Vertical Asymptotes: The Vertical Lines that never intersect the function

The vertical lines that correspond to the zeroes of the denominator of a rational function are called the Vertical Asymptotes. They occur only for those values of ‘x’ that produce zero in the denominator but not in the numerator. If  0/0 occurs, then we simply say we have a ‘hole’ in the graph. The next thing that comes to the mind is, How to find Vertical Asymptote of a given function. It involves a simple method where we just need to follow few steps to find all the values of x for which the denominator equals zero.

Following are the steps involved in finding the Vertical Asymptotes:
1. Vertical asymptote is in the form of an equation in x, x=a where f(x) is a function and f(a) does not exist.  A vertical asymptote is a vertical line which never intersects the function f(x).
2. We know that a fraction is undefined if its denominator is equal to zero. So, to find the vertical asymptote of a rational function, we need to  find the value of x such that the denominator is equal to zero.
3. In the next step, we need to equate the denominator to zero.
4. Then, we need to solve the equation in x to get the value of a.
5. Finally we need to plug in the value of a in the equation x=a, which is the vertical asymptote of the given function.

Note:  We can get more than one vertical asymptote depending on the given function

Finding Vertical Asymptote of a rational function, f(x) =(x^2+2x+3)/(x^2-5x+6). Let us first find all the x values by setting the denominator (x^2-5x+6) equal to zero and solving for ‘x’. Factorizing (x^2-5x+6) we get (x-6)(x+1)=0 and hence x = 6 and x=-1 which will make the denominator zero. Hence the vertical asymptotes of the given function are, x = 6 and x=-1.

Let us now Find Vertical and Horizontal Asymptotes of f(x)= (2x^2-5x+3)/x^2-1. The vertical asymptote is got by equating the denominator to zero, x^2-1=0. On factorization, we get (x+1)(x-1)=0. This gives us x=1 and x=-1 as the vertical asymptotes. To find the horizontal asymptote we need to first find the degree of the numerator and the denominator. Here the degree of the numerator and the denominator is the same, so, the horizontal asymptote is given by y= the coefficient of the highest degree in the numerator divided by the coefficient of the highest degree in the denominator. That gives us y=2/1=2, which is the horizontal asymptote.

Know more about the Asymptote Calculator. This article gives basic information about Asymptotes.

Introduction to Statistics

Introduction Statistics: The word statistics seems to have been derived from the Latin word ‘status’ or the Italian word ‘statista’ or the German word ‘statistik’ or the French word ‘statistique’, each of which means a political state.  In a short statistic summary we can understand the Importance of statistics in some different disciplines like

Statistics in planning: Statistics in indispensable planning – may it be in business, economics or government level.The modern age is termed as ‘the age of planning’ and almost all the organisations in the government or business or management are resorting to planning for efficient working and for formulating policy decisions. 

Statistics in states: As has already been pointed out, in the old days Statistics was the science of State-craft and its objective was to collect data relating to manpower, crimes, income and wealth etc. for formulating suitable military and fiscal policies.

Statistics in Mathematics: Statistics is intimately related to and essentially dependent upon mathematics.

Statistics in economics: The interaction between Statistics and Economics was first observed by William Petty in his book but it took fairly long time for effective use of Statistics in formulation of economic theories and economic policies.

Statistics in business and management: Prior to industry revolution, when the production was at the handicraft stage, the business activities were very much limited and were confined only to small units operating in their own areas.

Statistics in accountancy and Auditing: Today, the science of Statistics has assumed such unprecedented dimensions that even subjects like Accountancy and Auditing have not escaped its domain.

Statistics in industry: In industry, statistics is extensively used in Quality Control. The main objective in any production processes it to control the quality of the manufactured product so that it conforms to specifications. Statistical tools are widely used by business enterprises for the promotion of new business. Before embarking upon any production process, the business house must have an idea about the quantum of the product to be manufactures. 


Statistics in Insurance: Probability theory on which modern theory of statistics is based is the backbone of the Insurance.


Statistics in Astronomy: Even in the ancient past the astronomers made recordings about the movements of heavenly bodies like stars and planets for the study of eclipses.


Statistics in Physical Sciences: The application of Statistics in Astronomy, which is a physical science, has already been discussed. In physical sciences, a large number of measurements are taken on the same item. 

Statistics Summary:
In short summary statistics, we can more learn that In the ancient times the scope of Statistics was primarily limited to the collection of the following data by the governments for framing military and fiscal policies:
(i) Age and sex wise population of the country
(ii) Property and wealth of the country. The former enabling the government to have an idea of the manpower of the country (in order to safeguard itself against any outside aggression) and the latter providing it with information for the introduction of new taxes and levies.

Absolute Value

Introduction:
In 1806, the term ‘modulus’, ‘unit of measure’ was introduced by Jean Robert Argand especially for the complex absolute value. In 1857 onwards the term absolute value was used by French. The absolute value is also called as the ‘numerical value’ or ‘the magnitude’.

