2.5: REVIEW- FAQ's about HIPAA
FAQs: HIPAA Portability and Nondiscrimination Requirements for Workers
U.S. Department of Labor Employee Benefits Security Administration
Q. What is the Health Insurance Portability and Accountability Act (HIPAA)?
HIPAA offers protections for workers and their families. The law provides additional opportunities to enroll in a group health plan if you lose other coverage or experience certain life events. HIPAA also prohibits discrimination against employees and their dependents based on any health factors they may have, including prior medical conditions, previous claims experience, and genetic information.
Taking Advantage of Special Enrollment Opportunities
Q. What is Special Enrollment?
Special enrollment allows individuals who previously declined health coverage to enroll for coverage. Special enrollment rights arise regardless of a plan's open enrollment period.
There are two types of special enrollment – upon loss of eligibility for other coverage and upon certain life events. Under the first, employees and dependents who decline coverage due to other health coverage and then lose eligibility or lose employer contributions have special enrollment rights. For instance, an employee turns down health benefits for herself and her family because the family already has coverage through her spouse's plan. Coverage under the spouse's plan ceases. That employee then can request enrollment in her own company's plan for herself and her dependents.
Under the second, employees, spouses, and new dependents are permitted to special enroll because of marriage, birth, adoption, or placement for adoption.
For both types, the employee must request enrollment within 30 days of the loss of coverage or life event triggering the special enrollment.
A special enrollment right also arises for employees and their dependents who lose coverage under a state Children's Health Insurance Program (CHIP) or Medicaid or who are eligible to receive premium assistance under those programs. The employee or dependent must request enrollment within 60 days of the loss of coverage or the determination of eligibility for premium assistance.
Q. What are some examples of events that can trigger a loss of eligibility for coverage?
Loss of eligibility for coverage may occur when:
- Divorce or legal separation results in you losing coverage under your spouse's health insurance;
- A dependent is no longer considered a "covered" dependent under a parent's plan;
- Your spouse's death leaves you without coverage under his or her plan;
- Your spouse's employment ends, as does coverage under his employer's health plan;
- Your employer reduces your work hours to the point where you are no longer covered by the health plan;
- Your plan decides it will no longer offer coverage to a certain group of individuals (for example, those who work part time);
- You no longer live or work in the HMO's service area.
These should give you some idea of the types of situations that may entitle you to a special enrollment right.
Q. How long do I have to request special enrollment?
It depends on what triggers your right to special enrollment. The employee or dependent must request enrollment within 30 days after losing eligibility for coverage or after a marriage, birth, adoption, or placement for adoption.
The employee or dependent must request enrollment within 60 days of the loss of coverage under a state CHIP or Medicaid program or the determination of eligibility for premium assistance under those programs.
Q. After I request special enrollment, how long will I wait for coverage?
It depends on what triggers your right to special enrollment. Those taking advantage of special enrollment as a result of a birth, adoption, or placement for adoption begin coverage no later than the day of the event.
For special enrollment due to marriage or loss of eligibility for other coverage, your new coverage will begin on the first day of the first month after the plan receives the enrollment request. If the plan receives the request on January 3, for example, coverage would begin on February 1.
Q. What coverage will I get when I take advantage of a special enrollment opportunity?
Special enrollees must be offered the same benefits that would be available if you are enrolling for the first time. Special enrollees cannot be required to pay more for the same coverage than other individuals who enrolled when first eligible for the plan. 3
Q. Can my new group health plan deny me benefits because I have a preexisting condition?
While HIPAA previously provided limits on preexisting condition exclusions, new protections under the Affordable Care Act (ACA) prohibit group health plans from imposing any preexisting condition exclusion. Under this protection, a plan generally cannot limit or deny benefits relating to a health condition that was present before your enrollment date in the plan.
Q. Where do I find out more about special enrollment in my plan?
A description of special enrollment rights should be included in the plan materials you received when initially offered the opportunity to enroll.
Q. How will I know if I am eligible for assistance with group health plan premiums under CHIP or Medicaid?
You need to contact your state's CHIP or Medicaid program to see if your state will subsidize group health plan premiums and to determine if you are eligible for the subsidy under these programs. For information on the program in your state, call 1-877-KIDSNOW (543-7669) or visit InsureKidsNow.gov on the Web. If you are eligible for this premium assistance, you need to contact your plan administrator or employer to take advantage of the special enrollment opportunity and enroll in the group health plan.
HIPAA's Protections from Discrimination
Image: Privacy Compliance. Glorium Technolgies Blog. 2023
Q. What are HIPAA's protections from discrimination?
Under HIPAA, you and your family members cannot be denied eligibility or benefits based on certain "health factors" when enrolling in a health plan. In addition, you may not be charged more than similarly situated individuals based on any health factors. The questions and answers below define the health factors and offer some examples of what is and is not permitted under the law.
Q. What are the health factors under HIPAA?
The health factors are:
- Health status;
- Medical conditions, including physical and mental illnesses;
- Claims experience;
- Receipt of health care;
- Medical history;
- Genetic information;
- Evidence of insurability (see below);
- Disability.
Conditions arising from acts of domestic violence as well as participation in activities like motorcycling, snowmobiling, all-terrain vehicle riding, horseback riding, and skiing are 4 considered "evidence of insurability." Therefore, a plan cannot use them to deny you enrollment or charge you more for coverage. (However, benefit exclusions known as "source of injury exclusions" could affect your benefits. These exclusions are discussed in more detail below.)
Q. Can a group health plan require me to pass a physical examination before I am eligible to enroll?
No. You do not have to pass a physical exam to be eligible for enrollment. This is true for individuals who enroll when first eligible, as well as for late and special enrollees.
Q. Can my plan require me to fill out a health care questionnaire in order to enroll?
Yes, as long as the questionnaire does not ask for genetic information (including family medical history) and the health information is not used to deny, restrict, or delay eligibility or benefits, or to determine individual premiums.
Q. My group health plan required me to complete a detailed health history questionnaire and then subtracted "health points" for prior or current health conditions. To enroll in the plan, an employee had to score 70 out of 100 total points. I scored only 50 and was denied a chance to enroll. Can the plan do this?
No. In this case the plan used health information to exclude you from enrolling in the plan. This practice is discriminatory, and it is prohibited.
Q. My group health plan booklet states that if a dependent is confined to a hospital or other medical facility at the time he is eligible to enroll in the plan, that person's eligibility is postponed until he is discharged. Is this permitted?
No. A group health plan may not delay an individual's eligibility, benefits, or effective date of coverage based on confinement to a hospital or medical facility at the time he becomes eligible. Additionally, a health plan may not increase that person's premium because he was in a hospital or medical facility.
Q. My group health plan has a 90-day waiting period before allowing employees to enroll. If an individual is in the office on the 91st day, health coverage begins then. However, if an individual is not "actively at work" on that day, the plan states that coverage is delayed until the first day that person is actually at work. I missed work on the 91st day due to illness. Can I be excluded from coverage?
No. A group health plan generally may not deny benefits because someone is not "actively at work" on the day he would otherwise become eligible. However, a plan may require employees to begin work before health plan coverage is effective. A plan may also require an individual to work full time (say, 250 hours per quarter or 30 hours per week) in order to be eligible for coverage.
Q. Can my group health plan exclude or limit benefits for certain conditions or treatments?
Group health plans can exclude coverage for a specific disease or limit or exclude benefits for certain treatments or drugs, but only if the restriction applies uniformly to all similarly situated individuals and is not directed at individual participants or beneficiaries based on a health factor they may have. (Plan amendments that apply to all individuals in a group of similarly situated individuals and that are effective no earlier than the first day of the next plan year after the amendment is adopted are not considered to be directed at individual participants and beneficiaries.).
However, compliance with this rule under HIPAA does not affect whether the plan provision or practice is permitted under other laws including the ACA such as the requirement to offer essential health benefits in the individual and small group markets.
Q. How do you determine "similarly situated individuals"?
HIPAA states that plans may distinguish among employees only on "bona fide employmentbased classifications" consistent with the employer's usual business practice. For example, part time and full time employees, employees working in different geographic locations, and employees with different dates of hire or lengths of service can be treated as different groups of similarly situated individuals.
A plan may draw a distinction between employees and their dependents. Plans can also make distinctions between beneficiaries themselves if the distinction is not based on a health factor. For example, a plan can distinguish between spouses and dependent children, or between dependent children age 26 and older based on their age or student status.
Q. I have a history of high claims. Can I be charged more than others in the plan based on my claims experience?
No. Group health plans cannot charge an individual more for coverage than a similarly situated individual based on any health factor.
However, be aware that HIPAA does allow an insurer to charge one group health plan (or employer) a higher rate than it does another. When an insurance company establishes its rates, it may underwrite all covered individuals in a specific plan based on their collective health status. The result can be that one employer health plan whose enrollees have more adverse health factors can be charged a higher premium than another for the same amount of coverage. Note that compliance with this rule under HIPAA does not affect whether the practice is permitted under the ACA including the rating requirements in the small group market.
Think of it this way: HIPAA's protections from discrimination apply within a group of similarly situated individuals, not across different groups of similarly situated individuals. For example, an employer distinguishes between full-time and part-time employees. It can charge part-time employees more for coverage, but all full-time employees must pay the same rate, regardless of health status.
Also, for insured plans, state law may govern rates for health coverage. More information is available at NAIC.org.
Q. I am an avid skier. Can my employer's plan exclude me from enrollment because I ski?
No. Participation in activities such as skiing would be "evidence of insurability," which is a health factor. Therefore, it cannot be used to deny eligibility.
Q. Can my health plan deny benefits for an injury based on how I got it?
It depends. A plan can deny benefits based on an injury's source, unless an injury is the result of a medical condition or an act of domestic violence.
Therefore, a plan cannot exclude coverage for self-inflicted wounds, including those resulting from attempted suicide, if they are otherwise covered by the plan and result from a medical condition (such as depression).
However, a plan may exclude coverage for injuries that do not result from a medical condition or from domestic violence. For example, a plan generally can exclude coverage for injuries in connection with an activity like bungee jumping. While the bungee jumper may have to pay for treatment for those injuries, her plan cannot exclude her from coverage for the plan's other benefits.
Q. My group health plan says that dependents are generally eligible for coverage only until they reach age 26. However, this age restriction does not apply to disabled dependents, who seem to be covered past age 26. Does HIPAA permit a policy favoring disabled dependents?
Yes. A plan can treat an individual with an adverse health factor (such as a disability) more favorably by offering extended coverage.
Q. Are all family members, including a spouse, covered by HIPAA?
If your group health plan permits coverage of family members ("dependents"), and if they participate in the plan, then they will have the same HIPAA protections as employees.
