3.1: Cultural Safety
It is important to conduct the complete subjective health assessment in a culturally safe manner. Cultural safety refers to the creation of safe spaces for clients to interact with health professionals without judgment, racial reductionism, racialization, or discrimination.
Cultural safety begins with an understanding of dominant cultural structures that serve to exclude persons based on their cultural practices (e.g., beliefs, attire, language) or cultural identity (e.g., ethnicity). For example, dominant structures overtly reproduce exclusion when services are only available in English or French. The system covertly reproduces exclusion when persons are made to feel different, or “other,” for their beliefs or views about health.
When seeking cultural health information or any subjective data from a client, it is important to bring a critical lens that acknowledges colonial history, power inequity, and injustice. Consider why you are asking information: is it for racial reductionism or to better understand the client health and illness profile? As described in Table 3.1 , today, a critical self-reflective approach is more appropriate, where the health professional questions their own assumptions and challenges stereotypes.
| Self-reflective questions | Rationale |
|---|---|
|
How do I describe my own culture? |
Reflecting on how you would respond to a question that you pose to a client is always good practice when self-analyzing. Consider whether you are clear or confused by the question. Dig deep to think about how you define and think about culture. We all practise culture, some being more closely aligned with dominant culture than others. For some, when they align with the dominant culture, they may perceive themselves as not having a culture or not recognizing their own culture. Reflect on what it means to be closely aligned or not with the dominant culture and how this may be perceived in your nursing practice. |
|
What assumptions or biases do I hold about culture? |
Bias becomes problematic when it is acted upon and taken up as legitimacy. Reflecting on how assumptions and bias have influenced your thinking about culture, including primary language, mainstream knowledge, and concepts about health and illness, is important for you to understand how to create culturally safe spaces. In addition to thinking about “setting aside” your bias or assumptions, give thought to the belief system that has contributed to your own biases. What has informed your thinking? |
|
How might historical legacies affect and inform the healthcare I provide? |
Historically, healthcare was a vehicle for imperial and religious practices, and these legacies have contributed to the context of healthcare today. Western healthcare embodies a belief system predicated on empirics, while elsewhere on the globe other practices are taken up. Recognizing that all clients bring a historical cultural context to their interactions with health professionals is helpful in creating cultural safe spaces. |
|
How does my use of language affect cultural safety? |
We all become indoctrinated to language. It is often emblematic of a greater belief system. This indoctrination influences how we practise (e.g., how we refer to people and our interactions with clients). For example, labelling someone by their race or religion reinforces “othering,” where their identity is imposed on them. “Objectified language” refers to language that leads to the objectification of someone (e.g., “the diabetic,” “the alcoholic”) implying that the person is devoid of nuance beyond their pathology and defined by a biomedical system. When thinking about your own language, consider its effects on the client. |
Test Yourself
Query \(\PageIndex{1}\)


23.4: Summary
23.1 Physical Assessment of the Newborn
It is important for the nurse to know how to complete a comprehensive and accurate physical assessment on a newborn. Performing the physical assessment with the new parents in the room is a good opportunity for the nurse to educate the parents on safety issues in handling and caring for their newborn. New parents and even experienced ones will look to the nurse for guidance before discharge about how to integrate the newborn into their family, how to feed and soothe their newborn, and how to recognize when their newborn needs help. The nurse’s assessment should always work from the least invasive to the more invasive to avoid irritating the newborn and to maintain thermoregulation. Assessment begins with passive observations, then moves into a hands-on physical examination. The nurse should have a broad knowledge base of expected findings in a newborn, including weights and measurements, vital signs, commonly elicited reflexes, and expected and unexpected variations to look out for. Nurses are the experts in newborn care and should advocate for safe, timely discharge home for the newborn and their family.
23.2 Estimation of Gestational Age of the Newborn and Newborn Behavioral Assessment
A complete newborn assessment must consider the gestational age of a newborn. While that can be obtained by the date of a pregnant person’s last menstrual period, there are times when that is not accurate or is unknown. At those times, the nurse can utilize the Ballard scoring tool to determine the newborn’s approximate gestational age, allowing the clinical team to be prepared for any age-related issues that the newborn may experience. One of the greatest responsibilities of the nurse in caring for a newborn and their family is health promotion and risk reduction, which is the purpose of estimating gestational age.
The Brazelton Neonatal Behavioral Assessment gives nurses an opportunity to assess the newborn’s personality but also to assess the bonding that is (or is not) occurring between the parents and the newborn. Early intervention and a referral to a social worker is key if there is a problem, and providing resources before the couplet leaves the hospital is best practice. Additionally, this assessment gives the parents time to observe and interact with their newborn’s personality for the first time, while allowing the nurse one more chance for anticipatory guidance and discharge education.


8.8: The Rashidun Caliphs
Muhammad did not formally appoint a successor, or khalifa in Arabic, and no clear replacement arose to lead the Muslim community forward at the time of his death. In fact, the umma divided into three groups, with each willing to appoint their own successor to the Prophet. Emerging as a vocal leader at this critical juncture, ‘Umar, one of Muhammad’s closest companions, convinced the majlis , or elders of the community, to elect Abu Bakr by consensus as a compromise candidate. Abu Bakr had been Muhammad’s closest friend; Muhammad’s marriage of political alliance to ‘A’isha , Abu Bakr’s daughter, further solidified their relationship.
The election of Abu Bakr (632 – 634) brought much-needed stability and an almost democratic form of government to Islam. As caliph, Abu Bakr held together the converts to Islam by deploying the forces at his disposal, thus cementing his authority among the Arabian tribes. He prevented any rebellious Muslim tribes from reverting to the worship of their traditional tribal gods, as they were wont to do. Abu Bakr died in 634, two years after the Prophet Muhammad had died.
The majlis chose ‘Umar (634 – 644), a close friend of Abu Bakr, to be the next caliph. ‘Umar had been the military power behind Abu Bakr. A dynamic and uncompromising leader, ‘Umar recognized the necessity of expansion northward to achieve various ends. First, he sought to subdue the security threat of raiding nomads, many of which remained a law unto themselves. Second, in his struggle to contain discontent, he used the cohesive element of jihad to unite the Muslim community against unbelievers and expand God’s dominion. (The Arabic term of jihad actually refers to a “struggle,” usually against spiritual impurity, often known as “greater jihad ,” and is associated with fulfilling God’s objectives here on earth. The “lesser jihad ,” alternatively, is a physical struggle against the unbelievers of the Dar al-Harb , or Abode of War, until it is absorbed into the Dar al-Islam , or Abode of Islam, where believers were free to practice their faith as members of the predominant faith. Of note is the fact that Muhammad did not consider jihad important enough to make one of the pillars of Islam.) Third, ‘Umar understood the importance of plunder for the nascent caliphate. Troops received four-fifths of the loot from conquest; the remainder of the revenue went to him to be dispersed amongst the neediest members in the Islamic community.
‘Umar directed the full might of Islam northward against the Eastern Roman Empire, sometimes referred to as the Byzantine Empire. In 634, their first encounter took place in southern Palestine. The ensuing Battle of Ajnadayn was a decisive victory for the Muslims and a major loss for Emperor Heraclius. Two years later, an outnumbered Muslim army defeated the Eastern Roman Empire yet again at the Battle of Yarmouk , located on the eponymous river, somewhere between Damascus and Jerusalem. In both instances, the Byzantines relied on their slow, heavy cavalry, whereas the Arabs capitalized on their light armor and their superior mobility. The Muslims realized that they could not just charge the East Roman lines; they showed their tactical superiority by flanking the Byzantines and executing a successful rearguard action instead. These victories opened up greater Syria to Muslim conquest. Antioch, Aleppo, and Jerusalem fell to the Muslims not long thereafter. ‘Umar appointed Mu‘awiya, a member of the Meccan Umayyad aristocracy to govern Syria at his behest.
Once he dealt with the increasingly vulnerable Byzantines in the Levant, ‘Umar directed his army to the east against the Sasanian Empire of Persia. In 636, fighting along the banks of the Euphrates River, a smaller Arab force triumphed over the Persians, at the Battle of Qadisiya . After successive days of exhaustive combat, the Muslims took advantage of environmental conditions and their light cavalry’s mobility when they chased a dust storm and took the Sasanids by surprise.
To save their empire, the Persians mounted a failed counterattack. In 642, Umar’s army eventually defeated the forces of the Sasanian Emperor Yazdagird III at the Battle of Nahavand, situated deep in Iran’s Zagros Mountains. Yazdagird fled to the east as a fugitive, and, in 651, met his death at the hands of a local miller who killed the emperor in order to rob him of his belongings.
In 639, General ‘Amr petitioned ‘Umar for permission to invade Egypt and eventually persuaded the caliph that he could easily take Egypt so gained his reluctant consent. In 641, he received a message from ‘Umar recalling his forces. The general ignored the order and seized Egypt with just a few hundred soldiers. With promises of toleration, ‘Amr convinced the Egyptian Coptic majority to side with him against the Greek Orthodox ruling minority, whose Patriarch Cyrus had been actively persecuting the Copts as followers of a Christian heresy that failed to recognize the Holy Trinity.
Clearly outnumbered Muslim armies thus successfully defeated two long-standing empires in the span of just a few decades. Several explanations help us understand the rapid expansion of Islam during this period. One concept, termed the vacuum theory, posits that the Byzantine and Persian empires had been severely weakened from near-continuous fighting, dating back decades prior to the rise of Islam, so they both suffered from the fatigue of war. Islam, therefore, occupied the vacuum of political power resulting from the collapse of these two exhausted empires.
