The Paradox of Synergy and Responsiveness Essay Example

The Paradox of Synergy and Responsiveness Essay Example The Paradox of Synergy and Responsiveness Essay The Paradox of Synergy and Responsiveness Essay The corporate realm is today made very complex by the fact that businesses are increasingly consolidated and large. This is to say that practices of creating multilayered firms in both the serve and the product industries have created business models that must depend on both large and small goals. For a large firm which holds many different types of operations, making the right decisions for how to best manage these individual units while still promoting a larger and singular identity can be a big determinant of success. This discussion is guided by the concept found in the De Wit and Meyer text, stating that such large firms must deal with the so-called Paradox of Synergy and Responsiveness. According to this paradox, it is a constant challenge for large firms to find a balance between the interest of consolidating its brand identity into a single, monolithic corporate structure and the interest in responding with sensitivity to the needs presented by individual factors. As the discussion and the selected case study example will show, individual factors such as customer orientation, financial performance and industry conditions. In the paradox between these two sometimes opposite demands, companies must often make hard decisions about how to pursue their operations. In the context of something such as the service industry, we can see that the achievement of this balance will be directly related to how a business manages its portfolio of operations. For the ever-growing service industry, this causes an interesting investigation into the way that unit management can be central to an operation. The approach taken by a company in the hospitality sector of the service industry, for example, should serve as a useful illustration of how organizations approach a market shaped by the De Wit and Meyer paradox. Introduction: In our consideration of the Marriott Corporation, which is an extremely successful service sector company, we can see that the way a company moves its operational assets around will be important to its management effectiveness. According to the article entitled Marriott Corporation: The Cost of Capital, the hospitality industry has been through both very fast growth and very sharp retraction. This has been particularly true for Marriott, which seized on an era of increased tourism and hospitality investment in order to catapult to the top of the market. However, changing market realities and the increased nuance of the hospitality industry would force Marriott to consider altering its corporate structure so it could make a change to a context where it could more effectively approach hurdles distinct to individual divisions. This would be a demonstration of its responsiveness, even as synergy had been the primary guide in its initial growth. The example here, based on the discuss at hand, will remark upon the strategies regarding capital investment, stakeholder responsibility and debt management which would effectively be approached by the corporation in managing its portfolio of units. The primary avenue to be taken by Marriott in achieving the goal of managing its operational orientation would be the creation of a new department with a more refined focus on financial strategies and investment properties relating to the specific aspect of hotel management (as opposed to the real-estate operation) For the stock-holder, this would mean more directed and informed decisions as well as a relative insulation from revenue trends, making for a more stable market in times of real estate instability. Again, we can see that Marriott would be turning toward a strategy of responsiveness. Such is to say that Marriot determined to protect the value of its existing capital by establishing a circumstance in which growth and synergy could no longer be identified as the primary interest. Instead, ensuring the retention of capital at a rate justifying its debt condition, Marriott would being a new phase in its history in which hurdle rates for achieving growth would be purposefully higher. In this way, even growth would be directed from corporate HQ by responsive rather than synthetic origins. In our research, we find that the Marriot transition was not without its costs. These would make Marriott go through some real economic changes that are part of the paradox between synthesis and responsiveness. By being forced to change operational gears, Marriott would experience a decline in earnings during its period of transformation. Due to the debt challenges noted, Marriott’s bond ratings, declined to a B level according to both SP and Moody’s in the face of real estate declines, have resulted in a higher long-term expense on new bond issues for the company. One of the key ways that Marriott would deal with its debt maturities in order to afford the transition would be in its mortgage financing of its ‘trophy’ properties, which were those continually successful marquis lodging facilities there were no longer in any mortgage debt. The greatest concern to manu financial institutions in this case was the condition that Marriott’s heavy dependency on invested debt might create a scenario where it would be incapable of covering its obligations. Naturally, this is a fear which is facing so many corporations on this scale today. That shows just how relevant the paradox discussed is. That is why the Marriott example is very useful, because it would use portfolio management strategies to deal with this financial risk. Marriott would basically create a spin-off in its new management company. A spin-off is the divestment of a company in which, rather than selling off aspects of itself in poor market conditions, it distinguishes one aspect of itself to be set apart as an independent business. This independent business will take on its own structural and financial parameters while retaining branding, technology and, in some instances, access to assets, of the parent company. This would be the essential response by Marriott to the distinct obstacles created by the changing market, particularly in consideration of the corporate retraction likely due to markedly rapid corporate growth. The divestment of the company into management and real estate firms would essentially serve to unlock the company’s assets to the benefit of the stakeholder, with the stock owners particularly served by this approach. We can therefore ultimately begin to resolve that the approach taken by Marriott would be a natural outcome in consequence to its singularly high rate of growth from a synergy centered strategy and what might be characterized as its saturation of its own market. The impact would require a responsiveness in a market approach to diminishing the corporations inherency toward this rate of growth. Though we can suggest that the consequences are likely to be somewhat severe to the organization’s personnel and resource orientation in the short term, it will undoubtedly benefit the value and extent of capital usability for individual facilities, thus improving quality organization-wide. De Wit, B. Meyer, R. (1999). Strategy Synthesis. International Thomson Business. Ruback, R. S. (1998). Marriot Corp. : The Cost of Capital. Harvard Business Publishing.