Note: The Genetic Information Nondiscrimination Act of 2008 (GINA) expands the HIPAA nondiscrimination provisions discussed above by generally prohibiting the use of genetic information to adjust group premiums or contributions, the collection of genetic information and requests for individuals to undergo genetic testing.
HIPAA and Wellness Programs
Image: Doctor . AI-generated by Arfas_STD via Freepik.com
Q. I've learned that my health plan will include a wellness program next year. What is a wellness program?
Wellness programs encourage employees to work out, stop smoking or generally adopt healthier lifestyles by offering some type of financial or other incentive. If a wellness program is part of a group health plan, it must comply with rules created by HIPAA and the ACA that prevent the employee from being impermissibly discriminated against based on a health factor.
There are two types of wellness programs - participatory and health-contingent. A participatory wellness program is one that offers a reward simply for participating in the program. For example, the program reimburses employees for all or part of the cost for membership in a fitness center. Participatory wellness programs are allowed under the nondiscrimination rules as long as they are available to all similarly situated individuals.
A health-contingent wellness program is one that rewards an employee for satisfying a standard related to a health factor. If the standard is an activity-only one, you need to perform or complete an activity, like walking or other exercise, to get the reward. If the standard is outcome-based, you must achieve a specific health outcome, like a certain result on a health screening, to get the reward. Health-contingent wellness programs must meet certain requirements.
Q. I belong to a group health plan that rewards individuals who volunteer to be tested for early detection of health problems, such as high cholesterol. Can a plan do this?
Yes, as long as the program is available to all similarly situated individuals. If the health plan offers a reward based on participation in the program and not on test results, the program is considered a participatory wellness program and the plan does not have to comply with the additional requirements applicable to health-contingent wellness programs. For instance, a health plan can offer a premium discount for those who voluntarily test for cholesterol, as long as the discount is available to everyone who takes the test and not just those who get a certain result. If the discount was based on individuals having certain results, additional requirements discussed below would apply.
Q. My plan's wellness program offers a lower deductible to those who participate in a specific walking program. How can I tell if this is permissible?
Because the reward (the lower deductible) is available to all who participate in a walking program, this is an activity-only health-contingent program. The program will be permissible if:
- Individuals have a chance to qualify for the reward at least once per year;
- The total reward for all of the plan's health-contingent wellness programs is not more than 30% of the cost of employee-only coverage in the plan. If dependents can participate, the reward cannot be more than 30% of the cost of the coverage in which an employee and dependents are enrolled. For wellness programs designed to prevent or 8 reduce tobacco use the allowable percentage is higher – the reward for those programs cannot be more than 50% of the cost of coverage;
- The walking program is reasonably designed to promote health or prevent disease;
- A reasonable alternative standard (or a waiver of the walking requirement) is offered to those for whom it is unreasonably difficult because of a medical condition, or medically inadvisable, to participate in the walking program; and
- The plan discloses the availability of a reasonable alternative standard (or the possibility of a waiver) in all materials describing the terms of the program.
Q. I would like to participate in my plan's wellness program. Under the program, to get a discount on my premiums, my body mass index (BMI) must be 26 or lower. Is there any way for me to get the premium discount if my BMI is higher than 26?
Yes. The reward is provided to those who achieve a specific health outcome (BMI of 26 or lower), so this is an outcome-based health-contingent wellness program. If your BMI is above 26, the plan must provide you with a reasonable alternative standard to qualify for the reward. The reasonable alternative standard could be activity-based such as completion of an educational program, participation in a diet program, or following the recommendations of your personal physician; it could also be another outcome-based standard, such as a one-point reduction in your BMI over a set period of time. If it is unreasonably difficult because of a medical condition, or medically inadvisable, for you to complete the alternative, the plan must work with you to find a second alternative based on your physician's recommendations.
In addition, as with an activity-only program, you must be given the chance to qualify for the reward at least once per year; the total reward for the plan's health-contingent wellness programs cannot be more than 30% (or 50% for tobacco-related programs) of the cost of employee-only coverage (or the cost of the coverage enrolled in if dependents can participate); and the plan must disclose the availability of a reasonable alternative standard (or the possibility of a waiver) in all materials describing the terms of the program. This notice must also be included in any disclosure that you did not satisfy the initial standard.
Q. Can a plan charge a lower premium for nonsmokers than it does for smokers?
The plan is offering a reward based on an individual's ability to stop smoking so this is an outcome-based program. For this type of wellness program to be permissible:
- Individuals must have a chance to qualify for the nonsmoker's discount at least once a year;
- The difference in premiums between nonsmokers and smokers cannot be more than 50% of the cost of employee-only coverage (or 50% of the cost of coverage if dependents can participate);
- The program must be reasonably designed to promote health and prevent disease;
- There is a reasonable alternative standard to those who do not meet the otherwise applicable standard. For example, the reasonable alternative standard could include discounts in return for attending educational classes or for trying a nicotine patch; and
- Plan materials describing the premium discount (and any disclosure that an individual did not satisfy the standard) describe the availability of a reasonable alternative standard to qualify for the lower premium.
Coordination with Other Laws
Image: Personal Data. Glorium Technolgies Blog. 2023
Q. Can states modify HIPAA's requirements?
State laws may complement HIPAA by allowing more protections than the Federal law. For example, states may increase the number of days parents have to enroll newborns, adopted children, and children placed for adoption or require additional circumstances that entitle you to special enrollment periods beyond those in the Federal law. However, these state laws only apply if your plan provides benefits through an insurance company or HMO (an insured plan). To determine if your plan offers insured coverage, consult your Summary Plan Description (SPD) or contact your plan administrator. You also can visit your state insurance commissioner's office or the National Association of Insurance Commissioners' Website (select your state) for more information.
Q. How can I use HIPAA in conjunction with COBRA to extend my health coverage?
COBRA is a law that can help if you lose your job or if your hours are reduced to the point where the employer no longer provides you with health coverage. COBRA can provide a temporary extension of your health coverage – as long as you and your family members, if eligible, belonged to the previous employer's health plan and generally the employer had 20 or more employees. Usually, you pay the entire cost of coverage (both your share and the employer's, plus a 2 percent administrative fee). As long as the prior plan exists, COBRA coverage lasts up to 18 months for most people, although it can continue as long as 36 months in some cases.
If you enroll in COBRA, HIPAA provides you with the opportunity to request special enrollment in a different group health plan if you have a special enrollment event, such as marriage, the birth of a child, or if you exhaust your continuation coverage. To exhaust COBRA, you must receive the maximum period of continuation coverage available (usually 18 months for job loss) without early termination. If you choose to terminate your COBRA early, or fail to pay your COBRA premiums, you generally will not be entitled to special enroll in other group health coverage.
Q. Do I have other special enrollment rights?
In addition to the special enrollment rights in a group health plan under HIPAA (described above), there are also special enrollment rights under the ACA for individual coverage including through the Health Insurance Marketplace. The Marketplace offers "one-stop shopping" to find and compare private health insurance and other options (such as Medicare and CHIP coverage). Losing your job-based coverage, marriage, birth, and adoption are a few of the special enrollment events that may allow you to purchase Marketplace or other coverage outside of the regular enrollment period.
To qualify for special enrollment, you must select a plan either within 60 days before losing your job-based coverage or within 60 days after losing your job-based coverage.
You can apply for Marketplace coverage online or get more information at HealthCare.gov or by calling 1-800-318-2596 (TTY users should call 1-855-889-4325). When you fill out a 10 Marketplace application, you also can find out if you and your family qualify for free or lowcost coverage from Medicaid and/or the Children's Health Insurance Program (CHIP).
Q. Where can I get more information on my rights under HIPAA?
The Employee Benefits Security Administration offers more information on HIPAA and other laws mentioned above. Visit the Employee Benefits Security Administration's Website to view the following publications. To order copies or to request assistance from a benefits advisor, contact EBSA electronically or call toll free 1-866-444-3272.
Source: " FAQs on HIPPA" Employee Benefits Security Administration. U.S. Department of Labor HIPP
NOTE: A printable copy of HIPPA Facts is attached below.


Skip to main content
-
biases
-
prejudicial attitudes or preconceptions that act as filters to the way one perceives, interprets, and reacts to the world
-
cultural awareness
-
the state of examining and exploring one’s cultural background, beliefs, and biases and consciously avoiding these biases when interacting with clients from other cultures
-
cultural competemility
-
the combination of cultural competence and cultural humility in a synergistic process that infuses cultural humility into the five components of the Campinha-Bacote process of cultural competence model, encouraging health care professionals to be culturally humble in the delivery of health care services (Campinha-Bacote, 2018)
-
cultural competence
-
the attitudes, knowledge, and skills necessary for providing quality care to diverse populations (American Association of Colleges of Nursing, 2008)
-
cultural desire
-
in the Campinha-Bacote process of cultural competence model, the motivation to engage in the process of becoming culturally competent and the commitment to providing culturally responsive care
-
cultural encounters
-
in the Campinha-Bacote process of cultural competence model, deliberate interactions with clients from different cultural backgrounds intended to modify or refine one’s existing beliefs about one’s own culture and help prevent stereotyping
-
cultural humility
-
acknowledging, respecting, and valuing cultural differences by taking responsibility for one’s own perceptions and ideas; cultivating a deep respect for all cultures represented; and striving to learn more about and understand different people’s perspectives
-
cultural identity
-
the unique combination of beliefs, values, attitudes, behaviors, customs, practices, experiences, and language that make up an individual’s sense of belonging to a particular group or culture
-
cultural knowledge
-
in the Campinha-Bacote process of cultural competence model, the foundation for understanding the diversity of cultural groups
-
cultural skill
-
in the Campinha-Bacote process of cultural competence model, the ability to collect relevant cultural data in a culturally appropriate manner, perform culturally sensitive assessments, and understand how biological and physiologic variations can influence the accuracy of the data
-
culturally and linguistically appropriate services (CLAS)
-
services that emphasize the importance of tailoring health care to an individual’s cultural and language preferences, respecting and responding to the health needs and preferences of all individuals to achieve health equity, and ensuring that everyone has access to high-quality care
-
culturally congruent care
-
care that is meaningful to, and aligns with, a client’s or population’s health beliefs and behaviors (Leininger, 1991, 1999)
-
culturally responsive care
-
the process of providing care to individuals or populations while fully considering their cultural backgrounds, values, and preferences; involves actively making sure that all aspects of health care are tailored to the needs and beliefs of the specific population
-
explicit bias
-
overt prejudicial attitudes or preconceptions that exist at a conscious, intentional level and are often expressed verbally
-
high-context culture
-
a communication style that relies more on nonverbal cues, context, and relationships between people than on verbal language, meaning that words alone may not convey the full meaning of a message
-
implicit bias
-
a type of prejudice that unconsciously affects a person’s behavior, decisions, and attitudes, even if it contradicts their expressed beliefs and values
-
interpretation
-
a process that occurs during oral communication in which the words spoken in one language are rendered into another spoken language in real time
-
Inventory for Assessing the Process of Cultural Competence Among Healthcare Professionals Revised (IAPCC-R)
-
a tool used to assess the cultural competence of health care professionals according to the five constructs of Campinha-Bacote’s model
-
linguistic competence
-
the knowledge, attitudes, and skills that enable nurses to care and advocate for clients who communicate in a language other than the one most commonly spoken or understood in the service setting
-
low literacy
-
the inability to read, write, and comprehend basic information
-
low-context culture
-
a direct communication style that uses clear and explicit language
-
othering
-
the process of excluding, marginalizing, or differentiating individuals or groups based on their perceived differences from the dominant group in a society
-
self-awareness
-
a process that involves understanding one’s own values and beliefs, how they are influenced by one’s cultural background, and how they influence one’s perceptions and behaviors
-
stereotyping
-
the act of ascribing certain characteristics or behaviors to a person based on their membership in a particular group
-
Transcultural Self-Efficacy Tool (TSET)
-
a scale consisting of 83 items in the cognitive, practical, and affective categories that is designed to measure the confidence of nurses and nursing students in providing transcultural nursing care (Jeffreys & Dogan, 2010)
-
translation
-
a process in which written text in one language is converted to written text in another language


3.1: Taking the First Step - Gathering Baseline Data
selected template will load here
This action is not available.
