HOW TO GRAPH RATIONAL FUNCTIONS 3

About the topic "How to graph rational functions 3"

"How to graph rational functions 3" is the question having had by almost all the students who study math in high schools.

Even though students can get this stuff on internet, they do not understand exactly what has been explained. 

To make the students to understand "How to graph rational functions 3", we have explained the above mentioned stuff step by step.

Rational function - Example

Before learning "How to graph the rational functions", first you have to be knowing the following stuff. 

1. Vertical Asymptote

2. Horizontal Asymptote

3. Slant Asymptote

4. Hole

To know more about the above mentioned stuff, please click the topics given above.

If you had already learned the above mentioned stuff, then you are ready to learn the stuff, "How to graph rational functions 3".

Now let us take an example and understand graphing rational functions.

Example :

Graph the rational function given below.

Solution :

Step 1:

First, we have to find hole, if any.

To find hole of the rational function, we have to see whether there is any common factor found at both numerator and denominator.

So, let us factor both numerator and denominator.

y = [(x-5)(x+1)] / (x-2)

In our problem, clearly there is no common factor found at both numerator and  denominator.So, there is no hole.

Step 2:

Now, we have to find vertical asymptote, if any. 

Most of the rational function will have vertical asymptote.

To find vertical asymptote, we have to make the denominator equal to zero.

When we do so,    x - 2 = 0  ====> x = 2

So, the vertical asymptotes are  x = 2  

Step 3 :

Now we have to find horizontal asymptote, if any.

In the rational function given above, the highest exponent of the numerator is 2  and denominator is 1.

Clearly, the highest exponent of the numerator is greater than the highest exponent of the denominator

In a rational function, if the highest exponent of the numerator is greater than the highest exponent of the denominator, there is no horizontal asymptote.

So, there is no horizontal asymptote for the given rational function.

Step 4 :

Now we have to find slant asymptote, if any.

Clearly, the exponent of the numerator is greater than the exponent of the denominator by one. So, there is a slant asymptote

To get the equation of the slant asymptote, we have to divide the numerator by the denominator using long division as given below.

In the above long division, the quotient is (x - 2).

Hence, the equation of the slant asymptote is y = x - 2

Step 5 :

In the given rational function, now we have to plug some random values for "x" and find the corresponding values of "y". 

We have already known that the vertical asymptote is x = 2. 

Now, we have to take some random values for x in the following intervals. 

x<2, x>2, but not x = 2. 

(Because, x = 2 is vertical asymptote)

Step 5 :

In the slant asymptote y= x - 2, we have to plug some random values for "x" and find the corresponding values of "y". 

Graph of the given rational function

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