What is Absolute value?
The absolute value is defined as the value of real number ‘a’ as its distance from zero or origin which is the numerical value with out sign. The absolute value is also known as modulus and it can be denoted by the notation |a|. Absolute value of any number is always positive or zero but never negative.
Example:
What is the absolute value of -12 and 12?
The absolute value of     |12| = 12
|-12| = 12
In various mathematical and physical contexts, the absolute value is closely related to notation of magnitude, distance and norms. Even the complex number, ordered rings, the quaternions, fields, and vector spaces have also the concept of absolute values.

Solving Absolute Value Equations

To solve absolute value equation, the first thing is to convert all negative sign into positive sign.  This is the little trick insolving the absolute value equations.  The equations are split in to two equations as positive and negative as per the properties of the absolute value. Then the solutions of two equations are found and the solutions are checked by substituting the answer back into the original equation to find out whether the obtained solutions are correct or not.

Solving absolute value inequalities

In solving absolute value inequalities two things such as picture and the absolute value definition should be kept in mind. For example |x| < 3, all the points should be with in 3 and -3 and away from the zero.

The absolute value inequalities can be mentioned as open circle. This equation should be written as 3

Example:
|2x+4| < 6
-6 < 2x+4 <6
In solving absolute value inequalities, the first thing is to spiltthe given equation in to two equations as positive and negative.
2x+4 < 6 2x+4>-6
2x < 2 2x > -10
X < 1 x > -5
So the solution is -5 < x < 1 (the points in between -5 to 1)

How to solve absolute value equations
The steps are:
• Split the given equation into two equations as positive and negative
• Find the solution of two equations
• Substitute the answer back into the original equation to find out whether the solution is correct or not.

Example:
|x+2| = 7
Solving absolute value equation first spilt in to two equations as positive and negative.
X+2 = 7 x+2 = -7
X = 7-2 x = -7-2
X= 5 x = -9
The solutions are = {-9, 5}
Check the answer,
X+2 =7 x+2 = -7
If x=5 if x= -9
5+2 = 7 -9+2 = -7
7 = 7 -7 = -7

Mixed Fraction

Mixed  Fractions    A combination of a whole number and a proper fraction is called a mixed fraction. Example of mixed fractions:-
1(1/4),2(1/2) etc.Conversion of mixed fractions into improper  fractions and vice-versa….
Step 1  Obtain the mixed fraction.
Let the mixed fraction be 2 3/5.
Step 2 Identify the whole number, numerator and the denominator of the proper fraction.
Step 3 Multiply the whole number with denominator of the proper fraction and then add the product into the proper fraction numerator.
Step 4 The numerator of the fraction is thus equal to the number obtained in step 3 and denominator same as the denominator of the fraction in step 2.
Thus,

Improper fraction=((Whole number * Denominator) + Numerator)/Denominator

Addition of mixed fractions:Let us see an example of adding mixed fractions .Suppose we have to add 1 ¼ and 2 ¼ ,to add two or more mixed numbers, we change them to improper fractions and then add.For adding mixed fraction we will follow these steps:There is two way of adding mixed numbers:- 1(1/4),2(1/2) For adding mixed fraction first we are going to convert mixed fraction to improper fraction 5/4 +9/4, Now we can see that we have common denominator ,so we will simply add numerator (5+9)/4=>14/4 now we will simply reduce it to lowest term and then convert it intomixed fraction=> 3 2/4 =>3(1/2)

Subtraction of mixed fractions:For subtracting  mixed numbers,we change them to  improper fractions and then subtract.
Let us take an example to understand subtraction of mixed number.suppose we have to subtract 2(3/4) - 1(1/2) ,
For subtracting mixed fractions first we are  going to convert mixed fraction into improper fraction => 11/4 -3/2. Now for unlike denominator we will nake equal denominator =>    11-6/4=5/4  , Then convert it into mixed fraction
1 1/4

Multiplication of mixed fractions:  For multiplying mixed fractions  we will convert mixed fraction to improper fraction ,then we wll multiply the numerator and denominator separately .Reduce them  to lowest fraction and again convert it into mixed fraction .Let us take an example of 2 ½ by 5.First we will covert mixed fraction  21/2 into improper fraction ,we will have 5/2,then we will multiply 5/2 by 5, we have 2 ½ * 5 = 5/2 * 5 = 5*5/2=25/2 ,convert 25/2 into mixed fraction we will have 12(1/2)

Division of mixed fractions : For dividing mixed fractions , again we will convert mixed fraction to improper fraction>Let us take an example of 51/2 / 11  wewill convert 5 ½ into improper fraction which gives 11/2, now  we will change 11 as 1/11 by converting sign from division to multiplication .11/2 x1/11=> 11/22,conert it into lowest form, we have1/2