of speaking
- more or less
Some common filler phrases have single-word alternatives, which are preferable.
| Replace a common filler phrase . . . | With a single word |
| at all times | always |
| at the present time | now |
| at this point in time | now |
| for the purpose of | for |
| due to the fact that | because |
| the reason being | because |
| in the final analysis | finally |
| last but not least | finally |
Avoid there is/there are and it is
Starting a sentence with there is , there are , or it is can be useful to draw attention to a change in direction. However, starting a sentence with one of these phrases often forces you into a wordy construction. Wordiness means the presence of verbal filler; it does not mean the number of words, the amount of description, or the length of a composition. (For more on these constructions, see Editing Focus: Sentence Structure .)
Wordy There is often uncertainty about whether or not employees are required to turn on their cameras during online meetings, and there are some employees who don’t. However, it is the expectation of employers that cameras be turned on.
Concise Employees are often uncertain whether they must turn on their cameras during online meetings, and some don’t . However, employers expect cameras to be turned on.
Parallelism
Within a sentence, parallelism —the repetition of a word or grammatical construction— creates symmetry and balance, makes an idea easier to remember, and sounds pleasing to the ear. In the first example below, the parallelism is established by the repetition of the phrase beginning with who . In the second example, the parallelism is created by the underlined nouns.
Unparallel After 25 years, the battle over the reintroduction of wolves continues between environmental activists, who support it , and hunters and people who own cattle ranches and are opposed .
Parallel After 25 years, the battle over the reintroduction of wolves continues between environmental activists, who support it , and cattle ranchers and hunters, who oppose it .
Unparallel Exercises that improve core strength include crunches , leg lifts , and when you do push-ups and planks .
Parallel Exercises that improve core strength include crunches , leg lifts , push-ups , and planks .
Variety
Varying the length and structure of sentences makes your writing more interesting to read.
Simple Sentences
A simple sentence has one idea expressed in a single main clause (also known as an independent clause). A main clause contains a subject and a predicate and can stand alone as a sentence. A simple sentence can be short or long, as shown in the examples below. The phrases in the long sentence add information, but the sentence remains a simple sentence nonetheless because it has only one clause.
The coronavirus spread around in the world in 2020.
School-age children and college students were pushed into virtual learning environments in March 2020, with schools closing for unspecified lengths of time.
Compound Sentences
A compound sentence contains two or more main clauses that are equally important to the meaning of the sentence. (A main clause contains a subject and a predicate and can stand alone as a sentence.) You can create compound sentences in the following ways:
- Compound Sentence Using a Coordinating Conjunction
Create a compound sentence by using a coordinating conjunction — for, and, nor, but, or, yet, or so (fanboys) —to join main clauses. To remember the coordinating conjunctions, use the mnemonic device fanboys .
Restaurants and small retailers experienced steep drops in revenue during the pandemic, and many were forced to close.
Restaurants and small retailers experienced steep drops in revenue during the pandemic, yet many survived the downturn.
- Compound Sentence Using a Semicolon
A semicolon can join two main clauses that are closely related in meaning. When using a semicolon, you must have a complete sentence before and after it.
Restaurants and small retailers experienced steep drops in revenue during the pandemic; many were forced to close.
- Compound Sentence Using a Semicolon and Transitional Word or Phrase
A transitional words or phrases such as however, in fact, meanwhile, therefore, consequently, as a result, instead, or furthermore indicates the relation of two or more equally important ideas in the main clauses.
Restaurants and small retailers experienced steep drops in revenue during the pandemic; however, many survived the downturn.
Complex Sentences
A complex sentence contains one main clause (a clause that contains a subject and a predicate and can stand alone as a sentence) and one or more subordinate clauses (also known as dependent clauses). Subordinate clauses begin with a subordinating word or phrase such as although, because, even if, when, whenever, since, as though, whether, as long as, until, or while . The main clause expresses the main idea of the sentence, and the subordinate clause expresses the less important idea. Like a main clause, a subordinate clause has a subject and verb; however, unlike a main clause, it cannot stand alone as a sentence. A subordinate clause punctuated as a sentence is a type of sentence fragment. The subordinate clauses in the following sentences are underlined.
Although the federal government provided financial assistance , the money came too late for many businesses. When schools and universities shut down in March of 2020 , students had to learn at home, a situation that proved challenging for many households .
Compound-Complex Sentences
A compound-complex sentence contains two or more main clauses (clauses that contain subjects and predicates and can stand alone as sentences) and one or more subordinate clauses (clauses that begin with a subordinating word such as although, because, even if, when, whenever, since, as though, whether, as long as, until, and while ). A compound-complex sentence is an effective structure to use when you want to express three or more ideas in a single sentence. The example sentence has two main clauses (double underline) and three subordinate clauses (single underline).
When school districts reopened , parents had to decide whether they wanted their children to attend classes in person , and they had to be ready for classes to move online if there were outbreaks of the coronavirus in their community .