In order to truly measure the learning, growth, and development of a child over time, there needs to be a point of reference , or a starting point . Baseline data provides a starting point. As recommended by the California Department of Education, in the Desired Results Developmental Profile (DRDP), to establish a baseline, children should be observed “within 60 calendar days of enrollment and [formally assessed] every 6 months thereafter” (DRDP, 2015, p. ii). More specifically, any time a child starts your program, or any time you introduce a new concept, topic or theme, it would be beneficial to gather baseline data. With each observation the goal is to track - What the child can do. Then, as we review each observation, we further consider what the child has the potential to do. Knowing specific details about how the child responds to the activities we have planned, and how the child is interacting with their peers, allows intentional teachers to make informed decisions that are in the best interest of the child. Baseline data allows us to create individualized activities and enriched learning opportunities and set up engaging environments where each child can feel empowered, challenged and well cared for.
Here is an example:
Aaron is 3.5 years old and this is his first time in preschool. To see whether Aaron can write his name, you would set out writing materials (markers or crayons, paper and possibly stencils), and you would observe Aaron at the writing center. To gather baseline data, you could use an Anecdotal Note to record how Aaron holds the markers – is he using the palmer grasp or the pincher grasp? Is he using his right or left hand? You would also want to note what Aaron created - did he write his name, draw a picture or scribble? If Aaron scribbled, he might not be ready to use lined paper to write his name or to journal a story. Rather than planning an activity that would require Aaron to write between the lines, it may be more beneficial to plan activities that would help him further develop his fine motor skills. Perhaps you would set out activities that would build his pincher grasp like play dough or stringing beads. Once you observe Aaron’s progression and his “signs of readiness”, then you would reintroduce writing between the lines.


41.3.11: Chapter 12
Review Questions
1.
a.
Mannitol is administered for cerebral edema and increased intracranial pressure. Elevating the head of the bed to a 30-degree angle and keeping the client’s head in a neutral position helps to decrease intracranial pressure.
2.
c.
desired dose
supply on hand
3.
b.
Sumatriptan is contraindicated in pregnancy, and contraception should be taken while on this drug.
4.
a.
Valproic acid is a valproate, which is used in the treatment of seizures and as a mood stabilizer.
5.
b.
desired dose
supply on hand
6.
d.
Rash is an adverse effect of ethosuximide.
7.
b.
Migraine headache symptoms can last for several hours up to several days.
8.
d.
The normal intracranial pressure for a client in a vertical position is 7–15 mm Hg.
9.
a.
Acetazolamide is an osmotic diuretic and is used in the treatment of cerebral edema, intracranial hypertension, and to lower increased intracranial pressure.
10.
d.
Levetiracetam is used in the treatment of seizures and causes adverse effects such as behavioral symptoms of hostility and aggression as well as psychotic symptoms of hallucinations and delirium.


4.3: Graphs of Polynomial Functions
- Identify polynomial functions.
- Identify end behavior of polynomial functions.
- Find x-intercepts of polynomial functions.
Try these questions prior to beginning this section to help determine if you are set up for success:
- Identify the degree and leading coefficient of the given polynomial: \(4x^3-7x^5+9x^4-11+2x-x^4\)
-
Solve:
- \(\quad 2j^3-3j^2=8j-12\)
- \(\quad 4c^2=12c\)
- Answer
-
- The degree is 5 and the leading coefficient is -7
If you missed this problem or feel you could use more practice, review [ 2.3: Polynomial Expressions ]
-
- \(\quad j=-2\text{, }\dfrac{3}{2}\text{, }2\)
- \(\quad c=0\text{, }3\)
If you missed this problem or feel you could use more practice, review [ 2.8: Solving Polynomial Equations , 2.24: Solving Quadratic Equations Using the Square Root Property , 2.25: Solving Quadratic Equations by Completing the Square , and 2.26: Solving Quadratic Equations Using the Quadratic Formula ]
Polynomial functions are used widely for modeling many real-life situations. In this section, we will investigate the common traits of this important family of functions.
A polynomial function is a function of the form
\[ f(x) = a_{n} x^{n} + a_{n-1}x^{n-1} + \ldots + a_{2}x^{2} + a_{1}x + a_0, \label{polynomialfunction}\nonumber \]
where \(a_0\), \(a_1\), ... , \(a_{n}\) are real numbers and \(n \geq 1\) is a natural number. The domain of a polynomial function is \((-\infty, \infty)\).
- The natural number \(n\) is called the degree of the polynomial \(f\).
- The term \(a_{n} x^{n}\) is called the leading term of the polynomial \(f\).
- The real number \(a_{n}\) is called the leading coefficient of the polynomial \(f\).
- The real number \(a_{0}\) is called the constant term of the polynomial \(f\).
Determine if the following functions are polynomials. If they are, identify the degree, leading term, leading coefficient and constant term.
- \(f(x) = 4x^5 - 3x^2 + 2x - 5\)
- \(g(x) = 12x - x^3\)
- \(p(x) = \dfrac{4x+x^3}{x}\)
- \(h(x) = \dfrac{4-x}{5}\)
- \(A(x) = (2x-1)^{3}(x-2)(3x+2)\)
Solution
- This is a polynomial function by definition, and we see that the degree is \(5\), the leading term is \(4x^5\), the leading coefficient is \(4\) and the constant term is \(-5\).
- The definition of a polynomial function has the highest power of \(x\) first. To that end, we re-write \(g(x) = 12x - x^3 =-x^3+12x\), and see that the degree of \(g\) is \(3\), the leading term is \(-x^3\), the leading coefficient is \(-1\) and the constant term is \(0\).
- Even though \(p(x) = \dfrac{x^3+4x}{x}\) simplifies to \(p(x) = x^2+4\), which certainly looks like a polynomial function, the domain of \(p\) excludes \(0\). Alas, \(p\) is not a polynomial function.
- \(h\) can be rewritten as follows: \(h(x) = \dfrac{4-x}{5} = \dfrac{4}{5} - \dfrac{x}{5} = -\dfrac{1}{5} x + \dfrac{4}{5}\). The degree of \(h\) is \(1\), the leading term is \(-\dfrac{1}{5} x\), the leading coefficient is \(-\dfrac{1}{5}\) and the constant term is \(\dfrac{4}{5}\).
- It may seem that we have some work ahead of us to get \(A\) in the form provided in the definition of a polynomial function. However, it is possible to gain the information requested about \(A\) without multiplying out the entire expression \((2x-1)^{3}(x-2)(3x+2)\). The leading term of \(A\) will be the term which has the highest power of \(x\). The way to get this term is to multiply the terms with the highest power of \(x\) from each factor together - in other words, the leading term of \(A(x)\) is the product of the leading terms of the factors of \(A(x)\). Hence, the leading term of \(A\) is \((2x)^3(x)(3x) = 24x^5\). This means that the degree of \(A\) is \(5\) and the leading coefficient is \(24\). As for the constant term, we can perform a similar trick. The constant term is obtained by multiplying the constant terms from each of the factors \((-1)^3(-2)(2) = 4\).
Now that we know how to identify polynomial functions, we can start to investigate their graphical behavior. Polynomial functions share two properties which help distinguish them from other animals (functions) in the algebra zoo: they are continuous and smooth. While these concepts are formally defined in Calculus, informally, polynomial graphs have no 'breaks', 'holes', or 'sharp turns' (also known as a 'cusp'). Below, in Figure \(\PageIndex{1}\), on the left we a graph of a function which is neither smooth nor continuous, and to its right we have a graph of a polynomial, for comparison.
Next, let's look at monomials (single term polynomials) to help us establish some graphical pattern that is similar with all polynomials.
End Behavior of Polynomial Functions
Figure \(\PageIndex{2}\) shows the graphs of \(f(x)=x^2\), \(g(x)=x^4\) and and \(h(x)=x^6\), which are monomial functions with even degrees. Notice that these graphs have similar shapes, very much like that of the basic quadratic function. They are commonly called basic even-power functions. However, as the power increases, the graphs flattens near the origin within the interval \((-1,1)\) and becomes steeper must faster outside of this interval.
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 \(x{\rightarrow}\infty\), we are describing a behavior; we are saying that \(x\) is increasing without bound.
With the even-degree monomial functions, 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 write this as
\[\text{as } x{\rightarrow}{\pm}{\infty}, \;f(x){\rightarrow}{\infty} \nonumber \]
Figure \(\PageIndex{3}\) shows the graphs of \(f(x)=x^3\), \(g(x)=x^5\), and \(h(x)=x^7\), which are all odd degree monomial functions, commonly called basic odd-degree power functions . Notice that these graphs look similar to the basic cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.
These examples illustrate that functions of the form \(f(x)=x^n\) where \(n\) is a positive integer reveal symmetry of one kind or another. First, in Figure \(\PageIndex{2}\) we see that functions of the form \(f(x)=x^n\), \(n\) even, are symmetric about the \(y\)-axis. In Figure \(\PageIndex{3}\) we see that functions of the form \(f(x)=x^n\), \(n\) odd, are symmetric about the origin.