The success of Muslim military strategy offers a second explanation. While Byzantine forces adopted a defensive stance on the battlefield, the Arabs employed more aggressive tactics, making use of their mobile light cavalry against their enemies’ heavily armored armies. Once victorious, the Arabs populated garrison cities on the frontier, called amsar , with Muslims. These military settlements provided security, served as logistical loci, and discouraged Muslim troops from mingling with the locals. The caliphs thereby prevented their warriors being assimilated into the communities of the conquered while also preventing soldiers from disturbing the peace. Fustat in Egypt, as well as Kufa and Basra in Iraq, were the largest of the amsar . From bases like these, the Arabs could expand and consolidate their hold over the frontier.
Religion also provided an impetus for the expansion of Islam. Fearing that internal tribal divisions threatened the early Islamic state, ‘Umar united the Muslims through their common Islamic theology and faced them against a common enemy. Dedicated to the expansion of Islam, Muslims used the concept of jihad as a way to unify the umma , or Islamic community, against a foreign foe. Faith motivated the troops, who were zealous and determined to fight.
Simple economics also served as a primary motivating factor in the expansion of Islam. For one, Muslim rulers applied the jizya , an annual tax levied on non-Muslims, to newly-conquered lands. The money derived from conquest functioned as a driving force in the growth of the caliphate. With the expectation of material reward, soldiers could earn money for their service. While the practice of dividing the spoils of war amongst the soldiers continued under ‘Umar, he also started offering salaries to his troops, determining salaries according to the length of service.
The Muslims further exploited the internal divisions of targeted societies, as exemplified in Egypt, where the Coptic Christian majority, together with a large Jewish minority in Alexandria, had suffered under the rule of an oppressive Greek Orthodox Christian minority but gained autonomy and toleration within an Islamic state. And in Syria, another monophysite Christian minority called the Syrian Orthodox Church, or Jacobites, collaborated with the Muslims and hastened the collapse of the Byzantines. All these factors led the early Islamic state to expand exponentially.
In 644, an Iranian captive from the Persian campaign stabbed ‘Umar to death. His successor, ‘Uthman (644 – 656), was an elderly man from the Umayyad Clan who won a contentious election over ‘Ali. ‘Ali possessed all of the ‘Alid bona fides. ‘Ali was not only son of Muhammad’s early protector, Abu Thalib; he was also the Prophet’s cousin and son-in-law. He had married Muhammad’s daughter Fatima; together, they had two sons, Hasan and Husayn. ‘Ali had also earned a well-deserved reputation as a virtuous Muslim. One of the first converts to Islam, he had journeyed with Muhammad on most of his expeditions and fought against the Meccans. Finally, ‘Ali also served as a valued advisor to the early caliphs on questions of dogma.
Two factions formed in the wake of questions over ‘Uthman’s succession, thus initiating the development of a division within Islam. One faction was a group of ‘Alids who believed that ‘Ali should inherit the mantle of Islam and referred to traditions suggesting that Muhammad had proclaimed to the faithful that ‘Ali should be his successor. The amsar followed the ‘Alids and later adopted the Shi‘a appellation. The other faction, the Umayyads contended that the method of appointing successors should be by consensus, as was done with the first caliphs. Mostly based in Mecca, they later identified as Sunnis. Over time, these factional differences became increasingly difficult to bridge.
Although ‘Uthman, one of the Prophet’s first converts, was a pious Muslim, he was a corrupt administrator. He displayed nepotistic tendencies that gave precedence to the Meccan elite, a practice that diverged from ‘Umar’s policies of favoring soldiers who had been the first to respond to the call to action. ‘Umar’s beneficiaries had usually originated from lesser tribes, those too weak to constitute a coherent threat to the establishment; by contrast, ‘Uthman’s appointees were members of the Meccan elite who generally pursued policies benefiting the Umayyad merchants of Mecca.
Government also began to disintegrate under ‘Uthman’s rule, as opposition and instability plagued his tenure as caliph. He managed to offend three separate groups of Muslims. The first of these were the older, pious Muslims, who hailed from Medina. They resented how the hated Umayyads had taken over the same umma that they had previously persecuted and had once tried to destroy. Second were the Quran reciters. When ‘Uthman commissioned and authorized a single official version of the holy text, an act for which he received many accolades, the Quran reciters lost the opportunity for gainful employment. Third were a disgruntled contingent of ‘Alids who called for ‘Uthman to resign and advocated the election of ‘Ali. Their discontent culminated in 656, when resentful devotees of ‘Ali from Egypt broke into ‘Uthman’s home in Medina and assassinated him, purportedly while he was reading the Quran. They then hastily arranged for the election of ‘Ali as ‘Uthman’s successor.
Thrice rejected by the majlis in favor of the first three caliphs, ‘Ali (656 – 661) reluctantly accepted the position of leader of the Islamic community. His selection represented a victory for the faction of legitimists disappointed in the earlier choice of ‘Uthman. ‘Ali assumed the role of caliph amid high expectations, for he was a pious and generous man. Yet the caliphate suffered under his rule. During this time of instability, he constantly had to suppress revolts. For example, tensions between the supporters of ‘Ali and the family of ‘Uthman eventually erupted into the first civil war in Islam. In 656, at the Battle of the Camel , ‘Ali engaged the combined forces of the Prophet’s favored wife, ‘A’isha , and her associates, Talha and Zubayr, who were both relatives of ‘Uthman. Because ‘Ali had failed to bring the dead caliph’s assassins to justice; these three together demanded satisfaction for his death.
The conspirators challenged ‘Ali near the garrison city of Basra, in southern Iraq, before he had the chance to move the caliphate from Medina to the sympathetic military settlement of Kufa. A first, diplomacy seemed to prevail, as ‘Ali sought to avoid bloodshed by negotiating. He succeeded in convincing the three to lay down their arms; however, a group later known as Kharijis conspired to undermine their reconciliation and set fire to the tents in both camps in the dead of the night. Pandemonium ensued. Because of this single impetuous action, both parties thought the other side had flouted the agreement, committing a violation of trust. During the ensuing battle, ‘A’isha was pushed into the middle of the fray on the back of a camel, as was Arab custom.  The repercussions of his victory reverberated across the Islamic world, as older Muslim men castigated ‘A’isha for her part in the conflict and suggested that women should not play a role in public life.
This threat was not the only one ‘Ali faced, for he also had to contend with Mu‘awiya, ‘Uthman’s cousin and former governor of Syria. Conspicuously absent from ‘Ali’s new administration, Mu‘awiya refused to pay homage to ‘Ali and asserted his own independence in Syria. He also echoed the accusations of ‘A’isha, Talha, and Zubayr, as members of Mu‘awiya’s Umayyad Clan had expressed dismay about the quick election of ‘Ali, and questions still lingered over the new caliph’s part in ‘Uthman’s death. ‘Ali’s failure to act against ‘Uthman’s assassins proved his culpability, Mu‘awiya and the Umayyads, and Mu‘awiya asserted the traditional Arab custom of exacting revenge on one’s enemies.
His conflict with ‘Ali culminated in 657 when they met at the Battle of Siffin , on the Euphrates River in northern Syria. After months of clashes, ‘Ali agreed to arbitration with Mu‘awiya. Still preferring negotiation over bloodshed, ‘Ali had been of the opinion that Muslims should never take up arms against fellow Muslims. His willingness to negotiate with Mu‘awiya, however, caused some of Ali’s own soldiers to defect and adopt the appellation of Kharijis, from kharaja , meaning “to depart.” The first sect in Islam, they departed from Ali because they believed that “judgement belongs to God alone” (Quran 6:57); they saw ‘Ali’s willingness to negotiate with Mu‘awiya as somehow reducing the role of God in determining a successor. In lieu of arbitration, they thought that God would determine the rightful successor by influencing the outcome on the field of battle.


7.4: Creating an Academic Plan
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All of this self and career exploration and analysis will allow you to create an academic plan to achieve the educational credential(s) required for the career you plan to pursue. You will use this academic plan to map out your career path, at least while you’re in college. Your academic plan will list all of the classes you’re required to take to earn the credential you’re pursuing, whether you’re planning to earn a diploma, certificate or associate degree. In fact, if you're using financial aid, you are required to have an approved academic plan in Navigate/Degree Works. After earning your associate degree, you may plan to transfer to earn your bachelors degree, and some of you will continue for your masters and doctoral degrees.
Your academic plan will consist of pre-requisite courses, general education courses and major requirement courses. Prerequisites are courses required before taking other higher-level courses, and some examples may be developmental math, reading and writing plus 100-level introductory courses. Taking prerequisites first at the beginning of your academic plan will help you move forward evenly and strategically ensuring you have the basic knowledge upon which to build in your higher-level classes.
General education courses ensure you have a well-rounded education by learning about topics related and unrelated to your major. Examples of general education courses include first-year seminar, liberal arts, social sciences, physical sciences and physical education. General education courses provide the “soft skills” employers look for in addition to career-specific knowledge. Soft skills such as effective written and oral communication, critical thinking and information literacy (how to do effective research) often are viewed by employers as just as important as the career-specific knowledge required in an occupation.
Major requirement courses cover the specific content required for your career field such as business, psychology, human services, nursing and welding. These courses teach you the specific skills you will use in your day-to-day work. These courses often come in the second half of your academic plan although some majors may require introductory level major courses in your first or second semesters.
You do have some help creating your academic plan. Every HACC student has an assigned academic advisor who is charged with explaining your academic requirements and assisting you to make decisions about what classes to take when. Your academic advisor may be a professor in your intended field or a professional advisor in a centralized office. Consult with your academic advisor each semester to ensure you’re following the most effective course sequence and to ensure your major requirements haven’t changed. See the next section for details about how academic advising is structured at HACC.


