An asymptote helps determine actions or the shapes of things, but it is not really a part of the graph. It is simply an imaginary line that helps you graph a rational function. As the curve approaches toward an asymptote, it gets closer and closer to the asymptote but never actually touches it. Thus, the asymptote helps determine where the graph of the function can or cannot go. That being said, there are three types of asymptotes: vertical, horizontal and oblique asymptotes. But we will only discuss vertical asymptotes and horizontal asymptotes, and see how to figure out which is what actually.
A horizontal asymptote is a constant value on a graph which a function approaches but does not actually reach. It indicates what actually happens to the curve as the x-values get very large or very small. In the graphical examples above, the curve approaches a constant value b, but never actually reaches, y = 0.
The line y = b is a horizontal asymptote of the graph of ‘f’ if f(x) —-> b as x —-> ∞ or x —-> – ∞
To find a horizontal asymptote of a rational function, the degree of the polynomials in the numerator and the denominator is to be considered.
As the denominator of a fraction can never be zero, having the variable on the bottom if a fraction can be a problem. Some domain value of ‘x’ makes the denominator zero and the function will jump over this value in the graph, creating a vertical asymptote. They are vertical lines drawn in lightly or with dashes to show that they are not part of the graph.
If the real number ‘a’ is a zero of the denominator q(x), then the graph of f(x) = p(x)/q(x), where p(x) and q(x) have no common factors, has the vertical asymptote, x = a.
– A horizontal asymptote is a constant value on a graph which a function approaches but does not actually reach. It indicates what actually happens to the curve as the x-values get very large or very small. Vertical asymptotes, on the other hand, are invisible vertical lines which correspond to the zero in the denominator of a rational fraction. They are vertical lines drawn in lightly or with dashes to show that they are not part of the graph.
– To determine a horizontal asymptote of a rational function, the degree of the polynomials in the numerator and the denominator is to be considered. If the denominator has the highest variable power in the function equation, the horizontal asymptote is automatically the x-axis or y = 0. If both the numerator and denominator have an equal degree, then make a fraction of their coefficients to determine the horizontal asymptote equation. To determine the vertical asymptotes of a rational function, set the denominator of the fraction equal to zero.
– Let’s find out the asymptotes of the function
Y =3x^{2}+9x-21 ∕ x^{2}-25
To find the vertical asymptotes, set the denominator of the fraction equal to zero.
x^{2}-25 = 0
(x-5) (x+5) = 0
x = 5 and x = – 5
These two numbers are the two values that cannot be included in the domain, so the equations are vertical asymptotes. So, the two vertical asymptotes are, x = 5 and x = – 5.
Now, to determine the horizontal asymptote, look at the original equation. Here, the highest variable power is 2. As both the numerator and the denominator have the same degree of power, make a fraction of their coefficients:
y = 3x^{2}/x^{2}
y = 3/1
y = 3
So, the equation of the horizontal asymptote is, y = 3.
An asymptote helps determine actions or the shapes of things, but it is not really a part of the graph. Vertical asymptotes mark places where the function has no domain. You solve for the equation of the vertical asymptotes by setting the denominator of the fraction equal to zero. Horizontal asymptotes, on the other hand, indicate what happens to the curve as the x-values get very large or very small. To find a horizontal asymptote, you need to consider the degree of the polynomials in the numerator and the denominator.
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Power factor is always between 0 and 1 and can be determined by the lead or lag of current with regard to voltage. The terms ‘leading’ and ‘lagging’ refer to where the load current phasor lies in relation to the supply voltage phasor. They are determined by the sign of the phase angle between the current and voltage waveforms. Capacitive loads will, therefore, cause a leading power factor, whereas inductive loads will cause a lagging power factor. Power factors are often stated as leading or lagging. Let’s take a look at the differences between the two.