For these odd powered 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
\[\begin{align*} &\text{as }x{\rightarrow}-{\infty},\;f(x){\rightarrow}-{\infty} \\ &\text{as }x{\rightarrow}{\infty},\;f(x){\rightarrow}{\infty} \end{align*}\nonumber \]
The behavior of the graph of a function as the input values get very small \((x{\rightarrow}−{\infty})\) and get very large \(x{\rightarrow}{\infty}\) 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 various monomial functions in the form \(f(x)=kx^n\) where \(n\) is a non-negative integer depending on the power and the constant.
The end behavior describes above can be generalized for all polynomials as follows:
Suppose \(f(x) = a x^{n}\) where \(a \neq 0\) is a real number and \(n\) is an even natural number. The end behavior of the graph of \(y=f(x)\) matches one of the following:
- for \(a > 0\), as \(x \rightarrow -\infty\), \(f(x) \rightarrow \infty\) and as \(x \rightarrow \infty\), \(f(x) \rightarrow \infty\)
- for \(a < 0\), as \(x \rightarrow -\infty\), \(f(x) \rightarrow -\infty\) and as \(x \rightarrow \infty\), \(f(x) \rightarrow -\infty\)
Graphically:
Figure \(\PageIndex{5}\)
Suppose \(f(x) = a x^{n}\) where \(a \neq 0\) is a real number and \(n \geq 3\) is an odd natural number. The end behavior of the graph of \(y=f(x)\) matches one of the following:
- for \(a > 0\), as \(x \rightarrow -\infty\), \(f(x) \rightarrow -\infty\) and as \(x \rightarrow \infty\), \(f(x) \rightarrow \infty\)
- for \(a < 0\), as \(x \rightarrow -\infty\), \(f(x) \rightarrow \infty\) and as \(x \rightarrow \infty\), \(f(x) \rightarrow -\infty\)
Graphically:
Figure \(\PageIndex{6}\)
Describe the end behavior and determine a possible degree of the polynomial function in 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 small, the output values \(f(x)\) decrease without bound. We can describe the end behavior symbolically by writing
\[\text{as } x{\rightarrow}{\infty}, \; f(x){\rightarrow}{\infty} \nonumber \]
\[\text{as } x{\rightarrow}-{\infty}, \; f(x){\rightarrow}-{\infty} \nonumber \]
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 polynomial 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 \(\PageIndex{8}\).
- Answer
-
As \(x{\rightarrow}{\infty}\), \(f(x){\rightarrow}−{\infty}\); as \(x{\rightarrow}−{\infty}\), \(f(x){\rightarrow}−{\infty}\). It has the shape of an even degree polynomial function with a negative coefficient.
Given the function \(f(x)=−3x^2(x−1)(x+4)\), determine the leading term, degree, and end behavior of the function.
Solution
First, expand the function,
\[\begin{align*} f(x)&=−3x^2(x−1)(x+4) \\ &=−3x^2(x^2+3x−4) \\ &=−3x^4−9x^3+12x^2 \end{align*}\]
The leading term is \(−3x^4\); 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
\[\text{as }x{\rightarrow}−{\infty}, \; f(x){\rightarrow}−{\infty} \nonumber \]
\[\text{as } x{\rightarrow}{\infty}, \; f(x){\rightarrow}−{\infty} \nonumber \]
Given the function \(f(x)=0.2(x−2)(x+1)(x−5)\), determine the leading term, degree, and end behavior of the function.
- Answer
-
The leading term is \(0.2x^3\), so it is a degree 3 polynomial.
\[\text{as }x{\rightarrow}−{\infty}, \; f(x){\rightarrow}−{\infty} \nonumber \]
\[\text{as } x{\rightarrow}{\infty}, \; f(x){\rightarrow}{\infty} \nonumber \]
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 the intercepts and turning points. 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 \(\PageIndex{9}\). Turning points are local extrema values. We may approximate them using technology but it will take calculus to find them exactly. Hence, we will focus our time in this course are finding intercepts only.
Before we look at finding intercepts, let's note some important theory about graphical behavior.
Suppose \(f\) is a continuous function on an interval containing \(x=a\) and \(x=b\) with \(a<b\). If \(f(a)\) and \(f(b)\) have different signs, then \(f\) has at least one zero between \(x = a\) and \(x = b\); that is, for at least one real number \(c\) such that \(a < c < b\), we have \(f(c) = 0\).
The Intermediate Value Theorem is extremely profound; it gets to the heart of what it means to be a real number, and is one of the most often used and under appreciated theorems in Mathematics. With that being said, most students see the result as common sense since it says, geometrically, that the graph of a polynomial function cannot be above the \(x\)-axis at one point and below the \(x\)-axis at another point without crossing the \(x\)-axis somewhere in between.
Given the polynomial function \(f(x)=(x−2)(x+1)(x−4)\), determine the \(y\)- and \(x\)-intercepts.
Solution
The \(y\)-intercept occurs when the input is zero, so substitute 0 for \(x\).
\[ \begin{align*}f(0)&=(0−2)(0+1)(0−4) \\ &=(−2)(1)(−4) \\ &=8 \end{align*} \nonumber \]
The \(y\)-intercept is \((0,8)\).
The \(x\)-intercepts occur when the output is zero. The function is already factored which makes this equation very nice to solve:
\[ 0=(x−2)(x+1)(x−4) \nonumber \]
\[\begin{array}{l} {x-2 =0} \\ {x=2} \end{array} \qquad \text{ or } \qquad \begin{array}{l} {x+1=0} \\ {x =-1} \end{array}\qquad \text{ or } \qquad \begin{array}{l} {x-4 =0} \\ {x =4} \end{array} \nonumber \]
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 \(\PageIndex{10}\).
Given the polynomial function \(f(x)=2x^3−6x^2−20x\), determine the intercepts.
- Answer
-
\(x\)-intercepts \((0,0)\),\((–2,0)\), and \((5,0)\), \(y\)-intercept \((0,0)\);
Find the x-intercepts of \(f(x)=x^{6} -3x^{4} +2x^{2}\).
Solution
We can attempt to factor this polynomial to find solutions for \(f(x) = 0\).
\[x^{6} -3x^{4} +2x^{2} =0 \nonumber\] Factoring out the greatest common factor
\[x^{2} (x^{4} -3x^{2} +2)=0 \nonumber \] Factoring the inside as a quadratic in x\({}^{2}\)
\[x^{2} (x^{2} -1)(x^{2} -2)=0 \nonumber \] Then break apart to find solutions
\[\begin{array}{l} {x^{2} =0} \\[4pt] {x=0} \end{array}\qquad\text{ or } \qquad \begin{array}{l} {x^{2} -1=0} \\[4pt] {x^{2} =1} \\[4pt] {x=\pm 1} \end{array}\qquad \text{ or } \qquad \begin{array}{l} {x^{2} -2=0} \\[4pt] {x^{2} =2} \\ {x=\pm \sqrt{2} } \end{array} \nonumber \]
This gives us 5 x-intercepts: \((0,0)\), \((-\sqrt{2},0)\), \((–1,0)\), \((1,0)\), and \((\sqrt{2},0)\).
Given the polynomial function \(f(x)=x^4−4x^2−45\), determine the \(y\)- and \(x\)-intercepts.
Solution
The \(y\)-intercept occurs when the input is zero.
\[ \begin{align*} f(0) &=(0)^4−4(0)^2−45 \\[4pt] &=−45 \end{align*}\]
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.
\[\begin{align*} f(x)&=x^4−4x^2−45 \\ &=(x^2−9)(x^2+5) \\ &=(x−3)(x+3)(x^2+5)
\end{align*}\]
\[0=(x−3)(x+3)(x^2+5) \nonumber \]
\[\begin{array}{l} {x-3 =0} \\ {x=3} \end{array} \qquad \text{ or } \qquad \begin{array}{l} {x+3=0} \\ {x =-3} \end{array}\qquad \text{ or } \qquad \begin{array}{l} {x^2+5=0} \\ \text{no real solution} \end{array} \nonumber \]
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 \(f(x)=f(−x)\).
Find the intercepts of \(D(x)=3(x-2)^6-8\).
Solution
This is a basic even-power function that has has a horizontal shift right 2 units, vertical shift down 8 units, and a vertical stretch by 3. Since the basic shape is similar to a quadratic function, we know the graph opens up so that indicates that there are two x-intercepts. If you take a moment to sketch the function on graph paper using your knowledge of transformations, you will also notice that the x-intercepts should have x-coordinates close to 1 and 3. See Figure \(\PageIndex{12}\)
Finding the x-intercepts, \[\begin{align*} 0&=3(x-2)^6-8\\[4pt] 8&=3(x-2)^6\\[4pt] \dfrac{8}{3}&=(x-2)^6\\[4pt] \pm\sqrt[6]{\dfrac{8}{3}}&=\sqrt[6]{(x-2)^6}\\[4pt] \pm\sqrt[6]{\dfrac{8}{3}}&=x-2\\[4pt] 2\pm\sqrt[6]{\dfrac{8}{3}}&=x\\[4pt] \end{align*}\]The x-intercepts are \(\left(2-\sqrt[6]{\dfrac{8}{3}},0\right)\) and \(\left(2+\sqrt[6]{\dfrac{8}{3}},0\right)\).
Note that if you approximate the x-intercepts that they should be about where we anticipated them to be when we sketched the graph.
Finding the y-intercept, \[\begin{align*} D(0)&=3(x-2)^6-8\\[4pt]&=3(0-2)^6-8\\[4pt]&=3(-2)^6-8\\[4pt]&=3(64)-8\\[4pt]&=184 \end{align*}\]The y-intercept is \((0,184)\).
Local Extrema of Polynomials
Recall that local extrema are function occur where a function changes from increasing to decreasing or from decreasing to increasing. Turning points for a graph of a polynomial function occur at extrema values. Although calculus is required to prove the following, this is an important theorem to know about the number of local extrema values we can expect for a polynomial function.
A polynomial of degree \(n\) will have at most \(n-1\) local extrema.
Note that a polynomial of degree \(n\) can have less than \(n-1\) local extrema. For example, \(P(x)=2x^5\) is a polynomial of degree 5 but it has no local extrema.
Graphical Behavior Near X-Intercepts
If we graph the function \(f(x)=(x+3)(x-2)^{2} (x+1)^{3}\), as shown in Figure \(\PageIndex{13}\), notice that the behavior at each of the x-intercepts is different.
At the x-intercept at \((-3,0)\), which came from the \((x+3)\) factor of the polynomial, the graph passes directly through this horizontal intercept.
The factor \((x+3)\) is linear (has a power of 1), so the behavior near the intercept is like that of a line - it passes directly through the intercept. We call this a single zero, since the zero corresponds to a single factor of the function.