3.2: Nutritional Concerns
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That slower rate of growth is accompanied by a reduced appetite between the ages of 2 and 6. This change can sometimes be surprising to parents and lead to the development of poor eating habits. However, children between the ages of 2 and 3 need 1,000 to 1,400 calories, while children between the ages of 4 and 8 need 1,200 to 2,000 calories (Mayo Clinic, 2016a). 2
Caregivers who have established a feeding routine with their child can find the reduction in appetite a bit frustrating and become concerned that the child is going to starve. However, by providing adequate, sound nutrition, and limiting sugary snacks and drinks, the caregiver can be assured that 1) the child will not starve; and 2) the child will receive adequate nutrition. Preschoolers can experience iron deficiencies if not given well-balanced nutrition or if they are given too much milk as calcium interferes with the absorption of iron in the diet as well.
Caregivers need to keep in mind that they are setting up taste preferences at this age. Young children who grow accustomed to high fat, very sweet and salty flavors may have trouble eating foods that have more subtle flavors such as fruits and vegetables. Consider the following advice about establishing eating patterns for years to come (Rice, F.P., 1997). Notice that keeping mealtime pleasant, providing sound nutrition and not engaging in power struggles over food are the main goals.3
2. Children’s Development by Ana R. Leon is licensed under CC BY 4.0
Lifespan Development: A Psychological Perspective by Martha Lally and Suzanne Valentine-French is licensed under CC BY-NC-SA 3.0
3. Children’s Development by Ana R. Leon is licensed under CC BY 4.0


22.3: Assessments Related to Airway Suctioning
Subjective Assessment
If appropriate, perform a focused interview collecting a brief history of respiratory conditions and assess for feelings of shortness of breath (dyspnea), sputum production, and coughing.
Objective Assessment
Prior to suctioning, a baseline assessment for indications of respiratory distress and the need for suctioning should be obtained and documented, including, but not limited to, the following:
- Secretions from the mouth and/or tracheal stoma
- Auscultation of lung sounds
- Heart rate
- Respiratory rate
- Cardiac rhythm
- Oxygen saturation
- Skin color and perfusion
- Effectiveness of cough [1]
Prepare the patient by explaining the procedure and providing adequate sedation and pain relief as needed. Place the patient in semi-Fowler’s position if conscious or in a lateral position facing you if they are unconscious. While suctioning the patient, if signs of worsening respiratory distress occur, stop the procedure and request emergency assistance. The following should be monitored during and following the procedure:
- Lung sounds
- Skin color
- Breathing pattern and rate
- Oxygenation (pulse oximeter)
- Pulse rate
- Dysrhythmias if electrocardiogram is available
- Color, consistency, and volume of secretions
- Presence of bleeding or evidence of physical trauma
- Subjective response including pain
- Cough
- Laryngospasm (spasm of the vocal cords that can result in airway obstruction) [2]
After completing suctioning, the outcomes from the procedure should be evaluated and documented, including the following:
- Improvement of lung sounds
- Removal of secretions
- Improvement of pulse oximetry
- Decreased work of breathing
- Stabilized respiratory rate
- Decreased dyspnea
Be aware that the patient’s lung sounds may not clear completely after suctioning, but the removal of secretions should improve the patency of the patient’s airway.
Potential complications resulting from this procedure include nasal irritation/bleeding, gagging/vomiting, discomfort and pain, and uncontrolled coughing. Potential adverse reactions include mucosal hemorrhage, laceration of nasal turbinate, perforation of the pharynx, hypoxia/hypoxemia, cardiac dysrhythmias/arrest, bradycardia, elevated blood pressure, hypotension, respiratory arrest, laryngospasm, bronchoconstriction, bronchospasm, hospital-acquired infection, atelectasis, increased intracranial pressure, and pneumothorax.
- American Association for Respiratory Care. (2004). AARC clinical practice guideline: Nasotracheal suctioning - 2004 revision & update. Respiratory Care, 49 (9), 1080-1084. https://www.aarc.org/wp-content/uploads/2014/08/09.04.1080.pdf ↵
- American Association for Respiratory Care. (2004). AARC clinical practice guideline: Nasotracheal suctioning - 2004 revision & update. Respiratory Care, 49 (9), 1080-1084. https://www.aarc.org/wp-content/uploads/2014/08/09.04.1080.pdf ↵


5.2.3: Money and Debts
Lesson
Let's apply what we know about signed numbers to money.
Exercise \(\PageIndex{1}\): Concert Tickets
Priya wants to buy three tickets for a concert. She has earned $135 and each ticket costs $50. She borrows the rest of the money she needs from a bank and buys the tickets.
- How can you represent the amount of money that Priya has after buying the tickets?
- How much more money will Priya need to earn to pay back the money she borrowed from the bank?
- How much money will she have after she pays back the money she borrowed from the bank?
Exercise \(\PageIndex{2}\): Cafeteria Food Debt
At the beginning of the month Kiran had $24 in his school cafeteria account. Use a variable to represent the unknown quantity in each transaction below and write an equation to represent it. Then, represent each transaction on a number line. What is the unknown quantity in each case?
- In the first week he spent $16 on lunches. How much was in his account then?
- Then he deposited some more money and his account balance was $28. How much did he deposit?
- Then he spent $34 on lunches the next week. How much was in his account then?
- Then he deposited enough money to pay off his debt to the cafeteria. How much did he deposit?
- Explain why it makes sense to use a negative number to represent Kiran's account balance when he owes money.
Exercise \(\PageIndex{3}\): Bank Statement
Here is a bank statement.
- If we put withdrawals and deposits in the same column, how can they be represented?
- Andre withdraws $40 to buy a music player. What is his new balance?
- If Andre deposits $100 in this account, will he still be in debt? How do you know?
Are you ready for more?
The national debt of a country is the total amount of money the government of that country owes. Imagine everyone in the United States was asked to help pay off the national debt. How much would each person have to pay?
Summary
Banks use positive numbers to represent money that gets put into an account and negative numbers to represent money that gets taken out of an account. When you put money into an account, it is called a deposit . When you take money out of an account, it is called a withdrawal .
People also use negative numbers to represent debt. If you take out more money from your account than you put in, then you owe the bank money, and your account balance will be a negative number to represent that debt. For example, if you have $200 in your bank account, and then you write a check for $300, you will owe the bank $100 and your account balance will be -$100.
| starting balance | deposits and withdrawals | new balance |
|---|---|---|
| \(0\) | \(50\) | \(0+50\) |
| \(50\) | \(150\) | \(50+150\) |
| \(200\) | \(-300\) | \(200+(-300)\) |
| \(-100\) |
In general, you can find a new account balance by adding the value of the deposit or withdrawal to it. You can also tell quickly how much money is needed to repay a debt using the fact that to get to zero from a negative value you need to add its opposite.
Glossary Entries
Definition: Deposit
When you put money into an account, it is called a deposit .
For example, a person added $60 to their bank account. Before the deposit, they had $435. After the deposit, they had $495, because \(435+60=495\).
Definition: Withdrawal
When you take money out of an account, it is called a withdrawal.
For example, a person removed $25 from their bank account. Before the withdrawal, they had $350. After the withdrawal, they had $325, because \(350-25=325\).
Practice
Exercise \(\PageIndex{4}\)
The table shows five transactions and the resulting account balance in a bank account, except some numbers are missing. Fill in the missing numbers.
| transaction amount | account balance | |
|---|---|---|
| transaction 1 | \(200\) | \(200\) |
| transaction 2 | \(-147\) | \(53\) |
| transaction 3 | \(90\) | |
| transaction 4 | \(-229\) | |
| transaction 5 | \(0\) |
Exercise \(\PageIndex{5}\)
- Clare has $54 in her bank account. A store credits her account with a $10 refund. How much does she now have in the bank?
- Mai's bank account is overdrawn by $60, which means her balance is -$60. She gets $85 for her birthday and deposits it into her account. How much does she now have in the bank?
- Tyler is overdrawn at the bank by $180. He gets $70 for his birthday and deposits it. What is his account balance now?
- Andre has $37 in his bank account and writes a check for $87. After the check has been cashed, what will the bank balance show?
Exercise \(\PageIndex{6}\)
Last week, it rained \(g\) inches. This week, the amount of rain decreased by 5%. Which expressions represent the amount of rain that fell this week? Select all that apply.
- \(g-0.05\)
- \(g-0.05g\)
- \(0.95g\)
- \(0.05g\)
- \((1-0.05)g\)
(From Unit 4.2.3)
Exercise \(\PageIndex{7}\)
Decide whether or not each equation represents a proportional relationship.
- Volume measured in cups (\(c\)) vs. the same volume measured in ounces (\(z\)): \(c=\frac{1}{8}z\)
- Area of a square (\(A\)) vs. the side length of the square (\(s\)): \(A=s^{2}\)
- Perimeter of an equilateral triangle (\(P\)) vs. the side length of the triangle (\(s\)): \(3s=P\)
- Length (\(L\)) vs. width (\(w\)) for a rectangle whose area is 60 square units: \(L=\frac{60}{w}\)
(From Unit 2.3.2)
Exercise \(\PageIndex{8}\)
Add.
- \(5\frac{3}{4}+\left(-\frac{1}{4}\right)\)
- \(-\frac{2}{3}+\frac{1}{6}\)
- \(-\frac{8}{5}+\left(-\frac{3}{4}\right)\)
(From Unit 5.2.2)
Exercise \(\PageIndex{9}\)
In each diagram, \(x\) represents a different value.
For each diagram,
- What is something that is definitely true about the value of \(x\)?