The term ‘lagging power factor’ is used where the load current lags behind the supply voltage. It is a property of an electrical circuit that signifies that the load current is inductive, meaning inductive loads will cause a lagging power factor. In this regard, a lagging power factor can be corrected by adding capacitive loads. Common inductive loads include repulsion induction motors, which represent the most common form of three-phase motors and which always have a lagging power factor. Lagging power factor can be formally described as the current that reaches its peak value up to 90 degrees later than the voltage. All AC motors (except overexcited synchronous motors) and transformers operate at lagging power factor. Simply put, if the load is inductive then the power factor is lagging.
For capacitive circuits, where the load current leads the supply voltage, the term ‘leading power factor’ is used. It is a property of an electrical circuit that signifies that the load current is capacitive, meaning capacitive loads will cause a leading power factor. In this regard, a leading power factor can be corrected by adding inductive loads. A leading power factor means that the same terminal voltage can be maintained with a lower internal induced voltage. The power factor of a leading current is sometimes called a positive power factor. It can be formally described as the current that reaches its peak value up to 90 degrees ahead of the voltage. Simply put, if the load is capacitive then the power factor is leading.
– Power factor can be stated as leading or lagging to show the sign of the phase angle between the current and voltage waveforms. The term ‘lagging power factor’ is used where the load current lags behind the supply voltage. It is a property of an electrical circuit that signifies that the load current is inductive. For capacitive circuits, where the load current leads the supply voltage, the term ‘leading power factor’ is used. It is a property of an electrical circuit that signifies that the load current is capacitive. So, capacitive load are leading whereas inductive loads are lagging.
– The following graph shows a voltage with a leading and lagging current, plotted against time. The lagging current line represents a lagging current with a negative phase angle and a power factor less than 1, whereas the leading current line represents the leading current with a positive phase angle and a power factor less than 1. The leading current will reach its peak value before the voltage reaches its peak, where as the lagging current will reach its peak value after the voltage reaches its peak.
The terms ‘leading’ and ‘lagging’ refer to where the load current phasor lies in relation to the supply voltage phasor. They are determined by the sign of the phase angle between the current and voltage waveforms. The term ‘leading power factor’ is used where the load current leads the supply voltage, whereas the term ‘lagging power factor’ is used where the load current lags behind the supply voltage. A leading power factor signifies that the load current is capacitive in nature whereas a lagging power factor signifies that the load current is inductive. In this regard, a leading power factor can be corrected by adding inductive loads and a lagging power factor can be corrected by adding capacitive loads.
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Both commutative and associative properties are rules applied to addition and multiplication operations. These properties are laws used in algebra to help solve problems. The commutative property comes from the term “commute” which means move around and it refers to being able to switch numbers that you’re adding or multiplying. The associative property comes from the word “associate” or “group” and it refers to grouping of three or more numbers using parentheses, regardless of how you group them. The result remains the same, no matter how you re-group the numbers. Let’s take a look at the two properties to better understand how they work.
For example; we know that adding 2 and 5 gives the same answer as adding 5 and 2. The order of the numbers in an addition problem can be changed without changing the result. This thing about numbers and addition is called the commutative property of addition. So, we can say addition is a commutative operation. Similarly, multiplication is a commutative operation.
a + b = b + a
3 + 4 = 7 is the same as 4 + 3 = 7
The result will be the same regardless of the order of the numbers.
a × b = b × a
3 × 7 = 21 is the same as 7 × 3 = 21
Likewise, the result will be the same regardless of the order of the numbers.
Associative is yet another property we use has to do with re-grouping. For instance, when adding 2 + 3 + 5, we can either add 2 and 3 first and then add 5, or we can add 3 and 5 first and then the 2. Mathematically, it looks like this: 2 + 3 + 5 = 2 + (3 + 5) = (2 +3) + 5. Operations that behave in this manner are called associative operations. The result remains the same even if we change the grouping of numbers.
a + (b + c) = (a + b) + c = a + b + c
1 + (2 +3) = (1 +2) + 3 = 6
The result remains the same, no matter how you group the numbers.
a × (b × c) = (a × b) × c
2 × (3 × 4) = 2 × 12 = 24
(2 × 3) × 4 = 6 × 4 = 24
So, the grouping in the numbers does not change the result.