How to Find the Taurus Constellation

How to Find the Taurus Constellation The constellation Taurus is visible for skygazers beginning in late October and early November. Its one of the few constellations that looks somewhat similar to its name, even though its a stick figure. It contains a number of fascinating stars and other objects to explore. Look for Taurus in the sky along the ecliptic, near the constellations Orion and Aries. It looks like a V-shaped pattern of stars with long horns extending out across the sky.   Check out the constellations Perseus, Taurus, and Auriga to see the Pleiades, Hyades, Algol, and Capella. Carolyn Collins Petersen The Story of Taurus Taurus is one of the oldest star patterns known to sky watchers. The first known records of Taurus date back 15,000 years, when ancient cave painters captured its likeness on the walls of underground rooms at Lascaux, France. Many cultures saw a bull in this pattern of stars. Ancient Babylonians told tales of the supreme goddess Ishtar sending Taurus- known as the Bull of Heaven- to kill the hero Gilgamesh. In the ensuing battle, the bull is torn apart and his head is sent to the sky. The rest of his body is said to make up other constellations, including the Big Dipper. Taurus was viewed as a bull in ancient Egypt and Greece, too, and the name persisted into modern times. Indeed, the name Taurus comes from the Latin word for bull.   The Brightest Stars of Taurus The brightest star in Taurus is alpha Tauri, also known as Aldebaran. Aldebaran is an orange-colored supergiant. Its name comes from the Arabic Al-de-baran, meaning leading star, because it seems to lead the nearby Pleiades star cluster across the sky. Aldebaran is slightly more massive than the Sun and many times larger.  It has run out of hydrogen fuel in its core and is expanding as the core begins to convert helium.   The official IAU chart for the constellation Taurus.   IAU/Sky Publishing The two horn stars of the bull are called Beta and Zeta Tauri, also known as El Nath and Tianguan respectively. Beta is a bright white star, while Zeta is a binary star. From our point of view on Earth, we can see each of the two stars in Zeta eclipse each other every 133 days.   The constellation Taurus is also known for the Taurids meteor showers. Two separate events, the Northern and Southern Taurids, occur in late October and early November. The southern shower is the product of objects left behind by Comet Encke, while the Northern Taurids are created when materials from the Comet 2004 TG10 stream through Earths atmosphere and are vaporized.   Deep-Sky Objects in Taurus The Taurus constellation has a number of interesting deep-sky objects. Perhaps the best known is the Pleiades star cluster. This cluster is a collection of several hundred stars, but only the seven brightest can be seen without a telescope or binoculars. The Pleiades stars are hot, young blue stars that move through a cloud of gas and dust. They will continue to travel together for a few hundred million years before dispersing through the galaxy, each on its own path.   The Pleiades open star cluster, as seen by Hubble Space Telescope. NASA/ESA/STScI The Hyades, another star cluster in Taurus, makes up the V-shape of the bulls face. The stars in the Hyades form a spherical grouping, with the brightest ones making the V. They are mostly older stars, moving together through the galaxy in an open cluster. It will likely break apart in the distant figure, with each of its stars traveling along a separate path from the others. As the stars age, they will eventually die, which will cause the cluster to evaporate in several hundred million years.   The Hyades star cluster with the bright orange-red star Aldebaran (upper left) in the picture. The Hyades is a cluster that lies farther away from Aldebaran, which is in the same line of sight. NASA/ESA/STScI The other interesting deep-sky object in Taurus is the Crab Nebula, located near the horns of the bull. The Crab is a supernova remnant left over from the explosion of a giant star more than 7,500 years ago. The light from the explosion reached Earth in the year 1055 AD. The star that exploded was at least nine times the mass of the Sun and may have been even more massive. Crab Nebula in several wavelengths of light including visible and x-ray. The bright dot at the center is the Crab Nebula Pulsar, which is the rapidly spinning remains of the star that died in the ancient supernova explosion that created this object. NASA/HST/CXC/ASU/J. Hester et al. The Crab Nebula is not visible to the naked eye, but it can be seen through a good telescope. The best images have come from such observatories as the Hubble Space Telescope and the Chandra X-ray Observatory.