- What is something that could be true about the value of \(x\)?
(From Unit 5.1.1)


4.1: Introduction
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- Shanti Kumar
- University of California Press
In this chapter, I argue that Bollywood must be understood as a vital force of immaterial labor for the affective contagion of mass creativity in urban India. I focus on the some of the many reasons why politicians, policymakers, film stars, filmmakers, and business leaders in India are turning their attention to the infrastructure of cinema as a potential resource for attracting economic capital and creative labor in urban and semiurban areas. The fusion of cinematic infrastructure with urban architecture is most evident in Indian cities and towns that have, or are planning to have, a “film city” in their master plans for urban development. Recent examples of this popular trend include the inauguration near Kolkata with much fanfare of Prayag Film City by the Bollywood superstar Shah Rukh Khan in April 2012; the announcement in August 2012 by actor Jackie Shroff of his investment in a partnership to build a mini–film city in Ahmedabad; the proposal by the Bihar chief minister, Nitish Kumar, in November 2012 to build a Film City near Patna in response to intense lobbying by actors from the Hindi and Bhojpuri film industries; plans by the Uttar Pradesh chief minister, Akhilesh Yadav, in October 2012 to create an IT/Film City in Lucknow; and much-advertised plans by the corporate powerhouse Sahara India to build the Sahara Pariwar Film City in its Aambi Valley development project near Pune.
Figure 1: The inauguration of Prayag Film City by the Bollywood superstar Shah Rukh Khan in 2012 represented the fusion of cinematic infrastructure with the urban environment. Khan is shown here at the Berlin Film Festival in 2008. Photo credit: Siebbi, via Wikimedia Commons.
The mobilization of the film industry and its infrastructure, including cinema halls, shooting locations, and production facilities, for generating economic development and sustaining growth in urban, semiurban, and rural areas is hardly new in India. For instance, these proposals for film cities take their inspiration from the pioneering efforts of Ramoji Film City, built near Hyderabad in 1996, and Innovative Film City near Bengaluru, which opened in 2008. Many academic studies have detailed the central role that cinema halls, film studios, cinematic narratives, and Bollywood-inspired consumer culture have played for many decades now in producing and sustaining India’s nationalist visions and developmental goals. However, what is new about the recent spate of proposals for film cities is the way the immaterial infrastructure of Bollywood is being integrated into to the future designs and architectures of urban life as a whole in India.
Drawing on Nigel Thrift’s concept of “affective cities,” I examine how film cities—and plans for film cities—are being used in several cities and towns in India to produce and manage mass creativity by transforming urban life into social factories of immaterial labor. 1 As Maurizio Lazzarato defines it, “immaterial labor” is labor that produces the informational and cultural content of a commodity. 2 The informational aspect of immaterial labor refers to the ways digital technologies, computer networks, and cybernetic controls are becoming integral to the labor practices that workers used traditionally to perform in spaces such as the factory floor. The cultural aspect of immaterial labor involves the affect value of the practices of social life in areas such as fashion, tastes, traditions, and norms, which are usually not deemed relevant to matters of labor in the workplace. As information technologies have become central to all sorts of workplaces in recent times, immaterial labor has become more integral to practices of work and social life at large, according to Lazzarato. The result is that labor is increasingly becoming more “intellectual” in society, and the commodities created through practices of immaterial labor are not only goods made in a factory but also the products of “mass intellect” or “mass creativity” in social life.
I argue that Thrift’s concept of “affective cities” is a powerful framework for analyzing how practices of immaterial labor in urban life are shifting the focus of work from capital–labor relations (in spaces such as the factory floor) to capital–life relations (in society at large). Using Thrift’s concept of affective cities in relation to Lazzarato’s theory of immaterial labor, I examine how cities in India are trying to tap into the immaterial labor of Bollywood by mobilizing film cities for the production and management of mass creativity in urban life as a whole. In this context, immaterial labor in Bollywood is not strictly limited to what is traditionally understood as the creative process of making a film. Instead, it is the workers and consumers at large who produce a range of immaterial goods and services through the constant exchange of communication, information, and knowledge about the film commodity in the political, economic, technological, cultural, and affective realms of social relations. The film city, I argue, is a concrete embodiment of the many ways in which the immaterial infrastructure of Bollywood is being fused with the traditional architectures of cities and towns in India to meet the growing demands of—and desires for—mediated mobilities in the twenty-first century


6.3: Alternating Groups
Note that every \(k\)-cycle \((a_1a_2\ldots a_k)\in S_n\) can be written as a product of (not necessarily disjoint) transpositions:
\begin{equation*} (a_1a_2\ldots a_k)=(a_1a_k)(a_1a_{k-1})\cdots(a_1a_3)(a_1a_2). \end{equation*}
We, therefore, have the following theorem.
Theorem \(\PageIndex{1}\)
Every permutation in \(S_n\) can be written as a product of transpositions.
Definition: Even and Odd
We say that a permutation in \(S_n\) is even [resp., odd ] if it can be written as a product of an even [resp., odd] number of transpositions.
Theorem \(\PageIndex{2}\)
Every permutation in \(S_n\) is even or odd, but not both.
- Proof
-
We already know that every permutation in \(S_n\) is a product of transpositions, so must be even or odd. For proof that no permutation is both even and odd, see, for instance, Proof 1 or 2 of Theorem 9.15 on p. 91 in [1].
Lemma \(\PageIndex{1}\)
For each \(2\leq k\leq n\text{,}\) then a \(k\)-cycle is even if \(k\) is odd, and odd if \(k\) is even.
The proof of this is left as an exercise for the reader.
Example \(\PageIndex{1}\)
In \(S_3\text{,}\) the permutations \(e\text{,}\) \((123)=(13)(12)\text{,}\) and \((132)=(12)(13)\) are even, while the permutations \((12)\text{,}\) \((13)\text{,}\) and \((23)\) are odd.
Example \(\PageIndex{2}\)
List all of the even [resp., odd] permutations in \(S_4\text{.}\)
We have the following theorem, whose proof is left as an exercise for the reader.
Theorem \(\PageIndex{3}\)
The set of all even permutations in \(S_n\) is a subgroup of \(S_n\text{.}\)
Definition: Alternating Group
The alternating group on \(n\) letters is the subgroup \(A_n\) of \(S_n\) consisting of all of the even permutations in \(S_n\text{.}\)
We end with this theorem, whose proof can be found on p. 93 of [1].
Theorem \(\PageIndex{4}\)
\(|A_n|= \dfrac{(n!)}{2} \text{.}\)


10: Liquids and Solids
The great distances between atoms and molecules in a gaseous phase, and the corresponding absence of any significant interactions between them, allows for simple descriptions of many physical properties that are the same for all gases, regardless of their chemical identities. As described in the final module of the chapter on gases, this situation changes at high pressures and low temperatures—conditions that permit the atoms and molecules to interact to a much greater extent. In the liquid and solid states, these interactions are of considerable strength and play an important role in determining a number of physical properties that do depend on the chemical identity of the substance. In this chapter, the nature of these interactions and their effects on various physical properties of liquid and solid phases will be examined.
-
- 10.0: Prelude to Liquids and Solids
- In the liquid and solid states, these interactions are of considerable strength and play an important role in determining a number of physical properties that do depend on the chemical identity of the substance. In this chapter, the nature of these interactions and their effects on various physical properties of liquid and solid phases will be examined.
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- 10.1: Intermolecular Forces
- The physical properties of condensed matter (liquids and solids) can be explained in terms of the kinetic molecular theory. In a liquid, intermolecular attractive forces hold the molecules in contact, although they still have sufficient kinetic energy to move past each other. Intermolecular attractive forces, collectively referred to as van der Waals forces, are responsible for the behavior of liquids and solids and are electrostatic in nature.
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- 10.2: Properties of Liquids
- The intermolecular forces between molecules in the liquid state vary depending upon their chemical identities and result in corresponding variations in various physical properties. Cohesive forces between like molecules are responsible for a liquid’s viscosity (resistance to flow) and surface tension. Adhesive forces between the molecules of a liquid and different molecules composing a surface in contact with the liquid are responsible for surface wetting and capillary rise.
-
- 10.3: Phase Transitions
- Phase transitions are processes that convert matter from one physical state into another. There are six phase transitions between the three phases of matter. Melting, vaporization, and sublimation are all endothermic processes, requiring an input of heat to overcome intermolecular attractions. The reciprocal transitions of freezing, condensation, and deposition are all exothermic processes, involving heat as intermolecular attractive forces are established or strengthened.
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- 10.4: Phase Diagrams
- The temperature and pressure conditions at which a substance exists in solid, liquid, and gaseous states are summarized in a phase diagram for that substance. Phase diagrams are combined plots of three pressure-temperature equilibrium curves: solid-liquid, liquid-gas, and solid-gas. These curves represent the relationships between phase-transition temperatures and pressures. The intersection of all three curves represents the substance’s triple point at which all three phases coexist.
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- 10.5: The Solid State of Matter
- Some substances form crystalline solids consisting of particles in a very organized structure; others form amorphous (noncrystalline) solids with an internal structure that is not ordered. The main types of crystalline solids are ionic solids, metallic solids, covalent network solids, and molecular solids. The properties of the different kinds of crystalline solids are due to the types of particles of which they consist, the arrangements of the particles, and the strengths of the attractions bet
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- 10.6: Lattice Structures in Crystalline Solids
- The structures of crystalline metals and simple ionic compounds can be described in terms of packing of spheres. Metal atoms can pack in hexagonal closest-packed structures, cubic closest-packed structures, body-centered structures, and simple cubic structures. The anions in simple ionic structures commonly adopt one of these structures, and the cations occupy the spaces remaining between the anions.