1.7: Single Sign On (SSO) - External Authentication
There are two primary mechanisms for logging into LibreTexts libraries. The first is direct authentication that involves adding your username and password directly on the horizontal grey bar on the top of any LibreTexts page. The alternative is the Single Sign On (SSO) infrastructure that allows users to use campus credentials to authenticate. This is available for any user with an email that is either Google- or Microsoft enabled (most US campuses). To use the SSO, administrators have to disable your direct log in option for security reasons.
Below is a step-by-step procedure to log in (using a Kansas State University text as an example; other campuses will follow a similar procedure, but with different branding).
Step 1
On the LibreText resource homepage ( https://med.libretexts.org/Courses/Kansas_State_University/FNDH_400%3A_Human_Nutrition_(Lindshield)
select the SSO (single sign on) icon.
Step 2
-
Select Microsoft Active Directory (or Google if your campus supports that)
Step 3
On the next page sign in with your ksu.edu email address (or any campus email)
Step 4
Sign into to you K-State account like you normally would for email etc.
Step 5
On the next screen you can select “Don’t show this again” and click yes to hopefully reduce the number of times you have to sign in.
Step 6
You then are signed into LibreTexts, you can tell because the SSO icon is green.
Step 7 (Optional for ADAPT questions)
In order, to access the embedded questions, you need to sign in to ADAPT as prompted on the first question you encounter and try to access by clicking Campus Login.
- You will sign in similar to the process you did above except you will also need to copy and paste (or enter) the following access code: 3f57cc06ed9f
- After signing into ADAPT this first time, you should not have to use this access code any longer to sign into see and complete the embedded questions.


17.4: Sample Documentation
Sample Documentation of Expected Findings
Tube Feeding
Patient’s abdomen is soft, nondistended, and bowel sounds are present in all four quadrants. Head of the bed elevated to 45 degrees. Placement of tube verified with measurement of the tube at the nares at 55 cm and gastric aspirate had a pH 4. Patient tolerated 240 mL of tube feeding by gravity followed by a 30-mL water flush.
Tube Irrigation
Irrigated NG tube with 30 mL of water with no resistance. Patient tolerated flush without symptoms.
Tube Suctioning
NG tube connected to low-intermittent suction (LIS) at 60 mmHg. Output from the NGT is green with a volume of 100 mLs in eight hours.
Sample Documentation of Unexpected Findings
Tube Feeding
Patient’s abdomen is soft, nondistended, and bowel sounds are present in all four quadrants. Head of the bed elevated to 45 degrees. Placement of tube verified with measurements of tube at nares at 55 cm and gastric aspirate with a pH of 4. After 100 mLs of tube feeding was infused, the patient complained of feeling full and nauseated. Tube feeding infusion stopped and the head of the bed maintained at a 45-degree angle. Will reassess gastric volume residual in one hour and determine if tube feeding will be resumed.
Tube Irrigation
Attempted to irrigate NG tube with 30 mLs of water, but resistance was felt with procedure. Unable to inject water flush into the NGT. Dr. Smith notified of findings at 1320.
Tube Suctioning
Patient’s abdomen is slightly distended and bowel sounds are hypoactive in all four quadrants. Patient reports increased nausea over the last two hours. NGT connected to low-intermittent suction (LIS) at 60 mmHg. Output from the NGT is mahogany colored with a volume of 800 mLs in eight hours. Dr. Smith notified of findings at 1640.


2.3.2: "On Monsieur's Departure"
(ca. 1582)
I grieve and dare not show my discontent,
I love and yet am forced to seem to hate,
I do, yet dare not say I ever meant,
I seem stark mute but inwardly do prate.
I am and not, I freeze and yet am burned,
Since from myself another self I turned.
My care is like my shadow in the sun,
Follows me flying, flies when I pursue it,
Stands and lies by me, doth what I have done.
His too familiar care doth make me rue it.
No means I find to rid him from my breast,
Till by the end of things it be supprest.
Some gentler passion slide into my mind,
For I am soft and made of melting snow;
Or be more cruel, love, and so be kind.
Let me or float or sink, be high or low.


14.9: Introduction to Typical Communication Flows
What you’ll learn to do: differentiate between typical communication flows within an organization
How does communication move within an organization? Is there a pattern to the kinds of communication that come from senior management, from middle management and from workers? What do typical information flows tell us about the kind of information that is communicated within the organization? And what does it tell us about who controls the flow of information? This section describes how information in an organization follows established and predictable routes.
Contributors and Attributions
CC licensed content, Original
- Introduction to Typical Communication Flows. Authored by : John/Lynn Bruton and Lumen Learning. License : CC BY: Attribution


2.2: Graphs of the Sine and Cosine Functions
-
- Last updated
- Save as PDF
Learning Objectives
- Graph variations of \(y=\sin( x )\) and \(y=\cos( x )\).
- Use phase shifts of sine and cosine curves.
White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can be seen only when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow.
Light waves can be represented graphically by the sine function. In the chapter on Trigonometric Functions , we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions.