At the x-intercept at \((2,0)\), which came from the \((x-2)^{2}\) factor of the polynomial, the graph touches the axis at the intercept and then the graph bounces. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic – it bounces off the horizontal axis at the intercept. Since \((x-2)^{2} =(x-2)(x-2)\), the factor is repeated twice, so we call this a double zero. We could also say the zero has multiplicity 2.
At the x-intercept at \((-1,0)\), which came from the \((x+1)^{3}\) factor of the polynomial, the graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic, with the same “S” type shape near the intercept that the cubic parent function, \(x^{3}\), has. We call this a triple zero. We could also say the zero has multiplicity 3.
By utilizing these behaviors, we can sketch a reasonable graph of a factored polynomial function without needing technology.
If a polynomial contains a factor of the form \((x-h)^{p}\), the behavior near the horizontal intercept \(h\) is determined by the power on the factor.
For higher even powers 4,6,8 etc. … the graph will bounce off the horizontal axis but the graph will appear flatter with each increasing even power as it approaches and leaves the axis.
For higher odd powers, 5,7,9 etc … the graph will pass through the horizontal axis but the graph will appear flatter with each increasing odd power as it approaches and leaves the axis.
Sketch a graph of \(f(x)=-2(x+3)^{2} (x-5)\).
Solution
This graph has two horizontal intercepts. At -3, the factor it came from is squared, indicating the graph will bounce at this horizontal intercept. At 5, the factor t came from is linear, indicating the graph will pass through the axis at this intercept.
Additionally, we can see the leading term, if this polynomial were multiplied out, would be \(-2x^{3}\), so the long-run behavior the left side of the graph will go upward and the right side will go downward.
To sketch this we consider the following:
As \(x \to -\infty\) the function \(f(x) \to \infty\) so we know the graph starts in the 2\({}^{nd}\) quadrant and is decreasing toward the horizontal axis.
At (-3, 0) the graph bounces off the horizontal axis and so the function must start increasing.
At (0, 90) the graph crosses the vertical axis at the vertical intercept.
Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis since the graph passes through the next intercept at (5,0).
As \(x \to \infty\) the function\(f(x) \to -\infty\) so we know the graph continues to decrease and we can stop drawing the graph in the 4\({}^{th}\) quadrant.
Using technology and an appropriate viewing window we can verify the shape of the graph.
Write a formula for the polynomial function graphed in Figure \(PageIndex{16}\).
Solution
This graph has three horizontal intercepts: \(x\) = -3, 2, and 5. At \(x\) = -3 and 5 the graph passes through the axis, suggesting the corresponding factors of the polynomial will be linear. At \(x = 2\) the graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be 2\({}^{nd}\) degree (quadratic).
Together, this gives us:
\[f(x)=a(x+3)(x-2)^{2} (x-5) \nonumber\]
To determine the stretch factor, we can utilize another point on the graph. Here, the vertical intercept appears to be (0,-2), so we can plug in those values to solve for \(a\):
\[\begin{array}{l} {-2=a(0+3)(0-2)^{2} (0-5)} \\ {-2=-60a} \\ {a=\dfrac{1}{30} } \end{array} \nonumber\]
The graphed polynomial appears to represent the function
\[f(x)=\dfrac{1}{30} (x+3)(x-2)^{2} (x-5) \nonumber \]
Given the graph, write a formula for the function shown in Figure \(\PageIndex{17}\).
- Answer
-
Double zero at \(x= - 1\), triple zero at \(x = 2\). Single zero at \(x = 4\). \(f(x) = a(x - 2)^{3} (x+1)^{2} (x-4)\). Substituting (0,-4) and solving for \(a\), \(f(x) = -\dfrac{1}{8} (x-2)^{3} (x+1)^{2} (x-4)\)
Let's look at a couple more examples to apply all that we have been learning.
A box with no top, also called a tray, is to be fashioned from a \(10\) inch \(\times\) \(12\) inch piece of cardboard by cutting out congruent squares from each corner of the cardboard and then folding the resulting tabs. Let \(x\) denote the length of the side of the square which is removed from each corner.
Find the volume \(V\) of the box as a function of \(x\). Include an appropriate applied domain.
Use a graphing calculator to graph \(y=V(x)\) on the domain you found in part 1 and approximate the dimensions of the box with maximum volume to two decimal places. What is the maximum volume?
Solution
1. From Geometry, we know that \(\mbox{Volume} = \mbox{width} \times \mbox{height} \times \mbox{depth}\). The key is to find each of these quantities in terms of \(x\). From the figure, we see that the height of the box is \(x\) itself. The cardboard piece is initially \(10\) inches wide. Removing squares with a side length of \(x\) inches from each corner leaves \(10-2x\) inches for the width.\footnote{There's no harm in taking an extra step here and making sure this makes sense. If we chopped out a \(1\) inch square from each side, then the width would be \(8\) inches, so chopping out \(x\) inches would leave \(10-2x\) inches.} As for the depth, the cardboard is initially \(12\) inches long, so after cutting out \(x\) inches from each side, we would have \(12-2x\) inches remaining. As a function\footnote{When we write \(V(x)\), it is in the context of function notation, not the volume \(V\) times the quantity \(x\).} of \(x\), the volume is \[V(x) = x(10-2x)(12-2x) = 4x^3-44x^2+120x \nonumber \] To find a suitable applied domain, we note that to make a box at all we need \(x > 0\). Also the shorter of the two dimensions of the cardboard is \(10\) inches, and since we are removing \(2x\) inches from this dimension, we also require \(10 - 2x > 0\) or \(x < 5\). Hence, our applied domain is \(0 < x < 5\).
2. Using a graphing calculator, we see that the graph of \(y=V(x)\) has a relative maximum. For \(0 < x < 5\), this is also the absolute maximum. Using the 'Maximum' feature of the calculator, we get \(x \approx 1.81\), \(y \approx 96.77\). This yields a height of \(x \approx 1.81\) inches, a width of \(10 - 2x \approx 6.38\) inches, and a depth of \(12 - 2x \approx 8.38\) inches. The \(y\)-coordinate is the maximum volume, which is approximately \(96.77\) cubic inches (also written \(\mbox{in}^3\)).
In order to solve, we made good use of the graph of the polynomial \(y=V(x)\), so we ought to turn our attention to graphs of polynomials in general. Below are the graphs of \(y=x^2\), \(y=x^4\) and \(y=x^6\), side-by-side. We have omitted the axes to allow you to see that as the exponent increases, the 'bottom' becomes 'flatter' and the 'sides' become 'steeper.' If you take the the time to graph these functions by hand,\footnote{Make sure you choose some \(x\)-values between \(-1\) and \(1\).} you will see why.
We learned earlier in this course how to solve non-linear inequalities. Inequalities also tell us when a function will be positive and negative. Let's look at an example to ensure we understand the relationship between inequalities and graphs of functions.
Solve \((x+3)(x+1)^{2} (x-4)>0\)
Solution
As with all inequalities, we start by solving the equality \((x+3)(x+1)^{2} (x-4)=0\), which has solutions at x = -3, -1, and 4. We know the function can only change from positive to negative at these values, so these divide the inputs into 4 intervals.
We could choose a test value in each interval and evaluate the function \(f(x)=(x+3)(x+1)^{2} (x-4)\) at each test value to determine if the function is positive or negative in that interval
| Interval | Test \(x\) in interval | \(f\) (test value) | > 0 or < 0? |
|---|---|---|---|
| \(x < -3\) | -4 | 72 | > 0 |
| \(-3 < x < -1\) | -2 | -6 | < 0 |
| \(-1 < x < 4\) | 0 | -12 | < 0 |
| \(x > 4\) | 5 | 288 | > 0 |
On a number line this would look like:
From our test values, we can determine this function is positive when \(x < -3\) or \(x > 4\), or in interval notation, \((-\infty ,-3)\bigcup (4,\infty )\)
Given the function \(g(x)=x^{3} -x^{2} -6x\) use the methods that we have learned so far to find the vertical & horizontal intercepts, determine where the function is negative and positive, describe the end behavior, and sketch the graph without technology.
- Answer
-
Vertical intercept (0, 0), Horizontal intercepts (-2, 0), (0, 0), (3, 0)
The function is negative on (\(-\infty\), -2) and (0, 3)
The function is positive on (-2, 0) and (3,\(\infty\))
The leading term is \(x^{3}\) so as \(x\to -\infty\), \(g(x)\to -\infty\) and as\(x\to \infty\), \(g(x)\to \infty\)


2.4.E: Problems on Upper and Lower Bounds (Exercises)
Complete the proofs of Theorem 2 and Corollaries 1 and 2 for infima.
Prove the last clause of Note \(4 .\)
Prove that \(F\) is complete iff each nonvoid left-bounded set in \(F\) has an infimum.
Prove that if \(A_{1}, A_{2}, \ldots, A_{n}\) are right bounded (left bounded) in \(F,\) so is
\[
\bigcup_{k=1}^{n} A_{k}
\]
Prove that if \(A=(a, b)\) is an open interval \((a<b),\) then
\[
a=\inf A \text { and } b=\sup A.
\]
In an ordered field \(F,\) let \(\emptyset \neq A \subset F .\) Let \(c \in F\) and let \(c A\) denote the set of all products \(c x(x \in A) ;\) i.e.,
\[
c A=\{c x | x \in A\}.
\]
\[
\begin{array}{l}{\text { (i) if } c \geq 0 \text { , then }} \\ {\qquad \begin{array}{l}{\sup (c A)=c \cdot \sup A \text { and } \inf (c A)=c \cdot \inf A} \\ {\text { (ii) if } c<0 \text { , then }} \\ {\qquad \sup (c A)=c \cdot \inf A \text { and } \inf (c A)=c \cdot \sup A}\end{array}}\end{array}
\]
In both cases, assume that the right-side sup \(A\) (respectively, inf \(A )\) exists.
From Problem 5\((\text { ii })\) with \(c=-1,\) obtain a new proof of Theorem 1.
[Hint: If \(A\) is left bounded, show that \((-1) A\) is right bounded and use its supremum. \(]\)
Let \(A\) and \(B\) be subsets of an ordered field \(F .\) Assuming that the required lub and glb exist in \(F,\) prove that
(i) if \((\forall x \in A)(\forall y \in B) x \leq y,\) then \(\sup A \leq \inf B\);
(ii) if \((\forall x \in A)(\exists y \in B) x \leq y,\) then \(\sup A \leq \sup B\);
(iii) if \((\forall y \in B)(\exists x \in A) x \leq y,\) then \(\inf A \leq \inf B\).