Contributors and Attributions
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


11.17: Short Informative Messages
- Write a short informative message.
Short, informative messages can take a variety of forms. For example, a quick communication to a teammate can utilize IM technology and provide a vehicle for a quick response. These messages tend to be more conversational in structure and resemble a chat more than an email or other document (Figure 1).
Another form of a short informative message could be a Twitter message, or tweet. Tweets are better for communicating with larger audiences, possibly external as well; in fact, one of the primary uses of Twitter (by companies) is advertising and promoting their brands. (We discussed Twitter in more depth in Module 10: Social Media.)
If the situation requires a more formal approach for your message, sending an email is usually more appropriate. In this example, the communication is being sent to a company executive from a subordinate.
To: Dave McCann
CC:
Subject: Change Of Plans (the customer wants to meet before lunch tomorrow)
From: Jon Parks
Good morning Dave,
Our contact just sent me a message requesting that we meet in their lobby at 11:00 am instead of meeting for lunch at 12. Will that work for your schedule?
Thank you,
Jon
Jon Parks
Senior Account Manager
Code Software
Contributors and Attributions
- Short Informative Messages. Authored by : Robert Danielson. Provided by : Lumen Learning. License : CC BY: Attribution


7.5: Development and Learning
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- Last updated
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- Cindy Stephens, Gina Peterson, Sharon Eyrich, & Jennifer Paris
- College of the Canyons via College of the Canyons
“Development” and “learning” are two integrated concepts that we promote as teachers. As children are “learning” new concepts and skills, they are fostering their “development”. Our goal is to encourage the development of the “whole child” (physical, cognitive, social, emotional, spiritual) by providing learning experiences based on children’s interests and abilities, a concept known as “ intentional teaching ”.
Although children learn in an integrated manner (blending all areas of development together) these areas are often broken down for planning purposes. While we looked at this in chapter 5, Developmental Ages and Stages, to understand development, the same is true when planning curriculum.
Table \(\PageIndex{1}\) below shows the relationship between the domains of development and concepts of learning.
| Development | Learning |
|---|---|
| Cognitive | Science, Technology, Math |
| Language | Language and Literacy |
| Physical | Health, Safety, Nutrition, Self-Help Skills, Physical Education |
| Social, Emotional, Spiritual | Social Science, Visual and Performing Arts |
Vignette
Javier and Ji are playing in the block area. They have stacked several large blocks on top of each other. Twice the blocks have fallen and each time they have modified their plan slightly to make them stay. Once stable, Ji counts the blocks and Javier turns to the teacher and proudly says, “Look at our 5 story building, you should shop here.”
Pause to Reflect
Can you find development and learning for Javier and Ji in each of the categories listed in the table above?
If you are interested in further exploration, the California Learning Foundations that describe competencies and the companion California Preschool Curriculum Framework present strategies for early childhood educators. You can explore each of these:
- California Infant/Toddler Learning and Development Foundations (www.cde.ca.gov/sp/cd/re/documents/it foundations 2009.pdf)
- California Infant/Toddler Curriculum Framework ( https://www.cde.ca.gov/sp/cd/re/docu...mframework.pdf )
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California Preschool Learning Foundations
- Volume 1 addresses social-emotional development, language and literacy, English-language development, and mathematics ( https://www.cde.ca.gov/sp/cd/re/psfo...sp#psfoundvol1 )
- Volume 2 addresses visual and performing arts, physical development, and health ( https://www.cde.ca.gov/sp/cd/re/psfo...sp#psfoundvol2 )
- Volume 3 addresses history-social science and science ( https://www.cde.ca.gov/sp/cd/re/psfo...sp#psfoundvol3 )
-
California Preschool Curriculum Frameworks
- Volume 1 addresses social-emotional development, language and literacy, English-language development, and mathematics ( https://www.cde.ca.gov/sp/cd/re/psfr...sp#psframevol1 )
- Volume 2 addresses visual and performing arts, physical development, and health ( https://www.cde.ca.gov/sp/cd/re/psfr...sp#psframevol2 )
- Volume 3 addresses history-social science and science ( https://www.cde.ca.gov/sp/cd/re/psfr...sp#psframevol3 )
These will also be visited in more detail in ECE 104 Introduction to Curriculum.


2.3: Power Functions and Polynomial Functions
- Refresher Topics
-
-
Simplifying Expressions
- Laws of Exponents (negative exponents)
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Simplifying Expressions
- Foundational Topics
-
-
Geometry
- Common Areas
-
Geometry
In this section, you will:
- Identify power functions.
- Identify end behavior of power functions.
- Identify polynomial functions.
- Identify the degree and leading coefficient of polynomial functions.
- Compute the Difference Quotient for power and polynomial functions.
Figure \( \PageIndex{ 1 } \) (credit: Jason Bay, Flickr)
Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in Table \( \PageIndex{ 1 } \).
| Year | \[2009 \nonumber \] | \[2010 \nonumber \] | \[2011 \nonumber \] | \[2012 \nonumber \] | \[2013 \nonumber \] |
| Bird Population | \[800 \nonumber \] | \[897 \nonumber \] | \[992 \nonumber \] | \[1,083 \nonumber \] | \[1,169 \nonumber \] |
The population can be estimated using the function \(P(t)=−0.3 t^3 +97t+800\), where \(P(t)\) represents the bird population on the island \(t\) years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.
Identifying Power Functions
In order to better understand the bird problem, we need to understand a specific type of function. A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number. (A number that multiplies a variable raised to an exponent is known as a coefficient.)
As an example, consider functions for area or volume. The function for the area of a circle with radius \(r\) is\[A(r)= \pi r^2 \nonumber \]and the function for the volume of a sphere with radius \(r\) is\[V(r)= \dfrac{4}{3} \pi r^3. \nonumber \]Both of these are examples of power functions because they consist of a coefficient, \(\pi\) or \( \frac{4}{3} \pi \), multiplied by a variable \(r\) raised to a power.
A power function is a function that can be represented in the form\[f(x) = k x^p, \nonumber \]where \(k\) and \(p\) are real numbers, and \(k\) is known as the coefficient .
Is \(f(x)= 2^x\) a power function?
No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function , not a power function.
Which of the following functions are power functions?\[ \begin{array}{ll}
f(x)=1 & \text{Constant function} \\[6pt]
f(x)=x & \text{Identify function} \\[6pt]
f(x)= x^2 & \text{Quadratic function} \\[6pt]
f(x)= x^3 & \text{Cubic function} \\[6pt]
f(x)= \dfrac{1}{x} & \text{Reciprocal function} \\[6pt]
f(x)= \dfrac{1}{x^2} & \text{Reciprocal squared function} \\[6pt]
f(x)=\sqrt{ x } & \text{Square root function} \\[6pt]
f(x)= \sqrt[3]{x} & \text{Cube root function} \\[6pt]
\end{array} \nonumber \]
- Solution
-
All of the listed functions are power functions. The constant and identity functions are power functions because they can be written as \(f(x)= x^0\) and \(f(x)= x^1\), respectively. The quadratic and cubic functions are power functions with whole number powers. The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as \(f(x)= x^{-1}\) and \(f(x)= x^{-2} \). The square and cube root functions are power functions with fractional powers because they can be written as \(f(x)= x^{1/2} \) or \(f(x)= x^{1/3} \).
Which functions are power functions?\[ \begin{array}{rcl}
f(x) & = & 2 x^2 \cdot 4 x^3 \\[6pt]
g(x) & = & -x^5 +5 x^3 -4x \\[6pt]
h(x) & = & \frac{2 x^5 -1}{3 x^2 +4} \\[6pt]
\end{array} \nonumber \]
Focusing on Calculus - Revisiting the Difference Quotient
The ever-important Difference Quotient is often used in Calculus when working with power function.