Graphing Sine and Cosine Functions
Recall that the sine and cosine functions relate real number values to the \(x\)- and \(y\)-coordinates of a point on the unit circle.  Let’s start with the sine function. We can create a table of values and use them to sketch a graph. Table \(\PageIndex{1}\) lists some of the values for the sine function on a unit circle.
| \(x\) | \(0\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) | \(\dfrac{2\pi}{3}\) | \(\dfrac{3\pi}{4}\) | \(\dfrac{5\pi}{6}\) | \(\pi\) |
|---|---|---|---|---|---|---|---|---|---|
| \(\sin(x)\) | \(0\) | \(\frac{1}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{\sqrt{2}}{2}\) | \(\dfrac{1}{2}\) | \(0\) |
Plotting the points from the table and continuing along the x -axis gives the shape of the sine function. See Figure \(\PageIndex{2}\).
Notice how the sine values are positive between \(0\) and \(\pi\), which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between \(\pi\) and \(2\pi\), which correspond to the values of the sine function in quadrants III and IV on the unit circle. See Figure \(\PageIndex{3}\).
Now let’s take a similar look at the cosine function . Again, we can create a table of values and use them to sketch a graph. Table \(\PageIndex{2}\) lists some of the values for the cosine function on a unit circle.
| \(x\) | \(0\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) | \(\frac{2\pi}{3}\) | \(\dfrac{3\pi}{4}\) | \(\dfrac{5\pi}{6}\) | \(\pi\) |
|---|---|---|---|---|---|---|---|---|---|
| \(\cos(x)\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\dfrac{1}{2}\) | \(0\) | \(-\dfrac{1}{2}\) | \(-\dfrac{\sqrt{2}}{2}\) | \(-\dfrac{\sqrt{3}}{2}\) | \(-1\) |
As with the sine function, we can plots points to create a graph of the cosine function as in Figure \(\PageIndex{4}\).
Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval \([ −1,1 ]\).
In both graphs, the shape of the graph repeats after \(2\pi\), which means the functions are periodic with a period of \(2\pi\). A periodic function is a function for which a specific horizontal shift , \(P\), results in a function equal to the original function: \(f(x+P)=f(x)\) for all values of \(x\) in the domain of \(f\). When this occurs, we call the smallest such horizontal shift with \(P>0\) the period of the function. Figure \(\PageIndex{5}\) shows several periods of the sine and cosine functions.
Looking again at the sine and cosine functions on a domain centered at the \(y\)-axis helps reveal symmetries. As we can see in Figure \(\PageIndex{6}\), the sine function is symmetric about the origin. Recall from The Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because \(\sin(−x)=−\sin\space x\). Now we can clearly see this property from the graph.
Figure \(\PageIndex{7}\) shows that the cosine function is symmetric about the \(y\)-axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that \(\cos(−x)=\cos\space x\).
CHARACTERISTICS OF SINE AND COSINE FUNCTIONS
The sine and cosine functions have several distinct characteristics:
- They are periodic functions with a period of \(2\pi\).
- The domain of each function is \((−\infty,\infty)\) and the range is \([ −1,1 ]\).
- The graph of \(y=\sin\space x\) is symmetric about the origin, because it is an odd function.
- The graph of \(y=\cos\space x\) is symmetric about they- \(y\)-axis, because it is an even function.
Investigating Sinusoidal Functions
As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or cosine function is known as a sinusoidal function . The general forms of sinusoidal functions are
\[y=A\sin(Bx−C)+D\]
and
\[y=A\cos(Bx−C)+D\]
Determining the Period of Sinusoidal Functions
Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.
In the general formula, \(B\) is related to the period by \(P=\dfrac{2\pi}{|B|}\). If \(|B|>1\), then the period is less than \(2\pi\) and the function undergoes a horizontal compression, whereas if \(| B |<1\), then the period is greater than \(2\pi\) and the function undergoes a horizontal stretch. For example, \(f(x)=\sin(x)\), \(B=1\), so the period is \(2\pi\),which we knew. If \(f(x)=\sin(2x)\), then \(B=2\), so the period is \(\pi\) and the graph is compressed. If \(f(x)=\sin\left(\dfrac{x}{2}\right)\), then \(B=\dfrac{1}{2}\), so the period is \(4\pi\) and the graph is stretched. Notice in Figure \(\PageIndex{8}\) how the period is indirectly related to \(|B|\).
PERIOD OF SINUSOIDAL FUNCTIONS
If we let \(C=0\) and \(D=0\) in the general form equations of the sine and cosine functions, we obtain the forms
- \(y=A\sin(Bx)\)
- \(y=A\cos(Bx)\)
The period is \(\dfrac{2\pi}{|B|}\).
Example \(\PageIndex{1}\): Identifying the Period of a Sine or Cosine Function
Determine the period of the function \(f(x)=\sin\left(\dfrac{\pi}{6}x\right)\).
Solution
Let’s begin by comparing the equation to the general form \(y=A\sin(Bx)\).