– The commutative property comes from the term “commute” which means ‘move around’ and it refers to being able to switch numbers that you’re adding or multiplying regardless of the order of the numbers. The associative property, on the other hand, comes from the word “associate” or “group” and it refers to grouping of three or more numbers using parentheses, regardless of how you group them. The result will be the same, no matter how you re-group the numbers or variables.
– The commutative rule of addition states, a + b = b + a, which means adding a and b gives the same result as adding b and a. The orders can be changes without changing the result. This rule of addition is called the commutative property of addition. Similarly, multiplication is a commutative operation which means a × b will give the same result as b × a. The associative property, on the other hand, is the rule that refers to grouping of numbers. The associative rule of addition states, a + (b + c) is the same as (a + b) + c. Likewise, the associative rule of multiplication says a × (b × c) is the same as (a × b) × c.
– The commutative property of addition: 1 + 2 = 2 +1 = 3
The commutative property of multiplication: 2 × 3 = 3 × 2 = 6
The associative property of addition: 5 + (3 + 7) = (5 + 3) + 7 = 15
The associative property of multiplication: 5 × (2 × 4) = (5 × 2) × 4 = 40
In a nutshell, the commutative property is not to confuse with the associative property. The commutative property states that it’s okay to change of the order of the numbers in addition and multiplication operations because the result will be the same, no matter the order. The associative property, on the other hand, states that the result will be the same, no matter how you group the number or variables in addition/multiplication operations.
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Systematic error is the one that occurs in the same direction each time and it remains constant or changes in a regular fashion in repeated measurements of one and the same quantity. A systematic error remains constant throughout a set of readings and causes the measured quantity to be shifted away from the accepted or predicted value. Systematic errors occur because the experimental arrangement is different from that assumed in the theory and the correction factor which takes account of this difference is ignored. In many cases, such errors are caused by some flaw in the experimental apparatus. Systematic error can be eliminated by using proper technique, calibrating equipment and employing standards.
As its name implies, random error is the one that varies in a random manner and which is produced by unpredictable and unknown variations in the total experimental process. Any type of error that is inconsistent and does not repeat in the same magnitude or direction except by chance is considered to be a random error. Gulliksen defines random error in a statistical sense in terms of the mean error, the correlation between the error and the true score, and correlation between errors being zero. For example, wind speed may drop and pick up at different points in time resulting in variations in the results. Random error is discovered by performing measurements of same quantity repeatedly under the same conditions.
Errors can be divided into two primary kinds, systematic and random errors. Systematic error, as the name implies, is a consistent, repeatable error that deviates from the true value of measurement by a fixed amount. Systematic error is the one that occurs in the same direction each time due to the fault of the measuring device. On the contrary, any type of error that is inconsistent and does not repeat in the same magnitude or direction except by chance is considered to be a random error. Random errors are sometimes called statistical errors.
Random errors are discovered by performing measurements of the same quantity number of times under the same conditions and they involve the variability inherent in the natural world and in making any measurement. Systematic errors, on the other hand, can be discovered experimentally by comparing a given result with a measurement of the same quantity performed using a different method or by using a more accurate measuring instrument. Systematic errors give results that are either consistently above the true value or consistently below the true value.
Systematic errors are consistent and are caused by some flaw in the experimental apparatus or a flawed experimental design. Such errors are caused by faulty measuring devices that are either used incorrectly by individuals while taking the measurement or instruments that are imperfectly calibrated. Systematic errors are believed to be more dangerous than random errors. Random errors, on the other hand, are caused by unpredictable variations in the readings of a measurement device or by an observer’s inability to interpret the instrumental reading.
Systematic errors can be eliminated by using proper technique, calibrating equipment and employing standards. Systematic errors are usually produced by faulty human interpretations or changes in environment during the experiments, which are difficult to eliminate completely. Repeated measurements with the same instrument neither reveal nor do they eliminate a systematic error. In principal, all systematic errors can be eliminated, but there will always remain some random errors in any measurement. Random errors, however, can be reduced by taking average of a large number of observations.