\([\text { Hint for }(\mathrm{i}) : \text { By Corollary } 1,(\forall y \in B) \sup A \leq y, \text { so } \sup A \leq \inf B .(\text { Why? })]\)
For any two subsets \(A\) and \(B\) of an ordered field \(F,\) let \(A+B\) denote the set of all sums \(x+y\) with \(x \in A\) and \(y \in B ;\) i.e.,
\[
A+B=\{x+y | x \in A, y \in B\}.
\]
Prove that if \(\sup A=p\) and \(\sup B=q\) exist in \(F,\) then
\[
p+q=\sup (A+B);
\]
similarly for infima.
[Hint for sup: By Theorem \(2,\) we must show that
(i) \((\forall x \in A)(\forall y \in B) x+y \leq p+q(\text { which is easy })\) and
(ii')\((\forall \varepsilon>0)(\exists x \in A)(\exists y \in B) x+y>(p+q)-\varepsilon\).
Fix any \(\varepsilon>0 .\) By Theorem 2,
\[
(\exists x \in A)(\exists y \in B) \quad p-\frac{\varepsilon}{2}<x \text { and } q-\frac{\varepsilon}{2}<y .(\mathrm{Why} ?)
\]
Then
\[
x+y>\left(p-\frac{\varepsilon}{2}\right)+\left(q-\frac{\varepsilon}{2}\right)=(p+q)-\varepsilon,
\]
as required. \(]\)
In Problem 8 let \(A\) and \(B\) consist of positive elements only, and let
\[
A B=\{x y | x \in A, y \in B\}.
\]
Prove that if \(\sup A=p\) and \(\sup B=q\) exist in \(F,\) then
\[
p q=\sup (A B);
\]
similarly for infima.
[Hint: Use again Theorem 2\(\left(\mathrm{ii}^{\prime}\right) .\) For \(\sup (A B),\) take
\[
0<\varepsilon<(p+q) \min \{p, q\}
\]
and
\[
x>p-\frac{\varepsilon}{p+q} \text { and } y>q-\frac{\varepsilon}{p+q};
\]
show that
\[
x y>p q-\varepsilon+\frac{\varepsilon^{2}}{(p+q)^{2}}>p q-\varepsilon.
\]
For inf \((A B),\) let \(s=\inf B\) and \(r=\inf A ;\) choose \(d<1,\) with
\[
0<d<\frac{\varepsilon}{1+r+s}.
\]
Now take \(x \in A\) and \(y \in B\) with
\[
x<r+d \text { and } y<s+d,
\]
and show that
\[
x y<r s+\varepsilon.
\]
Explain!
Prove that
(i) if \((\forall \varepsilon>0) a \geq b-\varepsilon,\) then \(a \geq b\);
(ii) if \((\forall \varepsilon>0) a \leq b+\varepsilon,\) then \(a \leq b\).
Prove the principle of nested intervals: If \(\left[a_{n}, b_{n}\right]\) are closed intervals in a complete ordered field \(F,\) with
\[
\left[a_{n}, b_{n}\right] \supseteq\left[a_{n+1}, b_{n+1}\right], \quad n=1,2, \ldots
\]
then
\[
\bigcap_{n=1}^{\infty}\left[a_{n}, b_{n}\right] \neq \emptyset.
\]
[Hint: Let
\[
A=\left\{a_{1}, a_{2}, \ldots, a_{n}, \ldots\right\}.
\]
Show that \(A\) is bounded above by each \(b_{n}\).
Let \(p=\sup A .\) (Does it exist?)
Show that
\[
(\forall n) \quad a_{n} \leq p \leq b_{n},
\]
i.e.,
\[
p \in\left[a_{n}, b_{n}\right] . ]
\]
Prove that each bounded set \(A \neq \emptyset\) in a complete field \(F\) is contained in a smallest closed interval \([a, b]\) (so \([a, b]\) is contained in any other \([c, d] \supseteq A )\).
Show that this fails if "closed" is replaced by "open."
\([\text { Hint: Take } a=\inf A, b=\sup A]\).
Prove that if \(A\) consists of positive elements only, then \(q=\sup A\) iff
(i) \((\forall x \in A) x \leq q\) and
(ii) \((\forall d>1)(\exists x \in A) q / d<x\).
[Hint: Use Theorem 2. \(]\)


9.4: MAC Media Procedure
MacConkey Media Procedure
Name: _____________________________________________________
Course Section: ______________________________________________
The following activity will teach students how to inoculate an MAC agar plate. Post-inoculation, students will use their colony growth to determine if the bacteria are gram positive or gram negative and the bacterial rate of lactose fermentation.
| Materials |
|
| Procedure | |
|
Step 1:
|
| Procedure |
|
Step 2:
|
|
Step 3:
|
|
Step 4:
|
Results
In table below, record your observations for colonies grown on MAC media.
| Results Table | ||||
| Bacterial Species | Gram Positive or Negative | Colony/Media Color | Rate of Lactose Fermentation | |
|
Escherichia coli
|
|
|||
|
Serratia marcescens
|
|
|||
|
Staphylococcus epidermidis
|
|
General Questions
1. What adaptation allows enteric bacteria to survive in the human gut?
2.How does MacConkey agar inhibit the growth of gram positive bacteria?
3. Why does lactose fermentation change the color of the bacteria colonies and the MAC media?
4. You grow an unknown species of bacteria on an MAC plate. The colonies turn a white color while the media turns yellow. What kind of lactose fermenter is your unknown species?
Health-Related Questions
1. P. aeruginosa bacteria target which group of people?
2. Why is it important that a patient submit a clean urine sample when trying to determine if their UTI is caused by P. aeruginosa ?
3. You collect a urine sample from a patient with a UTI and culture the sample on an MAC plate. The bacteria grow in blue-green colored colonies, but there are no changes to the media color. Would you classify this UTI as being caused by P. aeruginosa ? Explain your answer.
Attributions
"Microbiology Labs II: Results, 12.6.1 MacConkey Agar" by Dr. Gary Kaiser , LibreTexts: Biology , Community College of Baltimore County, Catonsville Campus is licensed under CC BY 4.0
"Microbiology Labs II: Pure Cultures, 3.4 Use of Specialized Media" by Dr. Gary Kaiser , LibreTexts: Biology , Community College of Baltimore County, Catonsville Campus is licensed under CC BY 4.0
"Bio 221 Lab: Physiological Tests, 22.2 Selective and Differential Media-MacConkey, EMB, MSA" by Kelly Burke , LibreTexts: Biology , College of The Canyons is licensed under CC BY


16.8: The key exchange
Substitution ciphers
In the questions below, if it specifies an alphabetic cipher, then the original map used letters only: ABCDEFGHIJKLMNOPQRSTUVWXYZ. If it specifies an alphanumeric cipher, then the original map used letters and numbers: ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789
1. Encrypt the message “SEND SUPPLIES” using an alphabetic Caesar cipher with shift 7 (mapping A to H).
2. Encrypt the message “CANCEL CONTRACT” using an alphanumeric Caesar cipher with shift 16 (mapping A to Q).
3. Decrypt the message “2R1 ONO 5SN OXM O“ if it was encrypted using an alphanumeric Ceasar cipher with shift 10 (mapping A to K).
4. Decrypt the message “RJJY NSAJ SNHJ“ if it was encrypted using an alphabetic Ceasar cipher with shift 5 (mapping A to F).
For questions 5-8 use this substitution mapping:
Original: \(\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789}\)
Maps to: \(\mathrm{HLCO2BQF5WRTZN1G4D8IJ6SUVK3A0X9YME7P}\)
5. Use the substitution mapping to encrypt the message “DEAR DIARY”
6. Use the substitution mapping to encrypt the message “ATTACK AT SUNRISE”
7. Use the substitution mapping to decrypt the message “Z2DQ 2D1N”
8. Use the substitution mapping to decrypt the message “Z22 IHI3 YX3”
Transposition ciphers
9. Encrypt the message “Meet in the library at ten” using a tabular transposition cipher with rows of length 5 characters.
10. Encrypt the message “Fly surveillance over the northern county” using a tabular transposition cipher with rows of length 8 characters.
11. Decrypt the message “THE VHI NIE SAN SHT STI MQA DAN SDR S“ if it was encrypted using a tabular transposition cipher with rows of length 7 characters.
12. Decrypt the message “DOLR UTIR INON KVEY AZ“ if it was encrypted using a tabular transposition cipher with rows of length 6 characters.
13. Encrypt the message “Buy twenty million” using a tabular transposition cipher with the encryption keyword “RENT”.
14. Encrypt the message “Attack from the northeast” using a tabular transposition cipher with the encryption keyword “POWER”.
15. Decrypt the message “RYL OEN ONI TPM IEE YTE YDH WEA HRM S” if it was encrypted using a tabular transposition cipher with the encryption keyword “READING”.
16. Decrypt the message “UYH SRT ABV HLN SEE L” if it was encrypted using a tabular transposition cipher with the encryption keyword “MAIL”.
Shifting substitution ciphers
17. Encrypt the message “SEND SUPPLIES” using an alphabetic Caesar cipher that starts with shift 7 (mapping A to H), and shifts one additional space after each character is encoded.
18. Encrypt the message “CANCEL CONTRACT” using an alphabetic Caesar cipher that starts with shift 5 (mapping A to F), and shifts one additional space after each character is encoded.
Modular arithmetic
19. Compute
- \(15 \bmod 4\)
- \(10 \bmod 5\)
- \(257 \bmod 11\)
20. Compute
- \(20 \bmod 4\)
- \(14 \bmod 3\)
- \(86 \bmod 13\)
21. Determine if 4 is a generator modulus 11
22. Determine if 2 is a generator modulus 13
23. Use the modular exponent rule to calculate \(157^{10} \bmod 5\)
24. Use the modular exponent rule to calculate \(133^{8} \bmod 6\)
Diffie-Hell man-Merkle key exchange
25. Suppose you are doing a key exchange with Marc using generator 5 and prime 23. Your secret number is 7. Marc sends you the value 3. Determine the shared secret key.
26. Suppose you are doing a key exchange with Jen using generator 5 and prime 23. Your secret number is 4. Jen sends you the value 8. Determine the shared secret key.
RSA
27. Suppose that Alice has computed \(n=33, e=7,\) and \(d=3\). Show how Bob would encrypt the message 5 and how Alice would then decrypt it.
28. Suppose that Alice has computed \(n=55, e=7,\) and \(d=13\). Show how Bob would encrypt the message 8 and how Alice would then decrypt it.
Extensions
29. To further obscure a message, sometimes the usual alphabet characters are replaced with other symbols. Design a new set of symbols, and use it to encode a message. Exchange with a friend and see if they can decode your message.