In the given equation, \(B=\dfrac{\pi}{6}\), so the period will be
\[ \begin{align*} P&=\dfrac{2\pi}{|B|} \\[4pt] &=\dfrac{2\pi}{\dfrac{\pi}{6}} \\ &=2\pi ⋅ \dfrac{6}{\pi} \\[4pt] &=12 \end{align*}\]
Exercise \(\PageIndex{1}\)
Determine the period of the function \(g(x)=\cos(\frac{x}{3})\).
- Answer
-
\(6\pi\)
Determining Amplitude
Returning to the general formula for a sinusoidal function, we have analyzed how the variable \(B\) relates to the period. Now let’s turn to the variable \(A\) so we can analyze how it is related to the amplitude , or greatest distance from rest. \(A\) represents the vertical stretch factor, and its absolute value \(|A|\) is the amplitude. The local maxima will be a distance \(|A|\) above the vertical midline of the graph, which is the line \(x=D\); because \(D=0\) in this case, the midline is the x -axis. The local minima will be the same distance below the midline. If \(| A |>1\), the function is stretched. For example, the amplitude of \(f(x)=4 sin x\) is twice the amplitude of
\(f(x)=2\sin x\)
If \(| A |<1\), the function is compressed. Figure \(\PageIndex{9}\) compares several sine functions with different amplitudes.
AMPLITUDE OF SINUSOIDAL FUNCTIONS
If we let \(C=0\) and \(D=0\) in the general form equations of the sine and cosine functions, we obtain the forms
\[\begin{align} y=A\sin(Bx)\text { and } y=A\cos(Bx) \end{align}\]
The amplitude is \(A\), and the vertical height from the midline is \(|A|\). In addition, notice in the example that
\[|A| = amplitude = \dfrac{1}{2}∣maximum − minimum|\]
Example \(\PageIndex{2}\): Identifying the Amplitude of a Sine or Cosine Function
What is the amplitude of the sinusoidal function \(f(x)=−4\sin(x)\)? Is the function stretched or compressed vertically?
Solution
Let’s begin by comparing the function to the simplified form \(y=A\sin(Bx)\).
In the given function, \(A=−4\), so the amplitude is \(| A |=| −4 |=4\). The function is stretched.
Analysis
The negative value of \(A\) results in a reflection across the \(x\)-axis of the sine function, as shown in Figure \(\PageIndex{10}\).
Exercise \(\PageIndex{2}\)
What is the amplitude of the sinusoidal function \(f(x)=\frac{1}{2}\sin(x)\)? Is the function stretched or compressed vertically?
- Answer
-
\(\frac{1}{2}\) compressed
Analyzing Graphs of Variations of \(y = \sin\space x\) and \(y = \cos\space x\)
Now that we understand how \(A\) and \(B\) relate to the general form equation for the sine and cosine functions, we will explore the variables \(C\) and \(D\). Recall the general form:
\[y=A\sin(Bx-C)+D\qquad \text{ and } \qquad y=A\cos(Bx-C)+D\]
or
\[y=A\sin\left (B\left (x-\dfrac{C}{B} \right ) \right )+D \qquad \text{ and } \qquad y=A\cos\left (B\left (x-\dfrac{C}{B} \right ) \right )+D\]
The value \(\frac{C}{B}\) for a sinusoidal function is called the phase shift , or the horizontal displacement of the basic sine or cosine function. If \(C>0\), the graph shifts to the right. If \(C<0\), the graph shifts to the left. The greater the value of \(| C |\), the more the graph is shifted. Figure \(\PageIndex{11}\) shows that the graph of \(f(x)=\sin(x−\pi)\) shifts to the right by \(\pi\) units, which is more than we see in the graph of \(f(x)=\sin\left(x−\frac{\pi}{4}\right)\), which shifts to the right by \(\frac{\pi}{4}\) units.
While \(C\) relates to the horizontal shift, \(D\) indicates the vertical shift from the midline in the general formula for a sinusoidal function. See Figure \(\PageIndex{12}\). The function \(y=\cos(x)+D\) has its midline at \(y=D\).
Any value of \(D\) other than zero shifts the graph up or down. Figure \(\PageIndex{13}\) compares \(f(x)=\sin x\) with \(f(x)=\sin x+2\), which is shifted \(2\) units up on a graph.
VARIATIONS OF SINE AND COSINE FUNCTIONS
Given an equation in the form \(f(x)=A \sin (Bx−C)+D\) or \(f(x)=A \cos (Bx−C)+D\), \(\frac{C}{B}\) is the phase shift and \(D\) is the vertical shift.
Example \(\PageIndex{3}\): Identifying the Phase Shift of a Function
Determine the direction and magnitude of the phase shift for \(f(x)=\sin\left(x+\frac{\pi}{6}\right)−2\).
Solution
Let’s begin by comparing the equation to the general form \(y=A\sin(Bx−C)+D\).
In the given equation, notice that \(B=1\) and \(C=−\frac{\pi}{6}\). So the phase shift is
\[\begin{align*} \dfrac{C}{B}&= -\frac{\frac{\pi}{6}}{1}\\ &= -\frac{\pi}{6} \end{align*}\]
or \(\frac{\pi}{6}\) units to the left.