Compute the Difference Quotient for the given function.\[ f(x) = x^4 \nonumber \]
- Solution
-
Recall that the Difference Quotient for the function \( f \) is defined to be\[ \dfrac{f(x + h) - f(x)}{h}. \nonumber \]Therefore, in this case, we have\[ \begin{array}{rclcl}
\dfrac{f(x + h) - f(x)}{h} & = & \dfrac{(x + h)^4 - x^4}{h} & \quad & \left( \text{substituting} \right) \\[6pt]
& = & \dfrac{\left((x + h)^2\right)^2 - x^4}{h} & \quad & \left( \text{Laws of Exponents} \right) \\[6pt]
& = & \dfrac{\left((x + h)(x + h)\right)^2 - x^4}{h} & & \\[6pt]
& = & \dfrac{\left(x^2 + 2hx + h^2\right)^2 - x^4}{h} & \quad & \left( \text{distributing} \right) \\[6pt]
& = & \dfrac{\left(x^2 + 2hx + h^2\right)\left(x^2 + 2hx + h^2\right) - x^4}{h} & & \\[6pt]
& = & \dfrac{x^4 +4x^3 h + 6x^2 h^2 + 4xh^3 + h^4 - x^4}{h} & \quad & \left( \text{distributing} \right) \\[6pt]
& = & \dfrac{4x^3 h + 6x^2 h^2 + 4xh^3 + h^4}{h} & \quad & \left( \text{combining like terms} \right) \\[6pt]
& = & \dfrac{h\left(4x^3 + 6x^2 h + 4xh^2 + h^3\right)}{h} & \quad & \left( \text{factoring out the GCF from the numerator} \right) \\[6pt]
& = & \dfrac{\cancelto{1}{h}\left(4x^3 + 6x^2 h + 4xh^2 + h^3\right)}{\cancelto{1}{h}} & \quad & \left( \text{canceling like factors} \right) \\[6pt]
& = & 4x^3 + 6x^2 h + 4xh^2 + h^3 & \quad & \left( \text{simplifying} \right) \\[6pt]
\end{array} \nonumber \]
In Example \( \PageIndex{ 2 } \), we could have taken the alternative approach of using factoring techniques to simplify the numerator.\[ \begin{array}{rclcl}
\dfrac{f(x + h) - f(x)}{h} & = & \dfrac{(x + h)^4 - x^4}{h} & \quad & \left( \text{substituting} \right) \\[6pt]
& = & \dfrac{\left((x + h)^2\right)^2 - \left(x^2\right)^2}{h} & \quad & \left( \text{Laws of Exponents} \right) \\[6pt]
& = & \dfrac{\left((x + h)^2 - x^2\right)\left((x + h)^2 + x^2\right)}{h} & \quad & \left( \text{Difference of Squares} \right) \\[6pt]
& = & \dfrac{\left((x + h) - x\right)\left((x + h) + x\right)\left((x + h)^2 + x^2\right)}{h} & \quad & \left( \text{Difference of Squares} \right) \\[6pt]
& = & \dfrac{\left(h\right)\left(2x + h\right)\left(x^2 + 2hx + h^2 + x^2\right)}{h} & \quad & \left( \text{simplifying} \right) \\[6pt]
& = & \dfrac{ \cancelto{1}{h}\left(2x + h\right)\left(x^2 + 2hx + h^2 + x^2\right)}{\cancelto{1}{h}} & \quad & \left( \text{canceling like factors} \right) \\[6pt]
& = & \left(2x + h\right)\left(2x^2 + 2hx + h^2\right) & \quad & \left( \text{simplifying} \right) \\[6pt]
& = & 4x^3 + 6x^2 h + 4xh^2 + h^3 & \quad & \left( \text{distributing} \right) \\[6pt]
\end{array} \nonumber \]While both methods are valid and have their place, a student well-prepared for Calculus will inevitably want to try the second method.
Identifying End Behavior of Power Functions
Figure \( \PageIndex{ 2 } \) shows the graphs of \(f(x)= x^2\), \(g(x)= x^4\), and \( h(x)= x^6 \), which are all power functions with even, whole-number powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.
Figure \( \PageIndex{ 2 } \): Even-power functions
To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol \(\infty\) for positive infinity and \(- \infty\) for negative infinity. When we say that "\(x\) approaches infinity," which can be symbolically written as \(x \to \infty \), we are describing a behavior; we are saying that \(x\) is increasing without bound.
With the even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as \(x\) approaches positive or negative infinity, the \(f( x )\) values increase without bound. In symbolic form, we could write\[\text{as }x \to \pm \infty , f(x) \to \infty. \nonumber \]Figure \( \PageIndex{ 3 } \) shows the graphs of \(f(x)= x^3\), \(g(x)= x^5\), and \(h(x)= x^7 \), which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.
Figure \( \PageIndex{ 3 } \): Odd-power functions
These examples illustrate that functions of the form \(f( x )= x^n\) reveal symmetry of one kind or another. First, in Figure \( \PageIndex{ 2 } \) we see that functions of the form \(f( x )= x^n\), where \(n\) is even, are symmetric about the \( y \)-axis. In Figure \( \PageIndex{ 3 } \), we see that functions of the form \(f( x )= x^n\), where \( n \) is odd, are symmetric about the origin.
For these odd power functions, as \(x\) approaches negative infinity, \(f( x )\) decreases without bound. As \(x\) approaches positive infinity, \(f( x )\) increases without bound. In symbolic form, we write\[\text{as }x \to - \infty , f(x) \to - \infty \text{ and as }x \to \infty ,f(x) \to \infty. \nonumber \]The behavior of the graph of a function as the input values get very negative ( \(x \to - \infty\) ) and get very large ( \(x \to \infty\) ) is referred to as the end behavior of the function. We can use words or symbols to describe end behavior.
Figure \( \PageIndex{ 4 } \) shows the end behavior of power functions in the form \(f(x)=k x^n\), where \(n\) is a non-negative integer depending on the power and the constant.
Figure \( \PageIndex{ 4 } \)
Describe the end behavior of the graph of \(f(x)= x^8 \).
- Solution
-
The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As \(x\) approaches infinity, the output (value of \(f(x)\) ) increases without bound. Therefore, as \(x \to \infty \), \(f(x) \to \infty \). As \(x\) approaches negative infinity, the output increases without bound. In symbolic form, as \(x \to - \infty \), \(f(x) \to \infty \). We can graphically represent the function as shown in Figure \( \PageIndex{ 5 } \).
<Figure \( \PageIndex{ 5 } \)
Describe the end behavior of the graph of \(f(x)= -x^9 \).
- Solution
-
The exponent of the power function is 9 (an odd number). Because the coefficient is \(–1\) (negative), the graph is the reflection about the \( x \)-axis of the graph of \(f(x)= x^9 \). Figure \( \PageIndex{ 6 } \) shows that as \(x\) approaches infinity, the output decreases without bound. As \(x\) approaches negative infinity, the output increases without bound. In symbolic form, we would write\[\text{as }x \to - \infty , \, f(x) \to \infty ; \, \text{as } x \to \infty , \, f(x) \to - \infty. \nonumber \]
<Figure \( \PageIndex{ 6 } \)We can check our work by using the table feature on a graphing utility.
Table \( \PageIndex{ 2 } \) \(x\) \(f( x )\) –10 1,000,000,000 –5 1,953,125 0 0 5 –1,953,125 10 –1,000,000,000 We can see from Table \( \PageIndex{ 2 } \) that, when we substitute very small values for \(x\), the output is very large, and when we substitute very large values for \(x\), the output is very small (meaning that it is a very large negative value).
Describe in words and symbols the end behavior of \(f(x)=-5 x^4 \).
Identifying Polynomial Functions
An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius, \(r\), of the spill depends on the number of weeks, \(w\), that have passed. This relationship is linear.\[ r(w) =24+8w. \nonumber \]We can combine this with the formula for the area \(A\) of a circle.\[A(r) = \pi r^2. \nonumber \]Composing these functions gives a formula for the area in terms of weeks.\[A(w)=A(r(w)) =A(24+8w) = \pi (24+8w)^2. \nonumber \]Distributing gives the formula\[A(w)=576 \pi +384 \pi w+64 \pi w^2. \nonumber \]This formula is an example of a polynomial function . A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a natrual number power.
Let \(n \in \mathbb{N}\). A polynomial function is a function that can be written in the form\[ f(x)= a_n x^n + a_{n - 1} x^{n - 1} + \ldots + a_2 x^2 + a_1 x + a_0. \nonumber \]This is called the general form of a polynomial function . Each \( a_i \) is a coefficient and can be any real number, but \( a_n \neq 0\). Each product \( a_i x_i \) is a term of a polynomial function.
Which of the following are polynomial functions?\[ \begin{array}{rcl}
f(x) & = & 2 x^3 \cdot 3x+4 \\[6pt]
g(x) & = & -x( x^2 -4) \\[6pt]
h(x) & = & 5\sqrt{ x } +2 \\[6pt]
\end{array} \nonumber \]
- Solution
-
The first two functions are examples of polynomial functions because they can be written in the form \(f(x)= a_n x^n + a_{n - 1} x^{n - 1} + \ldots + a_2 x^2 + a_1 x + a_0 \) where the powers are non-negative integers and the coefficients are real numbers.
- \(f(x)\) can be written as \(f(x)=6 x^4 +4\).
- \(g(x)\) can be written as \(g(x)=- x^3 +4x\).
- \(h(x)\) cannot be written in this form and is therefore not a polynomial function.
Focusing on Calculus - Revisiting the Difference Quotient
Find and simplify the Difference Quotient for the following function.\[ P(x) = 2x^3 + x - 4 \nonumber \]
- Solution
-
\[ \begin{array}{rclcl}
\dfrac{P(x + h) - P(x)}{h} & = & \dfrac{2(x + h)^3 + (x + h) - 4 - \left( 2x^3 + x - 4 \right)}{h} & \quad & \left( \text{substituting} \right) \\[6pt]
& = & \dfrac{2(x + h)^3 + (x + h) - 4 - 2x^3 - x + 4}{h} & \quad & \left( \text{distributing the negative} \right) \\[6pt]
& = & \dfrac{2(x^3 + 3x^2h + 3xh^2 + h^3) + x + h - 4 - 2x^3 - x + 4}{h} & \quad & \left( \text{distributing} \right) \\[6pt]
& = & \dfrac{2x^3 + 6x^2h + 6xh^2 + 2h^3 + x + h - 4 - 2x^3 - x + 4}{h} & \quad & \left( \text{distributing} \right) \\[6pt]
& = & \dfrac{6x^2h + 6xh^2 + 2h^3 + h}{h} & \quad & \left( \text{combining like terms} \right) \\[6pt]
& = & \dfrac{h\left(6x^2 + 6xh + 2h^2 + 1\right)}{h} & \quad & \left( \text{factoring out the GCF from the numerator} \right) \\[6pt]
& = & \dfrac{\cancelto{1}{h}\left(6x^2 + 6xh + 2h^2 + 1\right)}{\cancelto{1}{h}} & \quad & \left( \text{canceling like factors} \right) \\[6pt]
& = & 6x^2 + 6xh + 2h^2 + 1 & \quad & \left( \text{simplifying} \right) \\[6pt]
\end{array} \nonumber \]
Identifying the Degree and Leading Coefficient of a Polynomial Function
Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form . The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term.
