### Home > APCALC > Chapter 2 > Lesson 2.2.1 > Problem2-51

2-51.

Sketch a graph of each of the functions below. Compare the equations and their graphs. Then write a complete set of approach statements for each.

For parts (a) and (b).
One of the functions has a hole and the other has an asymptote.

1. $y = \frac { ( x + 6 ) ( x - 1 ) } { x - 1 }$

Refer to Lesson 1.2.3, problems 1-44 & 1-45.

A complete set of approach statements include:

$x \rightarrow −\infty$
$x \rightarrow \text{(hole)}^−$
$x \rightarrow \text{(vertical asymptote)}^−$
$x \rightarrow \infty$
$x \rightarrow \text{(hole)}^+$
$x \rightarrow \text{(vertical asymptote)}^+$

1. $y = \frac { ( x + 6 ) ( x - 1 ) } { x - 2 }$

1. Explain why one graph has a hole while the other has a vertical asymptote.

How does algebra help determine the location of holes and vertical asymptotes?

2. What is the end behavior of each function?

Horizontal asymptotes, if they exist, are the same as end behavior. If there is no horizontal asymptote, find end behavior with polynomial division (ignore the remainder).