Just-in-Time Planning at Mutual Insurance Company of Iowa Case Study

Just-in-Time Planning at Mutual Insurance Company of Iowa - Case Study Example Its image could also be affected in the negative light and bring a halt to the current growing demands for its products as new customers shopping around for insurance products will opt for firms with a good of reputation in terms of faster processing of claims. Resources are overstretched and the company could end up losing new and existing business. Looking at the case at the case even without the advice of consultants or experts a number of assumptions can be made about the company’s personnel and equipment situation. Firstly the company and particularly the Des Moines facility for claim processing is facing human resources crisis. The increase in number of claims flowing in on a daily basis means increased amount of work for the same number of employees. One solution thus is the company to hire more personnel to meet the demands of growing workload. The inventory handling capacity should also increase in terms of equipment; the current equipments were not bought in anticipation of the workload the company is getting today. A permanent solution to this would be to get more modern equipment with a capacity to handle more work and also the facility should be expanded commensurately. The company is also faced with challenges surrounding administrative and workflow management. Claim documents have to pass through the hands of different persons for approval before a customer gets a verdict. The net effect of this is a lot of time taken to process just a single claim and also many people doing just the same work. Precisely this could be causing duplication of effort and therefore under-optimisation. The solution approach taken by Cook of streamlining workflow process and cross training employees will eliminate these problems as just the same employees will be capable of handling different types of duties. To help in restructuring the entire process of processing claims, Cook has established a taskforce and also outsourced an external

The First, Second, Third Punic Wars Essay Example | Topics and Well Written Essays - 750 words

The First, Second, Third Punic Wars - Essay Example The outcome of the wars established the enduring legacy of the Roman Empire as one of the greatest in the whole of history. The influence of the Punic Wars on Western Civilization The Punic Wars were important also for their influence on subsequent diplomatic and military strategies. Many theories pertaining to political and military strategy were conceived and codified during these three wars. These theories continued to be perused by later generations of leaders. The Punic Wars were also important for their impact on cultural and philosophical development in Europe. Since the Western Mediterranean region was such a cultural and intellectual melting pot, gaining control of it conferred prestige on the Roman Empire. The leaders of the Empire would in turn encourage the growth of arts and culture. It is no coincidence that the rise of the Roman Empire through victories in the Punic Wars happened during the Hellenistic era. It is as if the explosion of art, literature, philosophy, thea tre, architecture, music and science in Hellenistic Greece is a response to the ascendency of the Roman Empire through the Punic Wars. First Punic War: Winner, Loser, Gains and Losses At the beginning of the First Punic War, Rome only possessed a modest navy. On the other hand, Carthage held the most competent and experienced navy in the region. Since Rome can access Sicily only through its navy, Carthage was able to quell its initial forays. Though set back by these early defeats, the Roman military strategists rose to the occasion and started building a substantial fleet of ships to neutralize Carthaginian naval power. This enterprise proved to be a success and eventually Sicily and other contested territories was conceded by Carthage to Rome. The outcome of the First Punic War established Rome as a considerable imperial power in the Mediterranean region. As part of the reparations, Rome acquired a fair share of Carthage’s wealth, so much so that an indignant Carthaginian l eadership would carry its scars into the future. These hurt pride and perceived injustice would be the backdrop for further conflicts between the two empires. Rome and Carthage made several trade pacts after the war and they even agreed to an alliance to suppress King Pyrrhus of Epirus. As part of the war indemnity, Carthage was asked to release thousands of Roman prisoners of war. Large amounts of silver were also included as reparation. But Carthage’s economy and military were so devastated by the war that it was unable to fulfil its post-war pacts. This led to resentment from Rome and made further wars inevitable. Second Punic War: Winner, Loser, Gains and Losses The Second Punic War followed a similar pattern to that of the first. Although Carthage under the imaginative command of Hannibal made impressive forays into Roman held territory, the superior organization and adaptability of Roman forces eventually proved decisive. Hannibal’s crossing of Alps with an Eleph ant-ridden battalion was an impressive feat. Hannibal was able to dominate the country outside Rome on the back of his superior infantry. But the crucial fortress of Rome the city was never to be breached. Acting against Hannibal’s progress was the resolute support Rome received from its allies. Hence Carthage was once again defeated by the superior diplomacy, combat tactics and foresight of Roman leadership. But unlike the First Punic Wa