In principal, all systematic errors can be eliminated, but there will always remain some random errors in any measurement. Random errors, however, can be reduced by taking average of a large number of observations. Systematic errors are usually produced by faulty human interpretations or changes in environment during the experiments, which are difficult to eliminate completely. This is why systematic errors are potentially more dangerous than random errors. However, systematic errors can be eliminated by using proper technique, calibrating equipment and employing standards.
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Grounded theory is a methodical and inductive approach in gathering and analyzing emerging patterns in data. It seeks to interpret how human beings understand their world and the other beings who interact with them. Hence, the job of the grounded theorist is to verify the research participants’ reality and look into the socially-shared meanings which influence behaviors.
This is credited to American sociologists, Barney Glaser, and Anselm Strauss. With their research on dying patients, they developed the constant comparative method which evolved to grounded theory method. The following are the usual steps in grounded theory research:
The advantages of this theory include its organized and clear presentation, the liberty to be able to construct theories, and its applicability in various fields such as psychiatry, psychology, sociology, medicine, management, industry, and education. Hence, it has high ecological validity, novelty, and parsimony.
Ethnography came from the Greek words “ethos” which means “folk” or “nation” and “grapho” which means “write”. It is the methodological study of people and cultures which entails the researcher to observe participants from their perspectives. This design has many forms which includes life history, feminist, and confessional; two of its common forms are realist and critical. Realist ethnography uses a traditional approach from the third person’s perspective to promote objectivity. This is often used by cultural anthropologists and the researcher has the final say on how the information should be presented and interpreted. Critical ethnography advocates the causes of marginalized groups and aims to empower people. These ethnographers are usually politically minded and address issues on power, inequity, and repression.
The group’s culture is presented graphically and in writing; thus, ethnography can have a double meaning. The conceptual development of ethnography is attributed to Gerhard Friedrich Muller, a history and geography professor while the first known modern ethnographer is Bernardino de Sahagun, a Franciscan priest.
As a qualitative method, it observes practices and relationships without the strict employment of a deductive framework. An ethnographic study features the system of meanings in the existence of a cultural group. It is most appropriate for exploring beliefs, issues, language, and other cultural systems. The following are the general methods in conducting an ethnographic research:
Grounded theory is greatly influenced by symbolic interaction which seeks to gain more knowledge about the world by looking into how humans interact, specifically with the use of symbols such as language. On the other hand, ethnography is more holistic in approach and is not often assessed regarding philosophical standpoints.
The general aim of grounded theory is to study emerging patterns which lead to a theory while that of ethnography is to gain rich and holistic generalizations of a group’s behavior and their location.
Grounded theory is credited to American sociologists, Barney Glaser, and Anselm Strauss while ethnography’s concept development is attributed to Gerhard Friedrich Muller while the first known modern ethnographer is Bernardino de Sahagun.
Grounded theory has no distinct forms while ethnography has several which includes life history, feminist, and confessional; two of its common forms are realist and critical.
The usual steps in grounded theory research are data collection and review, theme coding, categorizing codes, and theory conceptualization while those of ethnography are population identification, theme selection, ethnography type specification, data collection and analysis, and generalizations.
The advantages of grounded theory include high ecological validity, novelty, and parsimony. Regarding ethnography, the benefits include addressing unpopular or ignored issues, and providing avenues for ethnographer’s creativity.
The criticisms of grounded theory include its being misunderstood as a “theory”, its vague notion of being “grounded”, and some have misgivings regarding its claim to develop inductive knowledge. The disadvantages of ethnography include the risk for bias since the ethnographer’s intuitions are tapped, its long duration and high cost since it may take time to establish trust with the participants, and some groups may be difficult to access.