30.To make an encryption harder to break, sometimes multiple substitution and transposition ciphers are used in sequence. For example, a method might specify that the first letter of the encryption keyword be used to determine the initial shift for a Caesar cipher (perhaps with a rotating cipher), and also be used for a transposition cipher. Design your own sequence of encryption steps and encrypt a message. Exchange with a friend and see if they can follow your process to decrypt the message.
31. When using large primes, computing values like \(67^{24} \bmod 83\) can be difficult on a calculator without using additional tricks, since \(67^{24}\) is a huge number. We will explore an approach used.
- Notice that \(67^{2} \bmod 83\) is fairly easy to calculate: \(67^{2} \bmod 83=4489 \bmod 83=7\). Since \(67^{4}\) mod \(83=\left(67^{2}\right)^{2}\) mod 83 can be rewritten using the modular exponent rule as \(\left(67^{2} \bmod 83\right)^{2} \bmod 83,\) this is also easy to evaluate:\(67^{4} \bmod 83=\left(67^{2} \bmod 83\right)^{2} \bmod 83=7^{2} \bmod 83=49\) This process can be continued to find \(67^{8}\) mod 83 as \(\left(6^{4}\right)^{2}\) mod 83 . Find this value, then find \(67^{16} \bmod 83\) and \(67^{32} \bmod 83\)
- There is a rule that \((a b) \bmod n=(a \bmod n)(b \bmod n) \bmod n\) Noting that \(17000=170 \times 100\), calculate 17000 mod 83 using the rule above.
- Note that \(67^{5}=67^{4} 67 .\) Use this, along with the rule from above and the results from part \(a\) to compute \(67^{5}\) mod 83
- Note that \(67^{7}=67^{4+2+1}=67^{4} 67^{2} 67^{1} .\) Compute \(67^{7}\) mod 83.
- Write \(67^{24}\) as a product of powers of \(67,\) and use this to compute \(67^{24}\) mod 83
32. Use the process from the previous question to evaluate \(23^{34} \bmod 37\).
33. To encrypt text messages with RSA, the words are first converted into a string of numbers, and then encrypted. Several characters are usually combined together to produce a message number smaller than the modulus, but approximately the same size. Look up an ASCII table to convert the message “SCALE THE WALLS” to numbers, then encrypt it using the RSA public key \(n=10823, e=5\). Since ASCII characters are two digits, pair up characters to form four-digit numbers before encoding. For example A is 65 and B is 66, so the character pair AB could be treated as the number 6566 and encrypted as 10148
34. Explore approaches to steganography that don’t require specialized software. Attempt to hide a message using one of these techniques, and see if a fellow student can detect the message.
35. When you visit a secure website, your web browser will report that the site’s identity has been verified by a third party, called a certificate authority. This is meant to assure you that you are visiting the actual company’s website. Research how these certificates work.


9.1: Introduction
As far as we can learn , John C. Fremont was the first to use the phrase ‘Coast Ranges’ for the mountains bordering the coast of California. Although mountains are practically continuous along the coast from Oregon to Southern California, yet, geographers have come to apply the term Coast Ranges much as Fremont used it, meaning that portion of the mountains bordering the Pacific Coast which lie between the Great Valley of California and the ocean.
The Coast Ranges are limited on the south by a line drawn westerly from the extreme southern end of the San Joaquin Valley, a line which closely accords with the northern boundary of Santa Barbara county. We find this marked topographically by the Cuyama Valley and the Santa Maria River, and further by the fact that to the south, the Transverse Ranges extend nearly east and west, while the Coast Ranges have a northwest and southeast direction.
The Coast Ranges do not constitute a simple mountain block like the main part of the Sierra Nevada. The folding of Earth's crust and the formation of fracture and earthquake lines in a general northwest and southeast direction have given origin to the series of parallel mountains and valleys. The disturbances of Earth's crust in this region have been many and severe, affecting different parts episodically.
At previous times, the Coast Ranges were submerged until nearly buried beneath the waters of the Pacific. They were then practically a group of islands and peninsulas separated from the Sierra Nevada by a broad, deep bay which occupied the Great Valley. Earthquakes and volcanic outbreaks, and changes in the level of the land seem never to have left the country at rest.
It was formerly believed that the Coast Ranges were newer than the Sierra Nevada and that the continent grew progressively westward from the Rocky Mountains. This is a mistake, for there was extensive land here long before there was any Sierra Nevada Range. The granite exposed in the Santa Lucia, Gavilan, and Santa Cruz Ranges, on the Farallon Islands and at Point Reyes, tells us there was land here long ago, and it was on this ancient land that the earliest known representatives of the Sequoia tree species flourished, as has already been mentioned.
The complexity of the geography of the Coast Ranges is increased by the fact that the underlying rocks vary greatly in their resistance to decay and erosion. Valleys may be formed where the rocks are soft, and peaks formed with more resistant rocks, as is illustrated in the case of the San Luis Buttes.  They represent ancient igneous eruptions that once emerged through the crust and now stand out because they are hard and the rocks around them are soft. They extend in a line from the town of San Luis Obispo northwesterly to the ocean and terminate in a great rock known as Morro Rock which rises bare and rugged to a height of nearly 600 feet.


14.17: Reading- Applying Marketing Principles in the Global Environment
The Marketing Mix in Global Marketing
The same marketing principles that lead to marketing success in domestic marketing can also apply to global marketing. With the rapidly growing force of globalization, the distinction between marketing within an organization’s home country and marketing within external markets is disappearing very quickly. With this in mind, organizations are modifying their marketing strategies to meet the challenges of the global marketplace while trying to sustain their competitiveness within home markets. These changes have also prompted brands to customize their global marketing mix for different markets, based on local languages, needs, wants, and values.
The four Ps of marketing—product, price, placement, and promotion—are all affected as a company moves through the different phases to become and maintain dominance as a global company.
Global Marketing Mix: Product Plus Promotion
For multinational corporations (MNCs), the interplay between product and promotion is important because it can enable a company to make minor adjustments to a single product and its promotion strategy rather than totally revamping the product and promotion for different markets. Coca-Cola is a strong example of this principle. The beverage brand uses two formulas (one with sugar and one with corn syrup) for all markets. The product packaging in every country incorporates Coca-Cola’s contour bottle design and signature ribbon in some shape or form. However, the bottle can also include the country’s native language and appear in identical sizes as other beverage bottles or cans in that country’s market.
Before launching promotional programs, global companies must first define their target markets and determine the products that will resonate most with those consumers and businesses. This research can help inform marketing leaders about what course to take—localization versus standardization strategy—as they learn more about the target market’s receptivity to their goods and services. In addition to pinpointing which price point and distribution channels would best serve a country’s markets, global marketers must decide whether and how to customize their products. Product introductions are also important. Promotional tactics for global audiences can range from television commercials to social-media marketing on Facebook or YouTube. It is the job of global marketers to create and place promotional efforts in settings where local consumers will be most receptive to receiving and acting on those messages. Consumers in each target market have different media habits and preferences, and understanding these behaviors is important for selecting the right promotional mix.
After product research, development, and creation, promotion is generally the largest line item in a global company’s marketing budget. Using integrated marketing communications can significantly increase efficiency and reduce promotional costs, as messages across multiple channels reinforce and amplify one another. For organizations that pursue a standardized approach to promoting products and building brands, promotion is the crucial component of the mix that enables a global company to send the same message worldwide using relevant, engaging, and cost-effective techniques.
While a standardized global promotion strategy enables global brands to engage in uniform marketing practices and promote a consistent brand and image, marketers must also be prepared for the challenge of responding to differences in consumer response to marketing mix elements. Marketers must also fend off the full spectrum of local and global competitors, using promotional strategies, branding, and product development to full competitive effect.
Global Marketing Mix: Promotion
Before a company decides to become global, it must consider social, cultural, economic, political, competitive, and other factors relative to the global expansion it is considering. Creating a worldwide marketing plan is no simple task. It is virtually impossible for a company to communicate one identical message in a unified voice to global markets unless a company holds the same position against its competition in all markets (e.g., market leader, low cost, etc.). This is rarely the case, so most global companies must be open to some level of localization and be nimble enough to adapt to changing local market trends, tastes, and needs.
Global marketers must balance four potentially competing business objectives when developing worldwide advertising: 1) building a brand while speaking with one voice, 2) developing economies of scale in the creative process, 3) maximizing local effectiveness of advertisements, and 4) increasing the company’s speed of implementation. Global marketers can use several approaches when executing global promotional programs: exporting executions, producing local executions, and importing ideas that travel.
To successfully implement these approaches, marketers must ensure that their promotional campaigns take into account how consumer behavior is shaped by internal conditions (e.g., demographics, knowledge, attitude, beliefs) and external influences (e.g., culture, ethnicity, family, lifestyle) in local markets. Areas for attention include:
- Language: Language differences are crucial in global marketing. There are nearly 3,000 languages in the world. Language differences have caused many problems for marketers in designing advertising campaigns and product labels. As discussed in this course’s discussion on naming, language can be problematic in the global naming process, with numerous examples of brand names that work well in some languages but have offensive or unfortunate meanings in other languages. Even countries that use the same language have words with different meanings. Consider the British terms “flat” ( apartment in U.S. English), “pants” ( underwear in U.S. English), and “lift” ( elevator in U.S. English). Marketing messaging and materials could easily go wrong if they are not adjusted to fit in-country dialect and usage. Language becomes even more significant if a country’s population speaks several languages. An additional language consideration for marketers is literacy rates. Depending on the target audience and market, literacy may be a significant issue. Some countries, primarily less-developed countries, still have low literacy rates, such as Afghanistan with just 38 percent adult literacy in 2015, and Haiti with 61 percent. India, with its burgeoning economy, reported a literacy rate of 72 percent in 2015. In many countries, literacy rates can differ widely between men and women in many countries, too.
- Colors: Colors may have different meanings in different cultures. This highlights the importance of careful testing of packaging and other visual elements intended for global audiences. For example, green is a sacred color in the Muslim faith, and it is not considered appropriate for packaging and branding purposes in Middle Eastern countries. In Japan, black and white are colors of mourning and should be avoided on product packaging. Purple is associated with death in some Hispanic nations.
- Values: An individual’s values arise from his or her education, moral or religious beliefs and are learned through experiences. For example, as a consumer market, Americans place a very high value on material well-being and are much more likely to purchase status symbols than people in India. Chinese consumers highly value the sense of honor, dignity, and pride, and in some situations they will pay price premiums to “save face” by spending what is perceived to be an appropriate amount to preserve their honor.
- Business norms: The norms of conducting business also vary from one country to the next. In France, for example, wholesalers do not like to promote products. They are mainly interested in supplying retailers with the products they need.