Generals Die in Bed - Plot Essay Example for Free

Generals Die in Bed Plot Essay When he thought war contained glory and glamour, he finds himself wrong when his comrades start to die, beginning with Brown. A while later, he is emotionally affected when he kills a German with his bayonet. His emotional status worsens when another of his friend dies. The narrator then goes on leave for 10 days in England, where a prostitute makes him forget about the war. When he comes back, an attempt to raid the Germans takes place where the rest of his friends, except Broadbent dies. The general tells the new team that the Germans sank a hospital ship, and organizes another raid, this time to kill everyone. The narrator has wounded his foot, and discover that Broadbent was mortally wounded too. Broadbent’s leg is hanging by a string of flesh, but then dies by blood loss. Then the war is over. The recruits are told that the general lied, the Germans didn’t sink a hospital ship. It was a ship filled with weapons. He then realizes war is basically a chess game for the generals, and the soldiers are just young boys, listening to the orders, with meaningless ideals Wikipedia

Environmental Impacts of Fossil Fuel Use Essay -- Environmental Issues

Environmental Impacts of Fossil Fuel Use One of the main issues involved with fossil fuels are the environmental impacts that occur from their use. These problems; such as acid rain, oil spills, climate change, global warming, etc., are not only occurring with fossil fuel usage, but are also increasing due to the increase in the use of fossil fuels. This essay will vaguely explain the area of environmental impacts from fossil fuel use, and will attempt to change, or further increase your understanding of the very serious environmental impacts that occur from fossil fuel use. One of the biggest environmental impacts which is steadily increasing in severity due to fossil fuel usage is global warming. Global warming is mainly caused due to the trapping of heat in the atmosphere due to increased amounts of carbon dioxide. Because of the burning of fossil fuels, there has been a twenty five percent increase in the amount of carbon dioxide in the atmosphere over the last 150 years (Clean Energy). This increase over the years has caused the global average surface temperature to raise ...

Ethical Analysis of Abortion

P. Ruiz Stevens Phil 3340-106 12/5/11 Ethical Analysis of Abortion Abortion could not be ethically justified because it is killing an innocent human being. It is arguable that a right to an abortion is a right to control one’s body and the death of fetus is an unavoidable consequence of choosing not to continue a pregnancy. That people have some claim to personal, bodily autonomy must be regarded as fundamental to the conception of any ethical, democratic, and free society. Given that autonomy exists as an ethical necessity, the question becomes how far the autonomy exists.If a woman consented to sex and/or didn’t properly use contraception, then she knew pregnancy might result. Being pregnant means having a new life growing inside. Whether the fetus is a person or not and, whether the state takes a position on abortion or not, it’s arguable that a woman has some sort of ethical obligation to the fetus. Most debates on the ethics of abortion focus on whether the fetus is a person. Even if it is not a person, however, this doesn’t mean it can’t have any moral standing.Maybe this obligation isn’t strong enough to eliminate abortion as an option, but it may be enough to limit when abortion can be ethically chosen or justified. According to the best interest principle, the best interest would be to have the baby so it can live a long and fulfilling life. It is argued that in these tragic cases the great value of the mental health of a woman who becomes pregnant as a result of rape or incest can be safe-guarded by abortion. It is also that a pregnancy caused by rape or incest is the result of a grave injustice and that the victim should not be obligated to carry the fetus to viability.This would keep reminding her of the violence for nine months and it would increase her mental anguish. â€Å"It is reasoned that the value of woman’s mental health is greater than the value of the fetus. In addition, it is maintained t hat the fetus is an aggressor against the woman’s integrity and personal life; it is only just and morally defensible to repel an aggressor even by killing him if that is the only way to defend personal and human values. † It is concluded then, that the abortion is justified in these cases.According to the best interests’ principle, in this case it might be ok for the mother to abort the fetus since she might end up resenting the fetus later in life. If life begins at conception, then it follows that all fertilized eggs are morally important. However the problem with that is that when one attempts to have children though normal reproduction it is estimated that â€Å"only 50 to 60 percent of conceptions advance to beyond twenty weeks of gestation. Of the pregnancies that are lost, 75% represent a failure of implantation and are therefore not realized as clinical pregnancies . † (Norwitz, E.R. . â€Å"Implantation and the survival of early pregnancy. † The New England Journal of Medicine vol. 34508 Nov. 2001 1400-1408) This indicates that the decision to attempt to have children leads to the death of many fertilized eggs, which, according to the pro-life position, are fully significant individuals. The death of these eggs is not justifiable the only motivation is to have children. Another objection to this argument would be what if the baby is malformed? We should not kill an unborn baby to alleviate the suffering of the mother any more then we should kill her infant to alleviate her suffering.Neither should we commit an abortion of a malformed fetus to prevent his or her suffering later in life. Being handicapped is not a capital crime. â€Å"The intentional destruction of health is not compassionate and it is not healthcare; is it assault. We must not be swayed from our pro-life ethic by emotional appeals that admittedly swell our eyes with tears. Truth and compassion prevent us from this fatal compromise. We must respond to all tragic circumstances of pregnancy from the unshakeable foundation of two indisputable premises: human life begins at conception, and it is always wrong to intentionally to kill an innocent human being.The unborn child’s right to life and liberty is given by his or her creator, not by his parents or by the state. The right to life is inalienable: that is, not to be trespassed upon another. In tragic circumstances such as rape or incest, we want to care for both the mother and her unborn baby. We want to relieve the suffering of the mother and her unborn baby. It is never right to intentionally kill an innocent human being, even if it does relieve another’s emotional or physical suffering. It is not up to a vote, and our obligation to submit unto divine judgments does not sway with our circumstances