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Beta measures the risk (volatility) of an individual asset relative to the market portfolio. Beta aims to gauge an investment’s sensitivity to market movements. It is a measure of the fund’s volatility relative to other funds. It is not an absolute measure of volatility; it measures a stock’s volatility relative to the market as a whole. Therefore, beta measures how movement in the stock price relates to the changes in the entire stock market. It is the average change in percentage in the value of the fund accompanying a 1% increase or decrease in the value of the S&P 500 index. For example, a stock with a beta of 1.5 goes up about 50% more than the index when the market goes down. Similarly, a stock with a beta of 2.00 experiences price swings double than those of the broader market. An S&P index fund, by definition, has a beta of 1.0.
Standard deviation is the most widely used statistical measure of spread which essentially reports a fund’s volatility. The volatility of a single stock is commonly measured by its standard deviation of returns over a recent period. The standard deviation of a stock portfolio is determined by the standard deviation of returns for each individual stock along with the correlations of returns between each pair of stock in the portfolio. It includes both the unique risk and systematic risk. Higher standard deviations are generally associated with more risk. If you scale the standard deviation of one market against another, you obtain a measure of relative risk. The funds with standard deviations of their annual returns greater than 16.5 are more volatile than average.
– Both Beta and Standard deviation are two of the most common measures of fund’s volatility. However, beta measures a stock’s volatility relative to the market as a whole, while standard deviation measures the risk of individual stocks. Standard deviation is a measure that indicates the degree of uncertainty or dispersion of cash flow and is one precise measure of risk. Higher standard deviations are generally associated with more risk. Beta, on the other hand, measures the risk (volatility) of an individual asset relative to the market portfolio.
– Beta is the average change in percentage in the value of the fund accompanying a 1% increase or decrease in the value of the S&P 500 index. An S&P index fund, by definition, has a beta of 1.0. A beta greater than 1.0 means greater volatility than the overall market, while a beta below 1.0 accounts for less volatility. Standard deviation is defined as the square root of the mean of the squared deviation, where deviation is the difference between an outcome and the expected mean value of all outcomes.
– A stock with a 1.50 beta is significantly more volatile than its benchmark. It is expected to go up about 50% more than the index when the market goes down. Similarly, a stock with a beta of 2.00 experiences price swings double than those of the broader market. Standard deviation can be used as a measure of the average daily deviation of share price from the annual mean, or the year-to-year variation in total return. Higher standard deviations are generally associated with more risk and lower standard deviations mean more return for the amount of risk acquired.
Both Beta and Standard deviation are two of the most common measures of fund’s volatility. However, beta is a measure of the fund’s volatility relative to other funds, while standard deviation describes only the fund in question, but not how it compares to the index or to other funds. Therefore, investment with higher standard deviations is generally associated with more risk, while investment with lower standard deviation yields modest returns. On the contrary, a beta greater than 1.0 means greater volatility than the overall market, while a beta below 1.0 accounts for less volatility.
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Geometry is fun. Geometry is all about shapes, sizes, and dimensions. Geometry is the kind of mathematics that deals with the study of shapes. It is easy to see why geometry has so many applications that relate to the real life. It is used in everything – in engineering, architecture, art, sports, and much more. Today, we will discuss triangle geometry, specifically triangle congruence. But first, we need to understand what it means to be congruent. Two figures are congruent if one can be moved onto the other in such a way that all their parts coincide. In other words, two figures are called congruent if they are the same shape and size. Two congruent figures are one and the same figure, in two different places.
It’s true than triangle congruence is the basic building block for many geometrical concepts and proofs. Triangle congruence is one of the most common geometrical concepts in High school studies. One major concept often overlooked in teaching and learning about triangle congruence is the concept of sufficiency, that is, to determine the conditions which satisfy that two triangles are congruent. There are five ways to determine if two triangles are congruent, but we are going to discuss only two, that is, ASA and AAS. ASA stands for “Angle, Side, Angle”, while AAS means “Angle, Angle, Side”. Let’s take a look at how to use the two to determine if two triangles are congruent.