Analysis
We must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation shows a minus sign before \(C\). Therefore \(f(x)=\sin(x+\frac{\pi}{6})−2\) can be rewritten as \(f(x)=\sin\left(x−\left(−\frac{\pi}{6}\right)\right)−2\). If the value of \(C\) is negative, the shift is to the left.
Exercise \(\PageIndex{3}\)
Determine the direction and magnitude of the phase shift for \(f(x)=3\cos\left(x−\frac{\pi}{2}\right)\).
- Answer
-
\(\frac{\pi}{2}\); right
Example \(\PageIndex{4}\): Identifying the Vertical Shift of a Function
Determine the direction and magnitude of the vertical shift for \(f(x)=\cos(x)−3\).
Solution
Let’s begin by comparing the equation to the general form \(y=A\cos(Bx−C)+D\).
In the given equation, \(D=−3\) so the shift is \(3\) units downward.
Exercise \(\PageIndex{4}\)
Determine the direction and magnitude of the vertical shift for \(f(x)=3\sin(x)+2\).
- Answer
-
\(2\) units up
How to: Given a sinusoidal function in the form \(f(x)=A\sin(Bx−C)+D\),identify the midline, amplitude, period, and phase shift
- Determine the amplitude as \(| A |\).
- Determine the period as \(P=\frac{2\pi}{| B |}\).
- Determine the phase shift as \(\frac{C}{B}\).
- Determine the midline as \(y=D\).
Example \(\PageIndex{5}\): Identifying the Variations of a Sinusoidal Function from an Equation
Determine the midline, amplitude, period, and phase shift of the function \(y=3\sin (2x)+1\).
Solution
Let’s begin by comparing the equation to the general form \(y=A\sin (Bx−C)+D\).
\(A=3\), so the amplitude is \(| A |=3\).
Next, \(B=2\), so the period is \(P=\dfrac{2\pi}{| B |}=\dfrac{2\pi}{2}=\pi\).
There is no added constant inside the parentheses, so \(C=0\) and the phase shift is \(\dfrac{C}{B}=\dfrac{0}{2}=0\).
Finally, \(D=1\), so the midline is \(y=1\).
Analysis
Inspecting the graph, we can determine that the period is \(\pi\), the midline is \(y=1\), and the amplitude is \(3\). See Figure \(\PageIndex{14}\).
Exercise \(\PageIndex{5}\)
Determine the midline, amplitude, period, and phase shift of the function \(y=\frac{1}{2}\cos \left(\frac{x}{3}−\frac{\pi}{3}\right)\).
- Answer
-
midline: \(y=0\); amplitude: \(| A |=\frac{1}{2}\); period: \(P=\frac{2\pi}{| B |}=6π\); phase shift: \(\frac{C}{B}=\pi\)
Example \(\PageIndex{6}\): Identifying the Equation for a Sinusoidal Function from a Graph
Determine the formula for the cosine function in Figure \(\PageIndex{15}\).
Solution
To determine the equation, we need to identify each value in the general form of a sinusoidal function.
\(y=A\sin (Bx−C)+D\)
\(y=A\cos (Bx−C)+D\)
The graph could represent either a sine or a cosine function that is shifted and/or reflected. When \(x=0\), the graph has an extreme point, \((0,0)\). Since the cosine function has an extreme point for \(x=0\), let us write our equation in terms of a cosine function.
Let’s start with the midline. We can see that the graph rises and falls an equal distance above and below \(y=0.5\). This value, which is the midline, is \(D\) in the equation, so \(D=0.5\).
The greatest distance above and below the midline is the amplitude. The maxima are \(0.5\) units above the midline and the minima are \(0.5\) units below the midline. So \(| A |=0.5\). Another way we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is \(1\), so \(| A |=\frac{1}{2}=0.5\). Also, the graph is reflected about the \(x\)-axis so that \(A=−0.5\).
The graph is not horizontally stretched or compressed, so \(B=1\); and the graph is not shifted horizontally, so \(C=0\).
Putting this all together,
\(g(x)=−0.5\cos (x)+0.5\)
Exercise \(\PageIndex{6}\)
Determine the formula for the sine function in Figure \(\PageIndex{16}\).
- Answer
-
\(f(x)=\sin(x)+2\)
Example \(\PageIndex{7}\): Identifying the Equation for a Sinusoidal Function from a Graph
Determine the equation for the sinusoidal function in Figure \(\PageIndex{17}\).
Solution
With the highest value at \(1\) and the lowest value at \(−5\), the midline will be halfway between at \(−2\). So \(D=−2\).
The distance from the midline to the highest or lowest value gives an amplitude of \(| A |=3\).