Aims of education Essay

Culture is activity of thought, and receptiveness to beauty and humane feeling. Scraps of information have nothing to do with it. A merely well-informed man is the most useless bore on God’s earth. What we should aim at producing is men who possess both culture and expert knowledge in some special direction. Their expert knowledge will give them the ground to start from, and their culture will lead them as deep as philosophy and as high as art. We have to remember that the valuable intellectual development is self- development, and that it mostly takes place between the ages of sixteen and thirty. As to training, the most important part is given by mothers before the age of twelve. A saying due to Archbishop Temple illustrates my meaning. Surprise was expressed at the success in after-life of a man, who as a boy at Rugby had been somewhat undistinguished. He answered, â€Å"It is not what they are at eighteen, it is what they become afterwards that matters.† In training a child to activity of thought, above all things we must beware of what I will call â€Å"inert ideas†-that is to say, ideas that are merely received into the mind without being utilised, or tested, or thrown into fresh combinations. In the history of education, the most striking phenomenon is that schools of learning, which at one epoch are alive with a ferment of genius, in a succeeding generation exhibit merely pedantry and routine. The reason is, that they are overladen with inert ideas. Education with inert ideas is not only useless: it is, above all things, harmful – Corruptio optimi, pessima. Except at rare intervals of intellectual ferment, education in the past has been radically infected with inert ideas. That is the reason why uneducated clever women, who have seen much of the world, are in middle life so much the most cultured part of the community. They have been saved from this horrible burden of inert ideas. Every intellectual revolution which has ever stirred humanity into greatness has been a passionate protest against inert ideas. Then, alas, with pathetic ignorance of human psychology, it has proceeded by some educational scheme to bind humanity afresh with inert ideas of its own fashioning. Let us now ask how in our system of education we are to guard against this mental dryrot. We enunciate two educational commandments, â€Å"Do not teach too many subjects,† and again, â€Å"What you teach, teach thoroughly.† The result of teaching small parts of a large number of subjects is the passive reception of disconnected ideas, not illumined with any s park of vitality. Let the main ideas which are introduced into a child’s education be few and important, and let them be thrown into every combination possible. The child should make them his own, and should understand their application here and now in the circumstances of his actual life. From the very beginning of his education, the child should experience the joy of discovery. The discovery which he has to make, is that general ideas give an understanding of that stream of events which pours through his life, which is his life. By understanding I mean more than a mere logical analysis, though that is included. I mean â€Å"understanding† in the sense in which it is used in the French

Make a Liquid Layers Density Column

Make a Liquid Layers Density Column When you see liquids stack on top of each other in layers, its because they have different densities from each other and dont mix well together. You can make a density column with many liquid layers using common household liquids. This is an easy, fun and colorful science project that illustrates the concept of density. Density Column Materials You can use some or all of these liquids, depending on how many layers you want and which materials you have handy. These liquids are listed from most-dense to least-dense, so this is the order in which you pour them into the column. HoneyCorn syrup or pancake syrupLiquid dishwashing soapWater (can be colored with food coloring)Vegetable oilRubbing alcohol (can be colored with food coloring)Lamp oil Make the Density Column Pour your heaviest liquid into the center of whatever container you are using to make your column. If you can avoid it, dont let the first liquid run down the side of the the container because the first liquid is thick enough it will probably stick to the side so your column wont end up as pretty. Carefully pour the next liquid you are using down the side of the container. Another way to add the liquid is to pour it over the back of a spoon. Continue adding liquids until you have completed your density column. At this point, you can use the column as a decoration. Try to avoid bumping the container or mixing its contents. The hardest liquids to deal with are the water, vegetable oil, and rubbing alcohol. Make sure that there is an even layer of oil before you add the alcohol because if there is a break in that surface or if you pour the alcohol so that it dips below the oil layer into the water then the two liquids will mix. If you take your time, this problem can be avoided. How the Density Column Works You made your column by pouring the heaviest liquid into the glass first, followed by the next-heaviest liquid, etc. The heaviest liquid has the most mass per unit volume or the highest density. Some of the liquids dont mix because they repel each other (oil and water). Other liquids resist mixing because they are thick or viscous. Eventually some of the liquids of your column will mix together.

