Graphing Techniques

Lesson 1: Asymptotes

The usual features of a graph's sketch includes:

  • Asymptotes

  • Axial intercepts

  • Stationary points

(1) Asymptotes

Asymptotes are imaginary boundary lines that affect the position and the shape of a graph. There are generally 3 types of asymptotes:

(A) Vertical asymptote: x = k

The vertical line x = k is a vertical asymptote if                           as            .

Generally vertical asymptotes can be obtained from the values of x which makes the equation (or y) undefined.

Example 1

Find the equation of the vertical asymptote(s) for each of the following graphs.

(A)                            

 

(B)

(C)

Solution

(A)                            

x = 3 is the vertical asymptote since               is undefined at x  = 3.

(B)   

(C)   

(B) Horizontal asymptote: y = b

The horizontal line y = b is a horizontal asymptote if              as                          .

Example 2

Find the equation of the horizontal asymptote(s) for each of the following graphs.

(A)    

(B)    

Solution

(A)    

(B)    

(C) Oblique asymptote: y = mx + c

The horizontal line y = b is an oblique asymptote if 

as                        .

Example 3

Find the equation of the asymptotes  and the graph of

Solution

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