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. Therefore, \(P=\dfrac{2\pi}{| B |}=6\). Using the positive value for \(B\),we find that
\[\begin{align*} B&=\dfrac{2\pi}{P}\\ &=\dfrac{2\pi}{6}\\ &=\dfrac{\pi}{3} \end{align*}\]
So far, our equation is either \(y=3\sin\left(\dfrac{\pi}{3}x−C\right)−2\) or \(y=3\cos\left(\dfrac{\pi}{3}x−C\right)−2\).For the shape and shift, we have more than one option. We could write this as any one of the following:
- a cosine shifted to the right
- a negative cosine shifted to the left
- a sine shifted to the left
- a negative sine shifted to the right
While any of these would be correct, the cosine shifts are easier to work with than the sine shifts in this case because they involve integer values. So our function becomes
\[\begin{align*} y&=3\cos \left (\frac{\pi}{3}x-\dfrac{\pi}{3} \right )-2 \qquad \text{or} \\ y&=-3\cos \left (\dfrac{\pi}{3}x+\dfrac{2\pi}{3} \right )-2 \end{align*}\]
Again, these functions are equivalent, so both yield the same graph.
Exercise \(\PageIndex{7}\)
Write a formula for the function graphed in Figure \(\PageIndex{18}\).
- Answer
-
two possibilities: \(y=4\sin\left(\dfrac{\pi}{5}x−\dfrac{\pi}{5}\right)+4\) or \(y=−4\sin\left(\dfrac{\pi}{5}x+\dfrac{4\pi}{5}\right)+4\)
Graphing Variations of \(y = \sin\space x\) and \(y = \cos\space x\)
Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create graphs from equations.
Instead of focusing on the general form equations
\(y=A\sin(Bx-C)+D \text{ and } y=A\cos(Bx-C)+D\)
we will let \(C=0\) and \(D=0\) and work with a simplified form of the equations in the following examples.
Given the function \(y=A\sin(Bx)\), sketch its graph.
- Identify the amplitude, \(| A |\).
- Identify the period, \(P=\dfrac{2\pi}{| B |}\).
- Start at the origin, with the function increasing to the right if \(A\) is positive or decreasing if \(A\) is negative.
- At \(x=\dfrac{\pi}{2| B |}\) there is a local maximum for \(A>0\) or a minimum for \(A<0\), with \(y=A\).
- The curve returns to the x -axis at \(x=\dfrac{\pi}{| B |}\).
- There is a local minimum for \(A>0\) (maximum for \(A<0\)) at \(x=\dfrac{3\pi}{2| B |}\) with \(y=–A\).
- The curve returns again to the x -axis at \(x=\dfrac{\pi}{2| B |}\).
Example \(\PageIndex{8}\): Graphing a Function and Identifying the Amplitude and Period
Sketch a graph of \(f(x)=−2\sin\left(\dfrac{\pi x}{2}\right)\).
Solution
Let’s begin by comparing the equation to the form \(y=A\sin(Bx)\).
-
Step 1.
We can see from the equation that \(A=−2\), so the amplitude is 2.
\(|A|=2 \)
-
Step 2.
The equation shows that \(B=\dfrac{\pi}{2}\), so the period is
\[\begin{align*} P&=\dfrac{2\pi}{\dfrac{\pi}{2}}\\ &=2\pi \cdot \dfrac{2}{\pi}\\ &=4 \end{align*}\]
- Step 3. Because \(A\) is negative, the graph descends as we move to the right of the origin.
- Step 4. The \(x\)-intercepts are at the beginning of one period, \(x=0\), the horizontal midpoints are at \(x=2\) and at the end of one period at \(x=4\).
The quarter points include the minimum at \(x=1\) and the maximum at \(x=3\). A local minimum will occur \(2\) units below the midline, at \(x=1\), and a local maximum will occur at \(2\) units above the midline, at \(x=3\). Figure \(\PageIndex{19}\) shows the graph of the function.
Exercise \(\PageIndex{8}\)
Sketch a graph of \( g(x)=−0.8\cos(2x)\). Determine the midline, amplitude, period, and phase shift.
- Answer
-
midline: \(y=0\); amplitude: \(| A |=0.8\); period: \(P=\dfrac{2\pi}{| B |}=\pi\); phase shift: \(\dfrac{C}{B}=0\) or none
How to: Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph
- Express the function in the general form \(y=A\sin(Bx−C)+D\) or \(y=A\cos(Bx−C)+D\).
- Identify the amplitude, \(| A |\).
- Identify the period, \(P=\dfrac{2\pi}{| B |}\).
- Identify the phase shift, \(\dfrac{C}{B}\).
- Draw the graph of \(f(x)=A\sin(Bx)\) shifted to the right or left by \(\dfrac{C}{B}\) and up or down by \(D\).
Example \(\PageIndex{9}\): Graphing a Transformed Sinusoid
Sketch a graph of \(f(x)=3\sin\left(\dfrac{\pi}{4}x−\dfrac{\pi}{4}\right)\).
Solution
- Step 1. The function is already written in general form: \(f(x)=3\sin\left(\dfrac{\pi}{4}x−\dfrac{\pi}{4}\right)\).This graph will have the shape of a sine function, starting at the midline and increasing to the right.
- Step 2. \(| A |=| 3 |=3\). The amplitude is \(3\).
