Graph Types

 

Graphs are pictorial representation of the characteristic behavior of mathematical functions. There can be numerous forms of graphical illustrations. 1st let us discuss about the functions of whose we sketch the graph. Function in a way is kind of relation that also has a set orderly pairs, but the difference between a relation and a function is that a function is always one – one & onto. For a relation to be one – one, every element in the domain set of the function must have a unique element in the range set. Onto means the complete domain of the function is covered. Depending upon the type of functions we deal with, the graph differs accordingly. For instance, suppose we have a linear function equation given as: h (x) = 7 x + 8, then the graph would represent a straight line. There can be other types of functions also like polynomial function, algebraic function, exponent function, logarithmic function etc. The types of graphs and the functions behaviors differ with the changing type of the functions. Suppose we have to graph the above linear function h (x) = 7x + 8, we would start by evaluating the domain of the function 1st. The domain in this case will be all real numbers and the corresponding range and therefore the coordinate pairs can be evaluated to plot the graph as follows: (0, 8), (-1, 1), (1, 15), (-2, -6) etc. Graph of the function can be drawn as follows:

 

Next we will discuss how to find the range of a function?

The range of a function can be evaluated by substituting the values of the domain of function like we did in the example we considered above to find out the coordinate pairs. These concepts are important from the perspective of iit jee syllabus.

 

Graphing

 

Here we will study different types of method for Graphing linear equation, function and relation, quadratic equation and many more. Here we will discuss graphing rational function.

Ratio of two polynomial equation is said to be rational function. As we know that numerator and denominator values are present in rational function, and value present in denominator of rational function is never equals to zero. For example: p² + 6 / p + 3, as it is the ratio of two polynomials. Now we will understand how to graph a simple rational function.

Suppose we have a rational function f (p) = p² – 4 / p² – 4p, then we can plot graph of rational function as shown below.

Solution: – Steps for graphing are given below.

Step 1: First of all check whether the function is rational function or not. According to definition of rational functions it is ratio of two polynomials. So rational function is:

=> f (p) = p² – 4 / p² – 4p,

 

Here in this given function we cannot take value of denominator as zero because when we put value of denominator as zero then whole function changes to infinity. y – Intercept will not be present in graph.

Step 2: Take numerator value equals to zero to obtain the value of ‘x’.

So we can write it as:

=> p² – 4 = 0, so here we have two values of 'p' that is p = + 2.

We can write above function as:

= p² – 4 = p (p – 4) = 0, so here we get value of 'p' as 0 and 4.

Step 3: Now take different values of 'p' to find more values.

If we take values of 'p' as 1, 3 and 5 then we get other values 1, -5 / 3 and 21 / 5 respectively. Graph of rational function is shown below:

 

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graphing trig functions

 

Trigonometric functions can be defined as functions of angles, they are also known as circular functions. Now we will understand the concept of graphing trig functions.

We will start with basic sine functions, suppose we have a function f (j) = sin (j). Value of amplitude for this function is given as 1, it is so because graph of sine function always moves one unit up and one unit down from given mid line of graph. Time period for sin function graph is given as 2⊼. It is because of wave of sine function changes after every 2⊼ units. Graph of trigonometric sine function is given below. Time period for sin function graph is 2⊼. Graph of trigonometric sine function is given below.

 

 

Above given graph is three times bigger than second graph and amplitude of this graph changes from 1 to 3. Here amplitude value of above function is ‘1’. Any value is given for 'J', it is multiplied with trigonometric function that results in amplitude value. In that case number is taken as 3, it is so because amplitude value is given as 3. So 0.5 cos (j) will have amplitude of ½, and – 2 cos (p), it has amplitude of 2 and also flipped upside down. In mathematics, commonly used trigonometric functions are sine, cosine and tangent function.

Now we will talk about trigonometric function description:

Sin = opposite / hypotenuse,                     cosine = adjacent / hypotenuse,

Tan = opposite / adjacent,                         cotangent = adjacent / opposite,

Secant = hypotenuse / adjacent,                cosecant = hypotenuse / opposite,

This is all about trigonometric functions.

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graph functions

In a function each input term is straight lined with one matching output term of that function. In mathematics, there are so many methods through which we can define function. Some functions are described by formula and some one by algorithm. Both the method tells that how to work out the output value for a given input value. Some of the functions are represented with the help of graph, so that types of function are said to be graph functions. In some of the cases function are represented with the help of table which gives us output values for selected input. (know more about graph functions, here)

We can plot the function graph by different methods. In some of the cases, first assume any function which contain value of two variable (suppose x and y), then put different values to find the values of other variable. In this way we can have a set of different values and at last put these values in the plane to get a graph. One other method to plot a function graph is shown below:

For example: let we have given a function:

F (x) = a, if a = 1,

b, if a = 2,

c, if a = 3. Then we have to graph a function using these values.