- Religious beliefs and holidays: In addition to affecting their values, a person’s religious beliefs can affect shopping patterns and products purchased. In the United States and other Christian nations, the Christmas holiday season is a major sales period. In China, the Chinese New Year brings out the shoppers. In India, a string of Hindu festivals including Dussehra and Diwali mark a holiday season that extends over multiple months.
Many other factors, including a country’s political or legal environment, economic status, and technological environment, can impact a brand’s promotional mix. Companies should monitor dynamics in their target markets and be ready to quickly respond and adapt to consumer needs and preferences.
Recommendations for Adjusting the Promotional Mix
When launching global advertising, public relations, or sales campaigns, global companies test promotional ideas using marketing research systems that provide results comparable across countries. These systems help marketers achieve economies of scale in marketing communications, since they reveal which messaging or creative elements contribute to a product’s market success. Marketing-research measures of nonverbal factors such as flow of attention, flow of emotion, and branding moments can provide insight into what is working in an advertisement or other marketing communication piece across multiple countries and languages.
The same recommendations about how to research and understand a target market in domestic settings apply to global settings. Marketing research is essential for marketers to build their understanding of which promotional tactics will be successful in any country or region. Informed experimentation and trial and error are also good teachers. Once marketers and brand managers discover what works (and what doesn’t) in the promotional mix, they can import this knowledge to infuse creative ideas into other markets. Likewise, companies can use this intelligence to modify various elements in their promotional mix that are receiving minimal or unfavorable response from global audiences.
Global Marketing Mix: Price
Pricing is the process of determining what a company will receive in exchange for its products. Many pricing considerations in global marketing are similar to domestic marketing. As marketers develop pricing strategy, they should keep the following goals in mind:
- Achieving the financial goals of the company and generating profits
- Matching the realities of the marketplace and consumer buying trends
- Supporting the designated positioning for a product, making it consistent with other elements of the marketing mix, product, promotion and placement
Similar to domestic marketing, in the global marketing mix, factors that affect pricing include manufacturing cost, distribution channels, marketplace, competition, market conditions, and quality of product. For instance, if distribution is exclusive to one channel partner, then prices are likely to be higher. High prices are required to cover high costs of manufacturing, shipping, extensive advertising, and promotional campaigns. If manufacturing costs go up due to the rise in price of some raw material, then prices will need to rise as well.
Global Pricing Considerations
Pricing considerations become more complicated in the global context when it comes to factors affecting global trade. Multinational companies must operate with different currencies, exchange rates, and interest rates. Pricing needs to account for risk associated with fluctuations in the relative value of different currencies in the markets where businesses operate. When the dollar is strong against a foreign currency, for example, imported American goods are more expensive relative to the local competition, so local sales may decrease. When a weak dollar makes product imports more expensive, the final good must carry a higher price tag to cover production costs.
Pricing can be affected by the cost of production (locally or internationally), natural resources (product ingredients or components), and the cost of delivery (e.g., the availability of fuel). For instance, if a country imposes a minimum wage law that forces the company to pay more to its workers, the price of the product is likely to rise to cover some of that cost. Natural resources, such as oil, may also fluctuate in price, changing the price of the final good. Pricing may be affected by government policy, such as trade tariffs and taxes, or costs associated with regulatory compliance and adherence to administrative or legal criteria of specific jurisdictions.
Global marketers must be also prepared to deal with other localized factors affecting pricing. Cultural expectations may dictate what consumers are willing to pay for some products and brands. A product’s positioning in relation to the local competition influences the brand’s ultimate profit margin. Global marketers must carefully consider how to position their products in global markets and decide whether their products are considered high-end, economical, or something in between according to cultural norms.
Global Marketing Mix: Place (Distribution)
Although other elements of the marketing mix are often more visible during the marketing process, place, or distribution, is essential in getting the product distributed to customers. Placement determines the channels used to distribute a product across different countries, taking in factors such as competition and they way similar brands are being offered to the target market. Regardless of its size or visibility, a global brand must adjust its country strategies to take into account placement and distribution in the marketing mix.
Global marketing presents more challenges around distribution, compared to domestic or local marketing. Consequently, brands competing in the global marketplace often conduct extensive research to accurately define the market, as well as the environments where consumers will find, buy, and use the product. A country’s transportation and economic infrastructure, customs, marketplace conditions, and the competitive landscape can all factor into strategic decisions around distribution. For example, not all cultures use or have access to vending machines to distribute beverages. In the United States, beverages are sold by the pallet via warehouse stores. But warehouse stores do not exist in all markets.
Placement decisions must also consider the product’s positioning in the marketplace. A global luxury brand would not want to be distributed via a discount chain in the United States. Conversely, low-end shoemakers would likely be ignored by shoppers browsing in an Italian boutique store. Global marketers must also consider how their products will be distributed across the different shopping venues unique to a particular country or market. Customizing these placement strategies for national and local markets while retaining a strong and consistent brand image can help companies gain significant competitive advantages.
Contributors and Attributions
- Revision and adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
- Global Marketing Mix. Provided by : Lumen Learning. Located at : courses.candelalearning.com/principlesmktg1x2kscope/chapter/13-3-global-marketing-mix/. License : CC BY-SA: Attribution-ShareAlike
- Global Marketing Mix: Price. Provided by : Lumen Learning. Located at : courses.candelalearning.com/principlesmktg1x2kscope/chapter/global-marketing-mix-price/. License : CC BY-SA: Attribution-ShareAlike
- Global Marketing Mix: Placement. Provided by : Lumen Learning. Located at : courses.candelalearning.com/principlesmktg1x2kscope/chapter/global-marketing-mix-placement/. License : CC BY-SA: Attribution-ShareAlike
- Chopstick Shelves. Authored by : Daniel Case. Located at : https://commons.wikimedia.org/wiki/File:Chopstick_shelves_at_Wal-Mart_store_in_Shenzhen,_China.jpg . License : CC BY-SA: Attribution-ShareAlike
- Globe Map. Authored by : Duangphorn Wiriya. Provided by : Unsplash. Located at : https://unsplash.com/photos/KiMpFTtuuAk . License : CC0: No Rights Reserved . License Terms : Unsplash License
- Lay's: Hot Chili Squid Flavor. Provided by : Lay's. Located at : https://www.lays.com . License : All Rights Reserved . License Terms : Fair Use


7.24A: Genetically Engineered Vaccines
- Evaluate genetically engineered vaccines
Genetic engineering, also called genetic modification, is the direct manipulation of an organism ‘s genome using biotechnology. New DNA may be inserted in the host genome by first isolating and copying the genetic material of interest using molecular cloning methods to generate a DNA sequence, or by synthesizing the DNA and then inserting this construct into the host organism. Genes may be removed, or “knocked out,” using a nuclease. Gene targeting is a different technique that uses homologous recombination to change an endogenous gene, and can be used to delete a gene, remove exons, add a gene, or introduce point mutations.
Genetic engineering alters the genetic makeup of an organism using techniques that remove heritable material, or that introduce DNA prepared outside the organism either directly into the host or into a cell that is then fused or hybridized with the host. This involves using recombinant nucleic acid (DNA or RNA) techniques to form new combinations of heritable genetic material, followed by the incorporation of that material either indirectly through a vector system or directly through micro-injection, macro-injection and micro-encapsulation techniques.
In medicine, genetic engineering has been used to mass-produce insulin, human growth hormones, follistim (for treating infertility), human albumin, monoclonal antibodies, antihemophilic factors, vaccines,and many other drugs. Vaccination generally involves injecting weak live, killed, or inactivated forms of viruses or their toxins into the person being immunized. Genetically engineered viruses are being developed that can still confer immunity, but lack the infectious sequences. Mouse hybridomas, cells fused together to create monoclonal antibodies have been humanised through genetic engineering to create human monoclonal antibodies.
The process of genetic engineering involves splicing an area of a chromosome, a gene, that controls a certain characteristic of the body. The enzyme endonuclease is used to split a DNA sequence and to split the gene from the rest of the chromosome. For example, this gene may be programmed to produce an antiviral protein. This gene is removed and can be placed into another organism. For example, it can be placed into a bacteria, where it is sealed into the DNA chain using ligase. When the chromosome is once again sealed, the bacteria is now effectively re-programmed to replicate this new antiviral protein. The bacteria can continue to live a healthy life, though genetic engineering and human intervention has actively manipulated what the bacteria actually is.
Despite the early success demonstrated with the hepatitis B vaccine, no other recombinant engineered vaccine has been approved for use in humans. It is unlikely that a recombinant vaccine will be developed to replace an existing licensed human vaccine with a proven record of safety and efficacy. This is due to the economic reality of making vaccines for human use. Genetically engineered subunit vaccines are more costly to manufacture than conventional vaccines, since the antigen must be purified to a higher standard than was demanded of older, conventional vaccines. Each vaccine must also be subjected to extensive testing and review by the FDA, as it would be considered a new product. This is costly to a company in terms of both time and money and is unnecessary if a licensed product is already on the market. Although recombinant subunit vaccines hold great promise, they do present some potential limitations.
In addition to being less reactogenic, recombinant subunit vaccines have a tendency to be less immunogenic than their conventional counterparts. This can be attributed to these vaccines being held to a higher degree of purity than was traditionally done for an earlier generation of licensed subunit vaccines. Ironically, the contaminants often found in conventional subunit vaccines may have aided in the inflammatory process, which is essential for initiating a vigorous immune response. This potential problem may be overcome by employing one of the many new types of adjuvants that are becoming available for use in humans. Recombinant subunit vaccines may also suffer from being too well-defined, because they are composed of a single antigen. In contrast, conventional vaccines contain trace amounts of other antigens that may aid in conferring an immunity to infectious agents that is more solid than could be provided by a monovalent vaccine. This problem can be minimized, where necessary, by creating recombinant vaccines that are composed of multiple antigens from the same pathogen.
Key Points
- All vaccines are genetically modified in a way. A gene may be programmed to produce an antiviral protein in a bacterial cell. Once sealed into the DNA, the bacteria is now effectively re-programmed to replicate this new antiviral protein.
- Recombinant engineered vaccines are being extensively explored, especially to eradicate infectious diseases, allergies, and cancers.
- Protocols for genetically engineered vaccines raise issues on their efficacy and overall benefit.
Key Terms
- FDA : Food and Drug Administration, an agency of the United States Department of Health and Human Services.
- vaccine : a substance given to stimulate the body’s production of antibodies and provide immunity against a disease, prepared from the agent that causes the disease, or a synthetic substitute.
- genetic engineering : The deliberate modification of the genetic structure of an organism. The term genetic modification is used as a synonym.