We often rearrange polynomials so that the powers are descending.
When a polynomial is written in this way, we say that it is in general form .
Identify the degree, leading term, and leading coefficient of the following polynomial functions.\[ \begin{array}{rcl}
f(x) & = & 3+2 x^2 -4 x^3 \\[6pt]
g(t) & = & 5 t^5 -2 t^3 +7t \\[6pt]
h(p) & = & 6p- p^3 -2 \\[6pt]
\end{array} \nonumber \]
- Solution
-
For the function \(f( x )\), the highest power of \(x\) is 3, so the degree is 3. The leading term is the term containing that degree, \(-4 x^3 \). The leading coefficient is the coefficient of that term, \(-4\).
For the function \(g( t )\), the highest power of \(t\) is \(5\), so the degree is \(5\). The leading term is the term containing that degree, \(5 t^5 \). The leading coefficient is the coefficient of that term, \(5\).
For the function \(h( p )\), the highest power of \(p\) is \(3\), so the degree is \(3\). The leading term is the term containing that degree, \(-p^3 ;\) the leading coefficient is the coefficient of that term, \(-1\).
Identify the degree, leading term, and leading coefficient of the polynomial \(f(x)=4 x^2 - x^6 +2x-6\).
Identifying End Behavior of Polynomial Functions
Knowing the degree of a polynomial function is useful in helping us predict its end behavior . To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as \(x\) gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree. See Table \( \PageIndex{ 3 } \).
| Polynomial Function | Leading Term | Graph of Polynomial Function |
|---|---|---|
| \(f(x)=5 x^4 +2 x^3 -x-4\) | \(5 x^4\) | |
| \(f(x)=-2 x^6 - x^5 +3 x^4 + x^3\) | \(-2 x^6\) | |
| \(f(x)=3 x^5 -4 x^4 +2 x^2 +1\) | \(3 x^5\) | |
| \(f(x)=-6 x^3 +7 x^2 +3x+1\) | \(-6 x^3\) |
Describe the end behavior and determine a possible degree of the polynomial function in Figure \( \PageIndex{ 7 } \).
Figure \( \PageIndex{ 7 } \)
- Solution
-
As the input values \(x\) get very large, the output values \(f(x)\) increase without bound. As the input values \(x\) get very negative, the output values \(f(x)\) decrease without bound. We can describe the end behavior symbolically by writing\[\text{as }x \to - \infty , \, f(x) \to - \infty ; \, \text{as } x \to \infty , \, f(x) \to \infty. \nonumber \]In words, we could say that as \(x\) values approach infinity, the function values approach infinity, and as \(x\) values approach negative infinity, the function values approach negative infinity.
We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.
Describe the end behavior, and determine a possible degree of the polynomial function in Figure \( \PageIndex{ 8 } \).
<Figure \( \PageIndex{ 8 } \)
Given the function \(f(x)=-3 x^2 (x-1)(x+4)\), express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.
- Solution
-
Obtain the general form by expanding the given expression for \(f( x )\).\[ \begin{array}{rcl}
f(x) & = & -3 x^2 (x-1)(x+4) \\[6pt]
& = & -3 x^2 ( x^2 +3x-4) \\[6pt]
& = & -3 x^4 -9 x^3 +12 x^2 \\[6pt]
\end{array} \nonumber \]The general form is \(f( x )=-3 x^4 -9 x^3 +12 x^2 \). The leading term is \(-3 x^4 \); therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (–3), so the end behavior is\[\text{as } x \to - \infty , \, f(x) \to - \infty ; \, \text{as } x \to \infty , \, f(x) \to - \infty. \nonumber \]
Given the function \(f(x)=0.2(x-2)(x+1)(x-5)\), express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.
Identifying Local Behavior of Polynomial Functions
In addition to the end behavior of polynomial functions, we are also interested in what happens in the "middle" of the function. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.
We are also interested in the intercepts. As with all functions, the \( y \)-intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one \( y \)-intercept, \((0, a_0 )\). The \( x \)-intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one \( x \)-intercept. See Figure \( \PageIndex{ 9 } \).
Figure \( \PageIndex{ 9 } \)
A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The \( y \)-intercept is the point at which the function has an input value of zero. The \( x \)-intercepts are the points at which the output value is zero.
Given the polynomial function \(f(x)=(x-2)(x+1)(x-4)\), written in factored form for your convenience, determine the \( y \)- and \( x \)-intercepts.
- Solution
-
The \( y \)-intercept occurs when the input is zero so substitute 0 for \(x\).\[ \begin{array}
f(0) & = & (0-2)(0+1)(0-4) \\[6pt]
& = & (-2)(1)(-4) \\[6pt]
& = & 8 \\[6pt]
\end{array} \nonumber \]The \( y \)-intercept is \( (0, 8) \).The \( x \)-intercepts occur when the output is zero.\[ 0=(x-2)(x+1)(x-4) \nonumber \]\[ \begin{array}{rccccc}
\implies & x-2=0 & \text{or} & x+1=0 & \text{or} & x-4=0 \\[6pt]
\implies & x=2 & \text{or} & x=-1 & \text{or} & x=4 \\[6pt]
\end{array} \nonumber \]The \( x \)-intercepts are \((2,0)\), \((–1,0)\), and \((4,0)\).We can see these intercepts on the graph of the function shown in Figure \( \PageIndex{ 10 } \).
<Figure \( \PageIndex{ 10 } \)
Given the polynomial function \(f(x)= x^4 -4 x^2 -45\), determine the \( y \)- and \( x \)-intercepts.
- Solution
-
The \( y \)-intercept occurs when the input is zero.\[ \begin{array}{rcl}
f(0) & = & (0)^4 -4 (0)^2 -45 \\[6pt]
& = & -45 \\[6pt]
\end{array} \nonumber \]The \( y \)-intercept is \((0,-45)\).The \( x \)-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.\[f(x)= x^4 -4 x^2 -45 =( x^2 -9)( x^2 +5) =(x-3)(x+3)( x^2 +5). \nonumber \]Therefore, we solve\[0=(x-3)(x+3)( x^2 +5). \nonumber \]This mean \(x-3=0\) or \(x+3=0\) (\(x^2 +5=0\) would require \( x^2 = -5 \), which has no real solution). Hence, \(x=3\) or \(x=-3\). The \( x \)-intercepts are \((3,0)\) and \((–3,0)\).
We can see these intercepts on the graph of the function shown in Figure \( \PageIndex{ 11 } \). We can see that the function is even because \(f( x )=f( -x )\).
<Figure \( \PageIndex{ 11 } \)
Given the polynomial function \(f(x)=2 x^3 -6 x^2 -20x\), determine the \( y \)- and \( x \)-intercepts.
Comparing Smooth and Continuous Graphs
The degree of a polynomial function helps us to determine the number of \( x \)-intercepts and the number of turning points. A polynomial function of \(n^{\text{th}}\) degree is the product of \(n\) factors, so it will have at most \(n\) roots or zeros, or \( x \)-intercepts. The graph of the polynomial function of degree \(n\) must have at most \(n–1\) turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.
While not technically true, we will, for the sake of simplicity, define a continuous function to be one that has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.
A polynomial of degree \(n\) will have, at most, \(n\) \( x \)-intercepts and \(n-1\) turning points.
Without graphing the function, determine the local behavior of the function by finding the maximum number of \( x \)-intercepts and turning points for \(f(x)=-3 x^{10} +4 x^7 - x^4 +2 x^3 \).
- Solution
-
The polynomial has a degree of \(10\), so there are at most \(10\) \(x\)-intercepts and at most \(10-1=9\) turning points.
Without graphing the function, determine the maximum number of \( x \)-intercepts and turning points for \(f(x)=108-13 x^9 -8 x^4 +14 x^{12} +2 x^3\).
What can we conclude about the polynomial represented by the graph shown in Figure \( \PageIndex{ 12 } \) based on its intercepts and turning points?
Figure \( \PageIndex{ 12 } \)
- Solution
-
The end behavior of the graph tells us this is the graph of an even-degree polynomial. See Figure \( \PageIndex{ 13 } \).
<Figure \( \PageIndex{ 13 } \)The graph has 2 \( x \)-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.
What can we conclude about the polynomial represented by the graph shown in Figure \( \PageIndex{ 14 } \) based on its intercepts and turning points?
<Figure \( \PageIndex{ 14 } \)
Given the function \(f(x)=-4x( x+3 )( x-4 )\), determine the local behavior.
- Solution
-
The \( y \)-intercept is found by evaluating \(f(0)\).\[ \begin{array}{rcl}
f(0) & = & -4(0)(0+3)(0-4) \\[6pt]
& = & 0 \\[6pt]
\end{array} \nonumber \]The \( y \)-intercept is \((0,0)\).The \( x \)-intercepts are found by determining the zeros of the function.\[0=-4x(x+3)(x-4) \implies x=0 \text{ or } x+3=0 \text{ or } x-4=0 \implies x=0 \text{ or } x=-3 \text{ or } x=4. \nonumber \]The \( x \)-intercepts are \((0,0),(–3,0)\), and \((4,0)\).
The degree is 3 so the graph has at most 2 turning points.
Given the function \(f(x)=0.2(x-2)(x+1)(x-5)\), determine the local behavior.
Access these online resources for additional instruction and practice with power and polynomial functions.


“Stay focused, go after your dreams, and keep moving toward your goals.”
– LL Cool J
College and Career: Key Connections
Think back to the time when you first began to contemplate college. Do you remember specific thoughts? Were you excited about the idea? What began to draw you into the web of college life? What compels you to be here now?
