# 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, ,

Q6:

Which of the following lines is a vertical asymptote of the graph of the function ?

• A
• B
• C
• D

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

Q8:

By sketching a graph, find the vertical asymptotes of the function .

• A,
• B,
• C,
• D,

Q9:

What are the two asymptotes of the hyperbola ?

• A
• B
• C
• D
• E

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