Question

How do you tell if a graph of a rational function has a hole in the graph or a vertical asymptote? Give an example of each case.

Answer 1

A rational function say

It can be written as a fraction of two polynomials where the denominator is not zero.

Asymptote

An asymptote is a horizontal/vertical oblique line whose distance from the graph of a function keeps decreasing and approaches zero, but never gets there.

Vertical Asymtote

An asymptote is a vertical oblique line whose distance from the graph of a function keeps decreasing and approaches zero, but never gets there.

lets consider an example say

set denominator =0

x-1=0

x=1

or

x+1=0

x=-1

So, the vertical asymptotes are the lines at x=1 and x=-1.

Hole

In a given rational function if a number causes the denominator and the numerator to be zero, then both the numerator and denominator can be factored and the common zero can be cancelled out. This means there is a hole in the function at this point.

lets consider an example say

To find a hole we look for common factors in the  denominator and the numerator .

here in this case

(x-2) is common in both  denominator and the numerator .

we need to take the common factor out and set that to zero

x-2=0

x=2

hus we get the x-coordinate of the hole. next we need to find y-coordinate.

Remember a hole is always a point (x,y).

From equation 1 we can write

substitute x=2

The hole is at