-
Step 3.
Since \(| B |=| \dfrac{\pi}{4} |=\dfrac{\pi}{4}\), we determine the period as follows.
\[\begin{align*} P&=\dfrac{2\pi}{|B|}\\ &=\dfrac{2\pi}{\dfrac{\pi}{4}}\\ &=2\pi \cdot \dfrac{4}{\pi}\\ &=8 \end{align*}\]
The period is \(8\).
-
Step 4.
Since \(C=\dfrac{\pi}{4}\), the phase shift is
\[\dfrac{C}{B}=\dfrac{\dfrac{\pi}{4}}{\dfrac{\pi}{4}}=1\].
The phase shift is \(1\) unit.
- Step 5. Figure \(\PageIndex{21}\) shows the graph of the function.
Exercise \(\PageIndex{9}\)
Draw a graph of \(g(x)=−2\cos\left(\dfrac{\pi}{3}x+\dfrac{\pi}{6}\right)\). Determine the midline, amplitude, period, and phase shift.
- Answer
-
midline: \(y=0\); amplitude: \(| A |=2\); period: \(P=\dfrac{2\pi}{| B |}=6\); phase shift: \(\dfrac{C}{B}=−\dfrac{1}{2}\)
Example \(\PageIndex{10}\): Identifying the Properties of a Sinusoidal Function
Given \(y=−2cos\left(\dfrac{\pi}{2}x+\pi\right)+3\), determine the amplitude, period, phase shift, and horizontal shift. Then graph the function.
Solution
Begin by comparing the equation to the general form.
\(y=A\cos(Bx−C)+D\)
- Step 1. The function is already written in general form.
- Step 2. Since \(A=−2\), the amplitude is \(| A |=2\).
- Step 3. \(| B |=\dfrac{\pi}{2}\), so the period is \(P=\dfrac{2\pi}{| B |}=\dfrac{2\pi}{\dfrac{pi}{2}}=2\pi⋅\dfrac{2}{\pi}=4\). The period is 4.
- Step 4. \(C=−\pi\),so we calculate the phase shift as \(\dfrac{C}{B}=−\dfrac{\pi}{\dfrac{\pi}{2}}=−\pi⋅\dfrac{2}{\pi}=−2\). The phase shift is \(−2\).
- Step 5. \(D=3\),so the midline is \(y=3\), and the vertical shift is up \(3\).
Since \(A\) is negative, the graph of the cosine function has been reflected about the \(x\)-axis.
Figure \(\PageIndex{23}\) shows one cycle of the graph of the function.
Using Transformations of Sine and Cosine Functions
We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, circular motion can be modeled using either the sine or cosine function .
Example \(\PageIndex{11}\): Finding the Vertical Component of Circular Motion
A point rotates around a circle of radius \(3\) centered at the origin. Sketch a graph of the \(y\)-coordinate of the point as a function of the angle of rotation.
Solution
Recall that, for a point on a circle of radius \(r\), the \(y\)-coordinate of the point is \(y=r \sin(x)\), so in this case, we get the equation \(y(x)=3 \sin(x)\). The constant \(3\) causes a vertical stretch of the \(y\)-values of the function by a factor of \(3\), which we can see in the graph in Figure \(\PageIndex{24}\).
Analysis
Notice that the period of the function is still \(2\pi\); as we travel around the circle, we return to the point \((3,0)\) for \(x=2\pi,4\pi,6\pi,\)....Because the outputs of the graph will now oscillate between \(–3\) and \(3\), the amplitude of the sine wave is \(3\).
Exercise \(\PageIndex{10}\)
What is the amplitude of the function \(f(x)=7\cos(x)\)? Sketch a graph of this function.
- Answer
Example \(\PageIndex{12}\): Finding the Vertical Component of Circular Motion
A circle with radius \(3\) ft is mounted with its center \(4\) ft off the ground. The point closest to the ground is labeled \(P\), as shown in Figure \(\PageIndex{26}\). Sketch a graph of the height above the 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 \(1\) ft above the ground, then increase up to \(7\) ft above the ground, and continue to oscillate \(3\) ft above and below the center value of \(4\) ft, as shown in Figure \(\PageIndex{27}\).
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. Let’s 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 \(1\), 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, we find that
\(y=−3\cos(x)+4\)
Exercise \(\PageIndex{11}\)
A weight is attached to a spring that is then hung from a board, as shown in Figure \(\PageIndex{28}\). As the spring oscillates up and down, the position \(y\) of the weight relative to the board ranges from \(–1\) in. (at time \(x=0\)) to \(–7\) in. (at time \(x=π\)) below the board. Assume the position of \(y\) is given as a sinusoidal function of \(x\). Sketch a graph of the function, and then find a cosine function that gives the position \(y\) in terms of \(x\).
- Answer
-
\(y=3\cos(x)−4\)
Example \(\PageIndex{13}\): Determining a Rider’s Height on a Ferris Wheel
The London Eye is a huge Ferris wheel with a diameter of \(135\) meters (\(443\) feet). It completes one rotation every \(30\) minutes. Riders board from a platform \(2\) meters above the ground. Express a rider’s height above ground as a function of time in minutes.