Reflections, Translations, and Rotations on SAT Math Coordinate Geometry Guide

Reflections, Translations, and Rotations on SAT Math Coordinate Geometry Guide SAT / ACT Prep Online Guides and Tips If it's always been a dream of yours to shift around graphs and points on the $x$ and $y$ axes (and why not?), then you are in luck! Points, graphs, and shapes can be manipulated in the coordinate plane to your heart's content. Want to scoot that triangle a little to the left? Flip it? Spin it? With reflections, rotations, and translations, a lot is possible. This will be your complete guide to rotations, reflections, and translations of points, shapes, and graphs on the SAT- what these terms mean, the types of questions you'll see on the test, and the tips and formulas you'll need to solve these questions in no time. Before You Continue Reflection, rotation, and translation problems are extremely rare on the SAT. If you're aiming for a perfect score (or nearly) and want to grab every last point you can, then this is the guide for you. But if you still need to brush up on your fundamentals, then your time and energy is better spent studying the more common types of math problems you'll see on the test. Remember, each question is worth the same amount of points, so it is better that you can answer two or three questions on integers, triangles, or slopes than to answer one question on rotations. So if you've got everything else nailed down tight (or you just really, really like coordinate geometry), then lets talk reflections, rotations, and translations! What is a Reflection? Just like how your image is reflected in a mirror, a graph or a flat (planar) object can be reflected in the coordinate plane. It can be reflected across the x-axis, the y-axis, or any other line, invisible or otherwise. This line, about which the object is reflected, is called the "line of symmetry." Most SAT reflection questions will ask you to identify a shape that is symmetrical about a line that you must imagine or draw yourself. These questions should be simple enough so long as you pay attention to the details. For example, The diagram below shows the Greek letter pi. Each side of the figure is reflected identically about a vertical line of symmetry. Of the letters shows bellow, which has both a vertical and a horizontal line of symmetry? A. B. C. D. E. Now, we are being asked for a letter that has BOTH a vertical AND a horizontal line of symmetry (even though the example, pi, only has a vertical line of symmetry). If you are going too quickly through the test, you might be tempted to find the letter with only a vertical line of symmetry like the example picture. Doing this, however, would lead you to select the wrong answer choice. So, now that we know that we must find a letter that is symmetrical both vertically and horizontally, let us examine our options. You can either draw lines of reflection in your mind or on the page, but we will draw it out here. Let us test our options by first giving them a vertical line of symmetry. If they fail the vertical test, then they will automatically be eliminated, with no need to test if they have a horizontal line of symmetry. (Remember, we are looking for a letter that has both.) So let us draw a potential vertical line of symmetry through each of our answer choices, starting with answer choice A. We can see that rho does not have a vertical line of symmetry, as each half is not a perfect reflection of one another. We can eliminate answer choice A. Each half of gamma is also not symmetrical with the other half. We can eliminate answer choice B. Mu is symmetrical about itself vertically and if you were going quickly through the test, you may be tempted to stop here. But we know we must also find a horizontal line of symmetry. Mu does not have a horizontal line of symmetry, so we can now eliminate answer choice C. Eta, as well, has a vertical line of symmetry. Let us see if it also has a horizontal one as well. Success! Eta is symmetrical, whether the line of symmetry is vertical or horizontal. We can stop here, as we have found our correct answer choice. Our final answer is D. Nature showing off its coordinate geometry skills. Clearly. What is a Rotation? Objects in the coordinate plane can also be rotated (turned) clockwise or counterclockwise. Imagine that we can adjust the object with our hands- it will spin, while still lying flat, like a piece of paper on a tabletop. To rotate an object, we must pick a point to act as the center point for our rotation. This center point of our rotation does NOT have to be the center of the shape, however; there must always be a center to our rotation, but we can pick any point to act as this center. Let us look at a visual demonstration of this. First, let's look at a shape that has a center of rotation at the center of the shape itself. Now we can see how the movement of the object changes as the center of rotation shifts. Here, we have a center of rotation as a point on the outline of the shape. But though any point can act as a center of rotation, you will almost always be asked to rotate an object "about the origin." This means that the origin (coordinates $(0,0)$) will act as your center of rotation. The angle about which the object moves is called the angle of rotation. As we rotate an object, the angle of rotation will be: Positive when we move the object counterclockwise Negative when the object is rotating clockwise. A positive angle of rotation. A negative angle of rotation. Through both objects ended up in the same place, one was rotated +180Â ° and the other was rotated -180Â °. If you are asked to rotate an object on the SAT, it will be at an angle of 90 degrees or 180 degrees (or, more rarely, 270 degrees). These are nice numbers that evenly