So the function value is (1, a), (2, b), (3, c). In cubical form function can be written as:

F (a) = a² – 9a, if we plot a graph of this function, we get curve mention below:

Using above values we can easily plot a function graph.

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How to Graph an Equation

Hi students, in mathematics, we will see different graph such as functional graph, quadratic graph. Here we will see How to Graph an Equation font face=“Lucida Grande, Lucida Sans Unicode, Calibri, Arial, Helvetica, Sans, FreeSans, Jamrul, Garuda, Kalimati”>. Linear equation is a type of equation that makes a straight line when it is plotted on a graph. Commonly linear equation is written in form y = mx + b, Here ‘m’ is the slope of line and ‘c’ denotes Y- intercept where line crosses y- axis. Here we will see How to Graph an Equation. Equation is a set or collection of different equations. Now we will understand process of graphing equations. Steps to be followed while graphing equations are shown below. (know more about Graph, here)

Step 1: for solving an equation first we take two equations. Let we have linear equations 3u + v = 10 and 6u – 3v = 8, now we will understand how to equation.

Step 2: Then calculate value of one variable from one equation and then put that value in second equation. So from first equation we can easily calculate the value of ‘v’. So it can be written as:

⇒ 3u + v = 10, on further solving we can write it as:

⇒ v = 10 – 3u, we can get the value of ‘v’. Now put the value of ‘v’ in second equation. On putting value of ‘v’ in equation we get:

⇒ 6u – 3v = 8; put value of ‘v’ to find the value of ‘u’.

⇒ 6u – 3 * (10 – 3u) = 8, on further solving we get:

⇒ 6u – 30 + 9u = 8,

Now add like terms if present in equation;

⇒ 15u = 38,

⇒ u = 2.53, Now put the value of ‘u’ in 1 st equation.

⇒ v = 10 – 3u,

⇒ v = 10 – 3 * 2.53,

⇒ v = 2.4.

So, value of ‘u’ and ‘v’ is 2.53 and 2.4, on plotting these values we get a straight line graph. This is how we can graph a equation.

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how to make a histogram

Geometrical mathematics is a field of mathematics that provides a wide area of study about geometric figures and shapes. In geometrical mathematics we basically deals with the measurement of figures, calculating their area, volume and so on. The most important use of geometrical mathematics is to represent the data in diagrammatically form. In real world some time we need to represent the data in a quick access way that helps in making decision. At that time a concept of graph comes in our mind. The graph is a way of representing the sample data in the form of bars, circle and so on.

Here we are going to discuss about one of type of graph that is histogram. In the same time a question arises from lots of students that says how to make a histogram. Then we can say that there are lots of steps are defined in mathematics that helps in understanding the concept of histogram. According to graph definition histogram can be define as graph that represent the frequency distribution of data. In another sense histogram can be represented in the form of  frequency that are shown in the form of  rectangle bars whose widths represent the class interval and whose areas are proportional to the corresponding data frequencies. In histogram we need to remember some points that are:

A ) Properly perform the analysis of the data.

B ) label the graph very clearly and carefully.

C ) Frequency calculation should be perform very carefully.

D ) bars are based on scale value so they drawn very carefully.

At the time of solving algebraic equation polynomial solver is a concept that helps in solving polynomial equation. Computer science syllabus cbse provide the enough material which helps student in their exams.

How to Solve Abscissa

Before study about the meaning of Abscissa coordinate first it is necessary to know about the coordinate plane. A rectangular plane that is obtained from two number lines i.e. vertical number line and the horizontal number line is known as coordinate plane. We know that horizontal line is denoted by the x – axis which is also known as abscissa and vertical line is known as ordinate which is the y – axis. In case of horizontal line positive numbers are taken at right hand side and negative number are taken at left hand side. In the vertical number line upper side contains positive numbers and lower side contains negative numbers. The meaning of abscissa is the distance from the y -axis in the Cartesian coordinate system which is measured parallel to the x – axis.

A point in a plane which has coordinates (5, 3), so ‘5’ is abscissa of this coordinate plane.

Abscissa definition can be understood by definition given below;

let (8, 3) here first value is ‘8’ is considered as the abscissa and it is plotted on the horizontal axis in two dimensional coordinate system. The second element is 3 in the ordered pair is considered as ‘ordinate’ which is plotted on vertical line i.e. on y – axis.