Solution
With a diameter of \(135\) m, the wheel has a radius of \(67.5\) m. The height will oscillate with amplitude \(67.5\) m above and below the center.
Passengers board \(2\) m above ground level, so the center of the wheel must be located \(67.5+2=69.5\) m above ground level. The midline of the oscillation will be at \(69.5\) m.
The wheel takes \(30\) minutes to complete \(1\) revolution, so the height will oscillate with a period of \(30\) minutes.
Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.
- Amplitude: \(67.5\), so \(A=67.5\)
- Midline: \(69.5\), so \(D=69.5\)
- Period:\(30\), so \(B=\dfrac{2\pi}{30}=\dfrac{\pi}{15}\)
- Shape: \(−\cos(t)\)
An equation for the rider’s height would be
\(y=−67.5\cos\left(\dfrac{\pi}{15}t\right)+69.5\)
where \(t\) is in minutes and \(y\) is measured in meters.
Media
Access these online resources for additional instruction and practice with graphs of sine and cosine functions.
- Amplitude and Period of Sine and Cosine
- Translations of Sine and Cosine
- Graphing Sine and Cosine Transformations
- Graphing the Sine Function
Key Equations
| Sinusoidal functions |
\(f(x)=A\sin(Bx−C)+D\) \(f(x)=A\cos(Bx−C)+D\) |
Key Concepts
- Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of \(2\pi\).
- The function \(\sin x\) is odd, so its graph is symmetric about the origin. The function \(\cos x\) is even, so its graph is symmetric about the y -axis.
- The graph of a sinusoidal function has the same general shape as a sine or cosine function.
- In the general formula for a sinusoidal function, the period is \(P=\dfrac{2\pi}{| B |}\). See Example \(\PageIndex{1}\).
- In the general formula for a sinusoidal function, \( | A |\) represents amplitude. If \(| A |>1\), the function is stretched, whereas if \(| A |<1\), the function is compressed. See Example \(\PageIndex{2}\).
- The value \(\dfrac{C}{B}\) in the general formula for a sinusoidal function indicates the phase shift. See Example \(\PageIndex{3}\).
- The value \(D\) in the general formula for a sinusoidal function indicates the vertical shift from the midline. See Example \(\PageIndex{4}\).
- Combinations of variations of sinusoidal functions can be detected from an equation. See Example \(\PageIndex{5}\).
- The equation for a sinusoidal function can be determined from a graph. See Example \(\PageIndex{6}\) and Example \(\PageIndex{7}\).
- A function can be graphed by identifying its amplitude and period. See Example \(\PageIndex{8}\) and Example \(\PageIndex{9}\).
- A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See Example \(\PageIndex{10}\).
- Sinusoidal functions can be used to solve real-world problems. See Example \(\PageIndex{11}\), Example \(\PageIndex{12}\), and Example \(\PageIndex{13}\).


11.1: Physical Fitness
selected template will load here
This action is not available.
Being physically active is one of the most important steps that Americans of all ages can take to improve their health. The benefits of physical activity occur in generally healthy people, in people at risk of developing chronic diseases, and in people with current chronic conditions or disabilities. Studies have examined the role of physical activity in many groups—men and women, children, teens, adults, older adults, people with disabilities, and women during pregnancy and the postpartum period. These studies have focused on the role that physical activity plays in many health outcomes, including:
There have also been additional studies by the Advisory Committee of the U.S. Department of Health and Human Services. The Advisory Committee rated the evidence of health benefits of physical activity as strong, moderate, or weak. To do so, the Committee considered the type, number, and quality of studies available, as well as consistency of findings across studies that addressed each outcome.
The Committee also considered evidence for causality and dose response in assigning the strength-of-evidence rating.


Skip to main content
-
A friend is required to do jury duty and knows you are study the criminal court process. He wants to understand the process better. He asks why there are so many requirements such as the right to a speedy trial, rules of evidence, and the right to a jury trial. Provide him with the reason we have so many protections.
-
Read about
Patterson v. New York,
432 U.S. 197 (1977). In Patterson, the defendant was on trial for murder. New York law reduced murder to manslaughter if the defendant proved extreme emotional disturbance to a preponderance of evidence. Did the US Supreme Court hold that it is constitutional to put this burden on the defense, rather than forcing the prosecution to disprove extreme emotional disturbance beyond a reasonable doubt? Which part of the Constitution did the Court analyze to justify its holding?
 Do you have to present a defense? Explain why you may decide not to have the defendant testify or provide evidence to the jury. Can the jury hold this against the defendant?
-
Legal defenses can mitigate or eliminate guilt in a crime. For example, the district attorney may prove the defendant committed the act (actus reus) but not
mens rea
(guilty mind). Explain how someone who killed another person could be found not guilty. Use a legal defense example that could reduce (mitigate) or eliminate guilt in the case of murder.