In this topic on career and college readiness, we examine key connections between your motivations to be in college and your ultimate success in achieving your goals. We also examine how your college experience prepares you for a specific career, as well as for attaining general skills that you can apply to multiple pursuits.
Activity: Motivations for Attending College
Objectives
-
Review some of the many motivations students have for entering college.
-
Identify your personal motivations as pathways to achieving goals.
Directions
-
Review the table below, which lists various motivations cited by other students.
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Identify your main motivations, and rank your top five.
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Reflect on your selections in terms of how they connect with short-term and long-term plans for the future.
Understanding your motivations is essential to helping you not only prioritize your plans for the future but also gain inspiration about directions you may not have yet charted. Ultimately, your motivations for being in college align you with roadways to fulfilling your goals and ambitions.
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MY TOP FIVE
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MOTIVATIONS FOR ATTENDING COLLEGE
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Gain more qualifications in my field
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Increase my earning potential; make more money
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Challenge myself
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Show others that I can succeed
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Start an independent life
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Satisfy my curiosity
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Have fun
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Change my career
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Do what my parents were not able to do
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Find a better lifestyle
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Build my confidence
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Expand my social contacts; bond with new friends
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Improve my network of business associates
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Gain exposure to a wide array of topics
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Attend campus events
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Make my family happy
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Fulfill my dreams
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Take classes at home or work or anywhere
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Take advantage of campus resources like the library and gym
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Join a sports team
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Join campus organizations
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Spend my time during retirement
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Have continued support via alumni programs
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Learn to study and work on my own
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Gain access to professors
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Link up with people who already excel in the ways I aspire to
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Get sports spirit
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Gain more access to entertainment like theater and bands
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Be more productive in life
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Explore myself
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Become well versed in many subjects
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Dig deeper into learning than I did in high school
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Expand my knowledge of the world
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Others?
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Am I College and Career Ready?
Knowing what you truly want to gain from your college experience is the first step toward achieving it. However, reaching your goals does not necessarily mean you are college and career ready.
What does it mean to be ready for college and a career? In general, you are a college- and career-ready student if you have gained the necessary knowledge, skills, and professional behaviors to achieve at least one of the following:
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Earn a certificate or degree in college
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Participate in career training
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Enter the workplace and succeed
For instance, if you are studying for a skilled trade license in college, or perhaps pursuing a bachelor of arts degree, you are college-ready if you have the reading, writing, mathematics, social, and thinking skills to qualify for and succeed in the academic program of your choice.
Similarly, you are a career-ready student if you have the necessary knowledge and technical skills needed to be employed in your desired field.
“Ultimately, college and career readiness demands students know more than just content, but demonstrate that they know how to learn and build upon that content to solve problems. They must develop versatile communication skills, work collaboratively and work competitively in a school or work environment. Ensuring that you possess both the academic and technical know-how necessary for a career beyond the classroom is a great step toward succeeding on whatever path you choose.”
– Washington, DC Office of the State Superintendent of Education
College and Career Readiness in Your State
So where are you on the readiness scale? You can find out how your state measures your readiness. Visit the Interactive State Map at the College and Career Readiness and Success Center of the American Institutes for Research Web site. The map leads you to definitions of college and career readiness for your state. It also provides metrics to measure readiness. And it provides information about programs and structures to help you and educators. You can compare states across one or more categories.
Student Voices on Being College and Career Ready
In the following video, a number of high school students and recent graduates reflect on college and career readiness and their futures. As you view the video, be thinking about how your short-term goals can connect with longer-range ambitions. You might also reflect on how your deepening experiences in college can lead to achieving your longer-term goals. After all, each new experience in your life builds upon the last. You may never truly “arrive” at a destination if indeed your life is an ongoing journey.
The Marriage of College and Career
The oldest institution of higher learning in the United States is widely acknowledged to be Harvard University. It was established in 1636 with the aim of providing instruction in arts and sciences to qualify students for employment. In the 1779 Constitution of Massachusetts submitted by Samuel Adams, John Adams, and James Bowdoin to the full Massachusetts Convention, the following language was used:
Art. I.—Whereas our wise and pious ancestors, so early as the year one thousand six hundred and thirty six, laid the foundation of Harvard-College, in which University many persons of great eminence have, by the blessing of GOD, been initiated in those arts and sciences, which qualified them for public employments, both in Church and State . . .
Is “public employment” preparation still the goal of higher education institutions today? Indeed, it is certainly one of the many goals! College is also an opportunity for students to grow personally and intellectually. In fact, in a 2011 Pew Research Center survey, Americans were split on their perceptions of the main purpose of a college education:
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47 percent of those surveyed said the purpose of college is to teach work-related skills.
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39 percent said it is to help a student grow personally and intellectually.
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12 percent said the time spent at college should be dedicated to both pursuits—teaching work-related skills and helping students grow personally and intellectually.
These statistics are understandable in light of the great reach and scope of higher education institutions. Today, there are some 5,300 colleges and universities in the United States, offering every manner of education and training to students.
What do employers think about the value of a college education? What skills do employers seek in their workforce? In 2014, Hart Research Associates conducted a survey on behalf of the Association of American Colleges and Universities. The survey revealed that the majority of employers believe that having field-specific knowledge as well as a broad range of knowledge and skills is important for recent college graduates to achieve long-term career success.
Employers also said that when they hire, they place the greatest value on skills and knowledge that cut across all majors. The learning outcomes they rate as most important include written and oral communication skills, teamwork skills, ethical decision-making, critical thinking, and the ability to apply knowledge in real-world settings.
Employment Rates and Salaries
Consider, too, the following statistics on employment rates and salaries for college graduates. College does make a big difference!
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The average college graduate earns about 75 percent more than a non-college graduate over a typical, forty-year working lifetime. (U.S. Census Bureau)
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In 2014, young adults ages 20 to 24 with a bachelor’s degree or higher had a higher employment rate (88.1 percent) than young adults with just some college (75.0 percent). (NCES)
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The employment rate for young adults with just some college (63.7 percent) was higher than the rate for those who had completed high school. (NCES)
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The employment rate for those who completed high school (46.6 percent) was higher than the employment rate for young adults who had not finished high school. (NCES)
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Employment rates were generally higher for males than females at each level of educational attainment in 2015. (NCES)
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Over the course of a forty-year working life, the typical college graduate earns an estimated $550,000 more than the typical high school graduate. (PEW)
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The median gap in annual earnings between a high school and college graduate as reported by the U.S. Census Bureau in 2010 is $19,550. (PEW)
Perhaps most important, an overwhelming majority of college graduates—86 percent—say that college has been a good investment for them personally (PEW).
Differences in Earnings between States
You may wish to use this Earnings and Educational Attainment (2011) interactive table to see how earnings for college graduates vs. high school–only graduates in your state compare with those in other states.
All in all, college imparts a wide and deep range of benefits. The short video
Why College,
below, shows that with a college degree you are more likely to
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Have a higher salary
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Have and keep a job
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Get a pension plan
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Be satisfied with your job
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Feel your job is important
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Have health insurance
Summary
Success in college can be measured in many ways: through your own sense of what is important to you; through your family’s sense of what is important; through your institution’s standards of excellence; through the standards established by your state and country; through your employer’s perceptions about what is needed in the workplace; training for and becoming an entrepreneur, small business owner, or your own boss; and in many respects through your own unfolding goals, dreams, and ambitions.
How are you striving to achieve your goals? And how will you measure your success along the way?
Licenses and Attributions:
CC licensed content, Original:
CC licensed content, Shared previously:
All rights reserved content:
-
Why College? Authored by: OregonGEARUP. Located at:
https://youtu.be/-N6nru0nThg
. License: All Rights Reserved. License Terms: Standard YouTube License.
Public domain content:
-
What Does College and Career Readiness Mean? Provided by: Office of the State Superintendent of Education. Located at: osse.dc.gov/service/what-does...readiness-mean. License: Public Domain:No KnownCopyright
.
-
Chart of Earnings and Unemployment. Provided by: Bureau of Labor Statistics. Located at:
http://www.bls.gov/emp/ep_chart_001.htm
. License: PublicDomain:No Known Copyright.
Adaptions: Foundations of Academic Success: Words of Wisdom essay removed (exists elsewhere in this work), relocated learning objectives, removed Earnings and unemployment rates by educational attainment graph, updated footnote references, modified footnote formatting. Removed job fair image.


12.10: Checking the Normality Assumption
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Testing the normality assumption is relatively straightforward. We covered most of what you need to know in Section 13.9. The only thing we really need to know how to do is pull out the residuals (i.e., the ϵ
ik
values) so that we can draw our QQ plot and run our Shapiro-Wilk test. First, let’s extract the residuals. R provides a function called
residuals()
that will do this for us. If we pass our
my.anova
to this function, it will return the residuals. So let’s do that:
my.anova.residuals <- residuals( object = my.anova ) # extract the residuals
We can print them out too, though it’s not exactly an edifying experience. In fact, given that I’m on the verge of putting myself to sleep just typing this, it might be a good idea to skip that step. Instead, let’s draw some pictures and run ourselves a hypothesis test:
hist( x = my.anova.residuals ) # plot a histogram (similar to Figure @ref{fig:normalityanova}a)
qqnorm( y = my.anova.residuals ) # draw a QQ plot (similar to Figure @ref{fig:normalityanova}b)
shapiro.test( x = my.anova.residuals ) # run Shapiro-Wilk test
##
## Shapiro-Wilk normality test
##
## data: my.anova.residuals
## W = 0.96019, p-value = 0.6053
The histogram and QQ plot are both look pretty normal to me. 212 This is supported by the results of our Shapiro-Wilk test (W=.96, p=.61) which finds no indication that normality is violated.