Algebra-2

To graph the function , we can analyze it in the following steps:

 

 1. Simplify the function

First, factor the numerator:

Then the function becomes:

For , we can cancel out the common factor , and we get  for .

 

 2. Asymptotes and holes

Hole:

Since we canceled out the factor  from the numerator and the denominator, there is a hole in the graph at . To find the -coordinate of the hole, we substitute  into the simplified function , which gives . So, there is a hole at the point .

 

Vertical asymptote:

There is no vertical asymptote in the traditional sense because after simplification, the function is well-defined everywhere except at  where we had the common factor cancellation.

 

Horizontal asymptote:

Since the function simplifies to  (a linear function), there is no horizontal asymptote as linear functions grow without bound as .

 

 3. Intercepts

x-intercept:

Set , for the simplified function , we have