ASA stands for “Angle, Side, Angle”, which means two triangles are congruent if they have an equal side contained between corresponding equal angles. If the vertices of two triangles are in one-to-one correspondence such that two angles and the included side of one triangle are congruent, respectively, to the two angles and the included side of the second triangles, then it satisfies the condition that the triangles are congruent. Because the two angles and the included side are equal in both the triangles, the triangles are called congruent.
AAS stands for “Angle, Angle, Side”, which means two angles and an opposite side. AAS is one of the five ways to determine if two triangles are congruent. It states that if the vertices of two triangles are in one-to-one correspondence such that two angles and the side opposite to one of them in one triangle are congruent to the corresponding angles and the non-included side of the second triangle, then the triangles are congruent. The non-include side is the side opposite to either one of the two angles being used. In simple terms, if two pairs of corresponding angles and the sides opposite to them are equal in both the triangles, the two triangles are congruent.
– ASA and AAS are two postulates that help us determine if two triangles are congruent. ASA stands for “Angle, Side, Angle”, while AAS means “Angle, Angle, Side”. Two figures are congruent if they are of the same shape and size. In other words, two congruent figures are one and the same figure, in two different places. While both are the geometry terms used in proofs and they relate to the placement of angles and sides, the difference lies in when to use them. ASA refers to any two angles and the included side, whereas AAS refers to the two corresponding angles and the non-included side.
– According to ASA congruence, two triangles are congruent if they have an equal side contained between corresponding equal angles. In other words, if two angles and an included side of one triangle are equal to the corresponding angles and the included side of the second triangle, then the two triangles are called congruent, according to the ASA rule. The AAS rule, on the other hand, states that if the vertices of two triangles are in one-to-one correspondence such that two angles and the side opposite to one of them in one triangle are equal to the corresponding angles and the non-included side of the second triangle, then the triangles are congruent.
– The main difference between the two congruence rules is that the side is included in the ASA postulate, whereas the side is not include in the AAS postulate.
Here, two angles (ABC and ACB) and the included side (BC) are congruent to the corresponding angles (DEF and DFE) and one included side (EF), which makes the two triangles congruent, according to the ASA congruence rule.
Here, two angles (ABC and BAC) and one non-included side (BC) of the first triangle are congruent to the corresponding angles (DEF and EDF) and the non-included side (EF) of the second triangle, which makes the two triangles congruent. AC and EF can also be the non-included sides of the two triangles respectively.
In a nutshell, ASA and AAS are two of the five congruence rules that determine if two triangles are congruent. ASA stands for “Angle, Side, Angle”, which means two triangles are congruent if they have an equal side contained between corresponding equal angles. AAS refers to “Angle, Angle, Side”, which means if two pairs of corresponding angles and the sides opposite to them are equal in both the triangles, the two triangles are called congruent. While both are basically same, the main difference between the two congruence rules is that side is included in the ASA rule, whereas side is not included in the AAS rule.
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Ascending refers to moving upwards, climbing up or increasing. The root word of ascending is ascend and ascending is used as a verb to describe the act of moving up or climbing upwards. Ascending a stairway or a mountain to reach the top is a good example. Ascending becomes an adjective to describe the order of numbers. Ascending order shows the increase in the value of numbers, metric measures and other mathematical concepts. Knowing how to arrange items in ascending order is a skill taught to children in school. The concept of ascending applies to the motion of an aircraft as it ascends into the sky, after take off. Ascending may be used figuratively to describe the climbing of social ladders or the rise of a monarch, as he or she, ascends the throne to rule the country. Ascending always refers to an upward motion or concept. Synonyms like arise, aspire, climb and mount help to understand the meaning of the word ascending.
Descending is the complete opposite action from ascending. It is a verb with the root word of descend. It can be used as an adjective to describe the descending order of numbers. Musically, a descending order of notes or scale, would take on a deeper lower pitch as the order of keys moves down the musical instrument. Descending is used in medical terms as well as in aviation and all for the opposite actions compared to ascending. Descending has a downward more depressing emotion of feeling ‘down’ or negatively feeling at a loss. Climbers descend from the mountain tops and planes descend before they land. Descending is completely the opposite of ascend in every aspect of its word usage.