divide the coordinate plane into 4 parts, and each of these degree measures has a standard rule of rotation. Let us look at these rotation rules. Note: if you're a little shaky on the different quadrants of the $xy$-coordinate plane and where $x$ and $y$ are positive and negative, you should take a couple of minutes to read through our article on the four graph quadrants before going to the next section of this guide. Put your cudgels away and we'll prove we're not fakirs. Rotation Rules and Formulas You can determine the new coordinates of your point if you rotate your object by a certain angle about the origin. [Note: these formulas only work when rotating a point or a series of points about the origin- they will not work if rotating the object about any other center of rotation.] Each of the three degree measures- 90, 180, or 270- will shift the coordinates of your original point to a different, calculable, position on the graph. If rotating counterclockwise (a positive angle of rotation), you can use these rules to find your new coordinate points. For example, let us start with a set of coordinates at $(4, 6)$ and rotate the point. Here we have our original coordinates of $(4, 6)$ For 90 degree rotations: $(a, b)$ = $(-b, a)$ If our original coordinates of (4, 6) are rotated 90Â °, the new coordinates will be (-6, 4). For 180 degree rotations: $(a, b)$ = $(-a, -b)$ If our original coordinates of $(4, 6)$ are rotated 180Â °, the new coordinates will be $(-4, -6)$. For 270 degree rotations: $(a, b)$ = $(b, -a)$ If our original coordinates of $(4, 6)$ are rotated 270Â °, the new coordinates will be $(6, -4)$. (And, of course, a 360 degree rotation will bring you right back to the beginning at $(a, b)$ again!) If our original coordinates of $(4, 6)$ are rotated 360Â °, the new coordinates will be the same, $(4, 6)$. â™ ª You spin me right round, baby, right round â™ ª What is a Translation? If we continue to think of the shape as a piece of paper lying flat on a table (on the coordinate plane), a translation is the act of sliding it along the coordinate plane in a particular direction. The shape can be translated up or down (or both!) any amount of distance along the plane. It maintains its shape and bearing, but is simply located elsewhere in the plane. The way to notate that a translation is to occur is by saying: $T_{a,b}(x,y)$ This means that your final coordinates for this point will be: $(x + a, y + b)$ For example, What is the new point for $T_{-3, 4}(2, -6)$? A. $(-5, 10)$B. $(-1, 2)$C. $(1, -2)$D. $(-5, -10)$E. $(-1, -2)$ We know that we must add together our translated points to the corresponding $x$ and $y$ values of our original coordinates. So: $T_{-3, 4}(2, -6)$ $(2 + -3, -6 +4)$ $(-1, -2)$ Our new coordinates for this point are at $(-1, -2)$. You can see why this is true if we look at it on a graph. We are starting at the coordinates $(2, -6)$. Now, we are traveling -3 spaces along the $x$-axis and +4 spaces along the $y$-axis. By tracing this, we can find our new coordinates. Our final answer is E, $(-1, -2)$. Typical Reflection, Rotation, and Translation Problems Again, these types of questions are extremely rare on the SAT, and the odds likely that you will not see any reflection, rotation, or translation problems at all on your test. That said, there are three different types of reflection/rotation/translation problems that will show up, when they appear at all. These questions will be either a reflection, rotation, or translation questions about: #1: Points#2: Shapes in the coordinate plane#3: Function graphs Let's look at all three. Points Points are the simplest objects to be rotated, reflected, or translated, because there is only one component- the single point. Any point on the coordinate plane will have an $x$-coordinate and a $y$-coordinate, but you will still have far fewer moving parts when dealing with a point rotation than with any other kind of rotation, reflection, or translation. Shapes Shapes are slightly more complicated to reflect or rotate than points are for the sheer reason that shapes are made up of several points (and the lines connecting those points). This means that any shape rotation/reflection/translation will require more consideration and care, in order to make sure all your pieces are properly aligned. It is often much easier, when working with modified shapes, to map out the positions of the points alone. Don't worry about the lines- mark the proper position of the new coordinates for the points and the lines will sort themselves out. For instance, let us say that we must rotate a trapezoid +90 degrees. The particular question may ask you to find the slope of one of the new lines of the rotated shape, identify a new coordinate point, or anything else. But first, we must rotate our figure. The easiest way to do this is to simply map the new coordinate points according to our rotating rules. We know that a 90 degree rotation will transform all of our coordinates from $(a, b)$ to $(-b, a)$, so let's find them. Each given coordinate point will transform like so: $(1, 1)$ = $(-1, 1)$ $(3, 4)$ = $(-4, 3)$ $(7, 4)$ = $(-4, 7)$ $(9, 1)$ = $(-1, 9)$ Now we can simply connect the lines and find our new trapezoid, allowing us to answer any question we need to about it. Function Graphs Finally, function graphs can be reflected or translated just like shapes and points, though NOT rotated. (Why can't functions be rotated? If a function were rotated, it would fail the vertical line test and no longer be a function.) A reflected function. A translated function. Functions cannot be rotated! This fails the vertical line test and so is no longer a function. Function Translations We can either translate our function vertically (up and down) or horizontally (left and right), or