# Worksheet: Horizontal and Vertical Asymptotes of a Function

In this worksheet, we will practice finding the horizontal and vertical asymptotes of a function.

**Q1: **

Find the vertical and horizontal asymptotes of the function .

- AThe function has no vertical asymptote and a horizontal asymptote at .
- BThe function has no vertical asymptote and a horizontal asymptote at .
- CThe function has a vertical asymptote at and no horizontal asymptote.
- DThe function has no vertical asymptote and a horizontal asymptote at .
- EThe function has a vertical asymptote at and no horizontal asymptote.

**Q2: **

What are the two asymptotes of the hyperbola ?

- A
- B
- C
- D
- E

**Q3: **

The graph of equation is a hyperbola only if . In that case, what are the two asymptotes?

- A
- B
- C
- D
- E

**Q4: **

By writing the expression in the form , determine the asymptotes of .

- A , ,
- B , ,
- C , ,
- D , ,
- E , ,

**Q5: **

On the left is the graph of and on the right is the graph of .

What are the coordinates of the intersection of the asymptotes of ?

- A
- B
- C
- D
- E

Find , , and so that with , we have .

- A , ,
- B , ,
- C , ,
- D , ,
- E , ,

**Q7: **

Consider the function .

What are the vertical and horizontal asymptotes of the graph ?

- A ,
- B ,
- C
- D ,
- E ,

Write in a simplified form. What are the vertical and horizontal asymptotes of the graph ?

- A , ,
- B , ,
- C , ,
- D , ,
- E , ,

Write in a simplified form. What are the vertical and horizontal asymptotes of the graph ?

- A , ,
- B , ,
- C , ,
- D , ,
- E , ,

What combination of horizontal and vertical shifts moves the intersection of the asymptotes of the graph to the origin ?

- Aa shift of to the left and a shift of 2 downward
- Ba shift of to the right and a shift of 2 upward
- Ca shift of to the left and a shift of 2 downward
- Da shift of to the left and a shift of 1 downward
- Ea shift of to the right and a shift of 2 upward

What is the dilation factor required to map the graph of onto the hyperbola ? Write this in the form .

- Aa dilation by a factor of so
- Ba dilation by a factor of so
- Ca dilation by a factor of so
- Da dilation by a factor of so
- Ea dilation by a factor of so

Applying a shift of 1 to the right, a shift of 3 upward, and then a dilation by a factor of 2 to the graph of produces the graph of . What is ?

- A
- B
- C
- D
- E

What sequence of transformations maps the graph of onto the hyperbola ?

- Aa shift of to the right, a shift of downward, and then a dilation by a factor of
- Ba shift of to the right, a shift of downward, and then a dilation by a factor of
- Ca shift of to the right, a shift of downward, and then a dilation by a factor of
- Da shift of to the right, a shift of downward, and then a dilation by a factor of
- Ea shift of to the right, a shift of downward, and then a dilation by a factor of

**Q10: **

Determine the vertical and horizontal asymptotes of the function .

- AThe vertical asymptote is at , and the horizontal asymptote is at .
- BThe vertical asymptote is at , and the horizontal asymptote is at .
- CThe vertical asymptote is at , and the horizontal asymptote is at .
- DThe function has no vertical asymptote, and the horizontal asymptote is at .
- EThe vertical asymptote is at , and the function has no horizontal asymptote.

**Q11: **

Find the vertical and horizontal asymptotes of the function .

- AThe function has vertical asymptotes at and and a horizontal asymptote at .
- BThe function has vertical asymptotes at and and no horizontal asymptotes.
- CThe function has vertical asymptotes at and and a horizontal asymptote at .
- DThe function has vertical asymptotes at and and no horizontal asymptotes.
- EThe function has vertical asymptotes at and and a horizontal asymptote at .

**Q12: **

Find the vertical and horizontal asymptotes of the function .

- AThe function has a vertical asymptote at and a horizontal asymptote at .
- BThe function has a vertical asymptote at and a horizontal asymptote at .
- CThe function has a vertical asymptote at and no horizontal asymptote.
- DThe function has a vertical asymptote at and no horizontal asymptote.
- EThe function has a vertical asymptote at and a horizontal asymptote at .

**Q13: **

Find the vertical and horizontal asymptotes of the function .

- AThe horizontal asymptote is , and there are no vertical asymptotes.
- BThere are no horizontal asymptotes, and the vertical asymptote is .
- CHorizontal asymptote at , and there are no vertical asymptotes.
- DThe horizontal asymptote is , and there are no vertical asymptotes.
- EThere are no horizontal asymptotes, and the vertical asymptote is .

**Q14: **

On the left is the graph of , which has a horizontal asymptote. On the right is the graph of .

What is the value of ?

- A
- B
- C
- D
- E

List all the asymptotes of .

- A , ,
- B , ,
- C , ,
- D ,
- E , ,

**Q15: **

Find and so that the function has the single vertical asymptote and the two horizontal asymptotes and . What is the range of this function?

- A , , and the range is all values of satisfying together with .
- B , , and the range is all values of satisfying .
- C , , and the range is all values of satisfying together with .
- D , , and the range is all values of satisfying together with .
- E , , and the range is all values of satisfying .

**Q16: **

Determine whether the following statement is true: The only polynomials whose graphs have a horizontal asymptote are the constant polynomials, those of degree 0.

- Afalse
- Btrue