Let we have to plotted the point (3, -5), this point lies in fourth quadrant. The point 3 represents the ‘abscissa’ and -5 represents the ‘ordinate’. The point is located in fourth quadrant. we know that in fourth quadrant abscissa is positive and ordinate is negative. So we can say abscissa meaning is x -coordinate. This is all about abscissa definition. In mathematics there are different types of Integration Rules. Before entering in the examination hall please go through the cbse sample papers so that it is very easy to revise all concepts. You can get them from here.

 

Learn Graphing Equations

By Graphing Equations, we mean that we need to plot the equations on the  graph paper. These equations can be used to represent the lines, circles, parabolas or some times they are used to get the solutions of the pair of the linear equations.

Let us start with graphing the  linear equation with only one variable. It will be more clear with the following example:

Suppose we have the equation say x + 5 = 9

Here we will first solve the linear equation, by taking the  constant terms on one side of the equation. For this we are going to subtract 5 from both sides of the equations.

 SO we get :

X + 5 – 5 = 9 – 5

Or x = 4

Now we conclude that the value of x will be a constant value i.e. 5, what ever be the value of x. Thus in order to plot this  equation on the graph paper, we will take the different values of y, but the value of x will remain same. Thus we say that the  coordinate of  different points on the plane can be as follows :

( 4, 0 ), ( 4 , 1),  ( 4 , 2 ),  ( 4 , 3) and so on. Thus we say that by marking these points on the graph paper, we come to the conclusion that the  line for the equation which we get is parallel to y axis and is 4 unit away from the axis through out. (Know more about Equations in broad manner, here,)

To learn how to get the solution for Improper Integrals, we can take the online help of the math tutor and understand the concept very easily. The tutors are designed in a friendly manner for the  learners.

We can take  cbse question bank from the cbse website for the different subjects, which can work as the guide line for the students appearing for the upcoming exams.

Box and Whisker Plot

A figure which is plot using a number line to represent the distribution of data is known as box and whisker. Along the number of lines a box and whisker plot is used to distribute a set of data. The end value of the whisker denotes the possible alternative values among them. (know more about Box plot, here)

The minimum and maximum value of data and standard deviation is used above and below the mean of the data. Some of the box plots are used for addition character to denote the mean of the given data.

Let’s see the steps for designing the box and whiskers graph.

To plot a box and whisker graphs we need to follow some steps which are mention below:

Step 1: To plot the box and whisker graph, we have to start the data by ordering.

Step 2: Then we have to put the values of box and whisker in the numeric order, if they are not ordered.

Step 3: After than we need to find the median of the given data, the median is used to divides the data into two halves.

Step 4: To divide the data into quarters, it is necessary to the median of these two halves.

Step 5: let the number of values is even then first median is the average value of two middle values, and then the median is an actual data point. It cannot be added in sub – median computation. It is sometimes also known as a box and whisker plot or box chart.

Now we will see the percentage increase calculator.

As we know that percentage increase calculator is an online mathematical tool which is used to calculate how much percentage a given relative number. The percentage number increase is the way to represent a change in percentage of a number with respect to the mention number. To get more information about percentage increase calculator then follow sample papers for CBSE.

Stem and Leaf Plot

Any stem and leaf plot or so called stem and leaf plot diagram appears almost similar to the bar type chart. Every number which is present in the data is divided in the stem and leaf plot diagram into the stem and the leaves. The stem of any number in the stem and leaf plot diagram contains every digit except the final digit whereas the leaf of any number in the stem and leaf plot diagram will just be a single digit always. (know more about Stemplot, here)

Let us consider one example in which certain data is given for which we have to construct the stem and leaf plot diagram. The data which is given contains the numbers which are as follows: 30, 34, 36, 38, 43, 46, 47, 48, 52, 55, 56, 59. Now we have to make the stem and leaf plot diagram of this data for which we will have to find the stem and the leaves of these numbers. As we have already discussed that stem of any number will contain every digit except the last digit in the number so the stem of this data will include the digits 3 in the 1^st row, 4 in the 2^nd row and 5 in the 3^rd row in the stem and leaf plot diagram. Also we know that the leaves are nothing but just consist of the single digit which is the final digit of the numbers so the leaves of this data will include the digits 0 4 6 8 in the 1^st row corresponding to the digit 3 of the stem, the digits 3 6 7 8 in the 2^nd row corresponding to the digit 4 of the stem and the digits 2 5 6 9 in the 3^rd row corresponding to the digit 5 of the stem in the stem and leaf plot diagram.

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