a combination of the two. The way we do this is by modifying our inputs and outputs (for more on how functions work, including inputs and outputs, check out our guide to SAT functions.) We can translate our function up or down by adding or subtracting from our output. Adding to output translates the graph up. Subtracting from the output moves the graph down. On the other hand: Adding to the input will shift the graph left Subtracting from the input will shift the graph to the right Function Reflections We can also reflect our function about a line of symmetry along the $x$ or $y$-axis. Making the output negative makes the function reflect across the $\bi x$-axis (inverts it about the $x$-axis). Making the input negative makes the function reflect across the $\bi y$-axis. If this is a lot of new information for you, don't stress. These types of questions are, again, so rare that the odds are you won't see them on your test. Only try to memorize these rules if you feel comfortable doing so. Strategies for Reflection and Translation Problems Though no two reflection/translation/rotation problems are exactly alike, there are a few tips and tricks to follow for any kind you may come across. #1: Draw Your Own Graphs Especially when dealing with a problem that requires a reflection or a translation, it is always a good idea to take a moment to sketch out a graph of the object's old and new positions in space. This allows you to work with the problem on the page instead of in your head, which is especially useful if you are asked to find information other than simply identifying a new coordinate point (a feat in and of itself!). For instance, you might be asked to find the slope of a reflected line, or the product of two translated $x$-coordinates, or anything else the SAT might think of. Without making your own drawings and diagrams, it can be easy to become confused, fall for bait answers, and lose precious points. #2: Drill Your Rotation Formulas When working with translations or reflections, it is simple enough to draw your own picture and line up your corresponding points, but when it comes to rotations, it can be much harder to visualize the movement of the point or the object. Even when you've mapped out the original point, rotations are often much trickier than they appear. Unless you have a paper cut-out of your point, shape, or function and want to spend your time spinning your scratch paper around in circles, it's better to simply memorize your rotation rules for 90, 180, and 270 degrees. #3: Double-check, double-check, (triple-check) Rotations, reflections, and translations may seem simple (and, indeed, the underlying principles are not overly complex), but the difficulty in solving these kinds of problems is in just how easy it is to mis-map a coordinate point or two. Nothing is more frustrating than when you know how to solve a problem, but go too quickly or too carelessly and so get the question wrong. So make sure you double-check that you've properly shifted your coordinates before you bubble in that final answer. Test Your Knowledge 1. 2. The graph $y = f(x)$ is shown below. What could be the graph of $y = f(x + 3)$? A. B. C. D. E. Answers: E, A Answer Explanations: 1. If we draw an imaginary vertical line through every letter in the answer options, we can see that all but one are symmetrical about that vertical line. Only the letter E has a different shape to it on each side of the vertical line. Our final answer is E. 2. We know that adding to the input or output will shift our graph and translate it somewhere else. In this case, we are adding to the input, which, you'll recall, translates our graph to the left. We are making no additional changes, so its vertical position will remain unchanged. The only answer choice that shows us a graph that maintains the vertical position and is shifted to the left is answer choice A. Here is the starting position of the function. And here it is shifted to the left in answer choice A. Our final answer is A. Yay! You did it! The Take-Aways Though quite rare to see, the surprise rotation or reflection question can throw a wrench in the works if you are unprepared for it. But nothing the SAT can put on the test is insurmountable (and, indeed, the test is designed to give you opportunities to succeed). Once you've got your understandings down tight and know not only the difference between all your terms, but how to properly take down any kind of coordinate geometry question the SAT can throw at you, you will be well on your way to earning that perfect score. What's Next? You've taken on one of the more obscure SAT math topics, but have you made sure that you have a solid understanding of all the rest of the math topics the SAT will test you on? As always, it is better to get as many points as possible (as accurately as possible), so now might be a good time to catch up on your understanding of circles, triangles, and integers, both basic and advanced. Want to know two of the most invaluable math strategies? Check out our guides on how to use plugging in numbers and plugging in answers to make sense of some of the trickiest SAT problems out there. Looking to get that perfect score? Our team has your back with our guide to getting a perfect 800 on the SAT math, written by a perfect-scorer. Want to learn more about the SAT but tired of reading blog articles? Then you'll love our free, SAT prep livestreams. Designed and led by PrepScholar SAT experts, these live video events are a great resource for students and parents looking to learn more about the SAT and SAT prep. Click on the button below to register for one of our livestreams today!

HRM Coursework Example | Topics and Well Written Essays - 500 words

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