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.

Sum of Kinetic and Potential Energy in the system remains constant.

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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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