In summing up the differences between the two words it is very clear that they are opposite in every situation. Wherever the word ascending is used, if it should be replaced by descending, the opposite order of events will be expected. Ascending a mountain means to climb up and descending brings the climber down. Ascending would imply a lifting up of emotion and spirits while descending implies a characteristic mood change, a feeling of depression and sadness.
Plato, in his work called Allegory of the Cave, used the terms ascend and descend figuratively to refer to the effect of education or the lack of it. He wrote, ‘It is the task of the enlightened not only to ascend to learning and see the good, but to be willing to descend again to those prisoners and to share their troubles.’ in this way he compares both words to each other and brings out their opposite characteristics in one statement highlighting an attitude of people with understanding and learning.
The use of a climax, or ascending turn of events, adds excitement to a literary piece. The use of an anti-climax, called bathos, is the dramatic use of a sudden drop from grandeur to descend into the common place. This is an anti climax. ‘I have neither gold nor silver or bronze – but I give you this chewing gum!’ is an example of bathos and anti-climax. A complete descending of emotion in the situation.
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A parallelogram is a two-dimensional geometrical structure that is made of four sides, of which the sides that are opposite are also parallel and of the same length.
The parallelogram is a geometrical two-dimensional shape, which is named for the fact that it has opposite parallel sides. A parallelogram can be divided into two triangles that are congruent, and the alternating interior angles formed by the line dividing the object into triangles, are equal. The sides that are corresponding of the two separate triangles will also be of equal length. The opposite angles of the parallelogram will also be the same. Both the sides and the angles that are opposite in a parallelogram will thus be equal. This is proved by virtue of the triangles and resulting findings. The successive angles of a parallelogram add up to 180 degrees. In other words, they are supplementary angles. If diagonal lines are drawn in a parallelogram, they will divide each other in half (i.e. bisect each other).
There is more than one type of parallelogram, and in fact, a rectangle, a rhombus, and a square are all types of parallelograms. A rectangle has two short sides and two longer sides, while a square has all four sides that are the same length.
There are many examples of parallelograms in the real world. A laptop computer and a book are both examples of everyday objects that have this shape. A square or rectangular wall or table is also an example of a parallelogram.
A quadrilateral is a two-dimensional geometrical structure that always has four sides and four corners. A quadrilateral is also frequently called a polygon that has four sides. The part of the term “quad” means four.
There are many different types of quadrilaterals that can be found. Each of these quadrilaterals has their own specific set of properties that further characterize and define the shape. The sum of all the angles that are found inside (interior) of a simple quadrilateral all adds up to 360 degrees. Opposite sides may or may not be equal in length depending on what type of quadrilateral we are concerned with. A square, for instance, would have sides that are all the same length, but a rectangle would not. Similarly, opposite sides of the quadrilateral may or may not be parallel. A rectangle, rhombus and square all have parallel opposite sides, but a kite does not have any opposite parallel sides at all. In fact, all parallelograms can also be characterized as being types of quadrilateral shapes. Some quadrilaterals are called complex structures because the sides cross over to form unusual shapes.
There are several geometric shapes that can be classified as types of quadrilaterals. In fact, all parallelograms are also classified as being quadrilaterals since they have four sides. This means that a rhombus, square and rectangle are quadrilaterals. In addition to these shapes, a trapezoid and kite are also types of quadrilaterals.
There are several examples of quadrilaterals from everyday life. A square table or a rectangular wall of a house, are shapes that are quadrilateral. A kite that somebody flies is also a quadrilateral. There are many more examples of quadrilaterals than there are of parallelograms because the opposite sides do not need to be parallel. The shape simply has to have four corners and four sides.
A parallelogram is a two-dimensional geometric structure that always has four sides, of which opposite sides are the same length, and parallel. A quadrilateral is simply a two-dimensional structure that has four sides to it.
In a parallelogram there is always a pair of sides that are opposite that are parallel. In a quadrilateral, there are only sometimes parallel sides present, and at other times opposite sides are never parallel.
