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In this lesson, we will learn how to find the horizontal and vertical asymptotes of given functions.
Q1:
Find the vertical and horizontal asymptotes of the function π ( π₯ ) = 3 π₯ β 1 5 π₯ + 3 2 2 .
Q2:
What are the two asymptotes of the hyperbola π¦ = 5 π₯ + 1 3 π₯ β 4 ?
Q3:
The graph of equation π¦ = π π₯ + π π π₯ + π is a hyperbola only if π β 0 . In that case, what are the two asymptotes?
Q4:
By writing the expression π π₯ + π π π₯ + π in the form π΄ π π₯ + π + π , determine the asymptotes of 5 π₯ β 1 3 π₯ β 3 + 2 + 1 2 π₯ 1 β 2 π₯ .
Q5:
Find the vertical and horizontal asymptotes of the function π ( π₯ ) = 3 π β 2 π₯ 2 .
Q6:
Find the vertical and horizontal asymptotes of the function π ( π₯ ) = β 2 π₯ 3 + 2 π₯ β 1 5 π₯ 2 l n .
Q7:
Find the vertical and horizontal asymptotes of the function π ( π₯ ) = 4 ( β π₯ + 5 ) l n l n .
Q8:
On the left is the graph of π ( π₯ ) = 2 π₯ + 1 3 π₯ + 4 and on the right is the graph of π¦ = 1 π₯ .
What are the coordinates of the intersection of the asymptotes of π¦ = π ( π₯ ) ?
Find π , π , and π so that with π ( π₯ ) = π π₯ , we have π ( π₯ ) = π ( π₯ + π ) + π .
Q9:
Which of the following lines is a vertical asymptote of the graph of the function π ( π₯ ) = π₯ β 8 π₯ + 2 π₯ β 1 5 3 2 ?
Q10:
The function π ( π₯ ) = 3 π₯ β 3 3 π₯ + 8 0 ( π₯ β 3 ) ( π₯ β 5 ) ( π₯ β 7 ) 2 is found to have a vertical asymptote at π₯ = 3 . To which of { β β , + β } do the values of π ( π₯ ) go as π₯ approaches 3 from the left and the right?
Q11:
On the left is the graph of π ( π₯ ) = 7 ( 1 β 2 ) β π₯ , which has a horizontal asymptote. On the right is the graph of π ( π₯ ) = 1 π ( π₯ ) .
What is the value of π΄ ?
List all the asymptotes of π ( π₯ ) = 1 π ( π₯ ) .
Q12:
Consider the function π ( π₯ ) = 4 π₯ + 7 2 π₯ β 5 .
What are the vertical and horizontal asymptotes of the graph π¦ = π ( π₯ ) ?
Write π οΌ π₯ + 5 2 ο in a simplified form. What are the vertical and horizontal asymptotes of the graph π¦ = π οΌ π₯ + 5 2 ο ?
Write π οΌ π₯ + 5 2 ο β 2 in a simplified form. What are the vertical and horizontal asymptotes of the graph π¦ = π οΌ π₯ + 5 2 ο β 2 ?
What combination of horizontal and vertical shifts moves the intersection of the asymptotes of the graph π¦ = π ( π₯ ) to the origin ( 0 , 0 ) ?
What is the dilation factor A required to map the graph of π¦ = π οΌ π₯ + 5 2 ο β 2 onto the hyperbola π¦ = 1 π₯ ? Write this in the form π΄ οΌ π οΌ π₯ + 5 2 ο β 2 ο = 1 π₯ .
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 π ( π₯ ) = π π₯ + π π π₯ + π produces the graph of π¦ = 1 π₯ . What is g?
What sequence of transformations maps the graph of π ( π₯ ) = 5 π₯ β 3 2 π₯ + 1 onto the hyperbola π¦ = 1 π₯ ?
Q13:
By sketching a graph, find the vertical asymptotes of the function π ( π₯ ) = 2 π₯ + 6 π₯ β 2 π₯ β 3 2 2 .
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