Vertical Learning

Introduction:

            Vertical learning specifies the concept of vertical line and vertical asymptotes. Vertical asymptotes are vertical lines passing through the zeroes of the denominator of a rational function; they can also arise in other contexts, such as logarithms. A vertical line is a line that goes straight top and bottom, equivalent to the y-axis of the coordinate plane.

 

Learning Types of asymptotes:

 

Learning asymptotes comprises of following types, 

     1) vertical asymptotes.

     2) Horizonatal asymoptotes.

     3) slant or oblique asymptotes.

            If the asymptote is equivalent and linear to y - axis then it is referred as vertical asymptote. A rational function r(x)/s(x) have a vertical asymptote at x=a, for any ‘a’ where s (a) is 0. If the asymptote is equivalent and linear to x - axis then it is referred as horizontal asymptote. When the degree of the denominator is lower than the degree of the numerator, there is no horizontal asymptote. The additive asymptote which is neither vertical nor horizontal is called slant or oblique asymptote.

The following diagram representation of vertical line shows given below:

              

The following diagram representation of vertical asymptote shows given below:

                        

 

Examples problems for Learning vertical asymptotes:

 

The following examples are very helpful for learning vertical asymptotes:

1) Find vertical asymptotes of function f(x) = (x+6)/(x2+6x+8)

Solution:

         Check the zeroes of the denominator:

         x2 + 6x + 8 = 0

        (x + 4)(x + 2) = 0

        x = –4 or x = –2

        Since denominator cannot be zero, vertical asymptotes will be at x = –4 and x = –2

 

2) Find vertical asymptotes of function f(x) = (x+6)/(x2+6x+8)

Solution:

       Check the zeroes of the denominator:

       x2 + 6x + 8 = 0

       (x + 4)(x + 2) = 0

       x = –4 or x = –2

       Since denominator cannot be zero, vertical asymptotes will be at x = –4 and x = –2

3) Find vertical asymptotes of f(x) =(x-3)/(x2+4)

Solution:

          make denominator zero, that is x2 +4=0

          x2 =-4

         There are no real solutions. So, there are no vertical asymptotes.

4) Find vertical asymptotes of f(x)=(x-3)/(x2-5)

Solution:

          Make denominator zero, that is x2-5=0

          x2 =5

          There are no real solutions. So, there are no vertical asymptotes.