A shape that is a parallelogram always has opposite sides that are of the same length. A shape that is a quadrilateral does not always have sets of opposite sides that are of the same length, sometimes they are not equal length.
A parallelogram is a shape in which the opposite angles are always of equal size. A quadrilateral is a shape in which the opposite angles are sometimes, but not always of equal size.
The following geometric shapes are all also classified as being types of parallelograms: square, rhombus, and rectangle. Geometric shapes that can be classified as being quadrilaterals include the square, rectangle, rhombus, trapezoid, kite and an assortment of complex shapes.
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Exponential growth is when the number of some entity increases rapidly in an exponential manner over time. An exponential growth mathematical function is one in which numbers multiply in size as time progresses. An exponent is part of the equation as well, so for instance, an equation could be y = 5*2^{x}. In this case, each number, starting at 5 is multiplied by 2 to an exponent power such as 2. The exponent is usually an integer greater than 1, such that when a number is raised to this power, it produces an even larger number.
Drawing a graph of this function would produce a curved line that goes upwards. The slope would be constantly changing as more numbers are put into the equation. To get an equation for the slope you would have to calculate the derivative using calculus. As the numbers on the x-axis of the graph, the time variable, become bigger so do the numbers on the y-axis, the size variable. The relationship between the variables is not an inverse one and slopes upwards.
Examples of exponential growth can be seen in populations of bacteria which divide very rapidly. Salmonella enterica serovar Typhimurium bacteria, for instance, have been extensively studied and shown to have a lag phase during which time they prepare to enter a pattern of exponential growth. The bacteria will divide and the population will grow exponentially until there are no more nutrients left.
Knowing the growth rate of bacteria under various conditions can be useful in enabling scientists to develop various antimicrobial agents. These antibiotics can then be tested and evaluated based on their impact on the exponential growth rate of the bacterial target.
Decay is when numbers decrease over time in an exponential fashion, thus the result looks something like a repeated division. An exponential equation is still involved but the exponent is such that the values keep decreasing or decaying over time. For example, let’s say we have an equation: y = 5*2^{x}. In this case, each number, starting at 5 is multiplied by 2 to an exponent power such as 1/2. The exponent is a fraction such that the numbers decrease in size when plugged into the equation.
Drawing a graph of this function would produce a curved line that goes downwards. The slope would be constantly changing as more numbers are put into the equation. To get an equation for the slope you would have to calculate the derivative using calculus. As the numbers on the x-axis of the graph, the time variable, become bigger so the numbers on the y-axis, the size variable become smaller. This is an inverse relationship between the two variables of time and size, and the graph slopes downwards.
A good example of decay is the value of a new car. When you first buy the car it is worth a lot of money, but as time goes on it depreciates and loses value so that if you were to sell the car you would get less for it than you paid in the beginning. In science, the radioactive decay of isotopes is a good example of a natural process of decay which occurs. The half-life of an isotope is the time it takes for half of the atom to decay.
Knowing the radioactive decay of certain isotopes has been very useful as it has enabled scientists to date fossils that have been found in sedimentary rock layers. This gives an indication of what life was present on earth during each geological time period.
In exponential growth, numbers increase in value over time in an exponential fashion. In decay, numbers decrease in value over time in an exponential fashion.
The exponent in the equation in the case of exponential growth is usually an integer, a number that is greater than 1. The exponent in the equation for decay is a fraction that is between 0 and 1.
In the case of exponential growth, the y-values on a graph will increase as the x-values increase. In the situation of decay, the y-values on the graph will decrease as the x-values increase.
The trend that is evident in exponential growth is increasingly large numbers over time. The trend in decay is the reverse of that seen with exponential growth and instead, it is increasingly small numbers over time.
Exponential growth rate examples include the growth rates of several types of bacteria when conditions are optimal and before the substrate is depleted. Decay examples include the decreasing value of a car (depreciation) over time and the radioactive decay of radioactive isotopes with time.
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