Find Intercepts and Asymptotes - Pre-Calculus
Card 0 of 64
Find the vertical and horizontal asymptotes of the function
Find the vertical and horizontal asymptotes of the function
Tap to see back →
The function
is already in simplified form.
To find the vertical asymptotes, we set the denominator equal to 
and solve for 
.
yields the vertical asymptotes
To find the horizontal asymptote, we examine the largest degree of 
between the numerator and denominator
Note that
Because the largest degree of 
in the numerator is less than the largest degree of 
in the denominator, or 
we find the horizontal symptote to be
The function
is already in simplified form.
To find the vertical asymptotes, we set the denominator equal to
yields the vertical asymptotes
To find the horizontal asymptote, we examine the largest degree of
Note that
Because the largest degree of
we find the horizontal symptote to be
Suppose the function below has an oblique (i.e. slant asymptote) at 
.
If we are given 
, what can we say about the relation between 
and 
and between 
and 
?
Suppose the function below has an oblique (i.e. slant asymptote) at
If we are given
Tap to see back →
We can only have an oblique asymptote if the degree of the numerator is one more than the degree of the denominator. This stipulates that 
must equal 
.
The slope of the asymptote is determined by the ratio of the leading terms, which means the ratio of 
to 
must be 3 to 1. The actual numbers are not important.
Finally, since the value of 
is at least three, we know there is no intercept to our oblique asymptote.
We can only have an oblique asymptote if the degree of the numerator is one more than the degree of the denominator. This stipulates that
The slope of the asymptote is determined by the ratio of the leading terms, which means the ratio of
Finally, since the value of
Find the 
-intercept and asymptote, if possible.
Find the
Tap to see back →
To find the y-intercept of 
, simply substitute 
and solve for 
.
The y-intercept is 1.
The numerator, 
, can be simplified by factoring it into two binomials.
There is a removable discontinuity at 
, but there are no asymptotes at 
since the 
terms can be canceled.
The correct answer is: 
To find the y-intercept of
The y-intercept is 1.
The numerator,
There is a removable discontinuity at
The correct answer is:
Find the 
-intercepts of the rational function
.
Find the
Tap to see back →
The 
-intercept(s) is/are the root(s) of the numerator of the rational functions.
In this case, the numerator is 
.
Using the quadratic formula,
the roots are 
.
Thus, 
are the 
-intercepts.
The
In this case, the numerator is
Using the quadratic formula,
the roots are
Thus,
Find the vertical asymptotes of the following rational function.
Find the vertical asymptotes of the following rational function.
Tap to see back →
Finding the vertical asymptotes of the rational function 
amounts to finding the roots of the denominator, 
.
It is easy to check, using the quadratic formula,
that the roots, and thus the asymptotes, are 
.
Finding the vertical asymptotes of the rational function
It is easy to check, using the quadratic formula,
that the roots, and thus the asymptotes, are
Find the y-intercept and asymptote, respectively, of the following function, if possible. 
Find the y-intercept and asymptote, respectively, of the following function, if possible.
Tap to see back →
Before we start to simplify the problem, it is crucial to immediately identify the domain of this function 
.
The denominator cannot be zero, since it is undefined to divide numbers by this value. After simplification, the equation is:
The domain is 
and there is a hole at 
since there is a removable discontinuity. There are no asymptotes.
Since it's not possible to substitute 
into the original equation, the y-intercept also does not exist.
Therefore, the correct answer is:
Before we start to simplify the problem, it is crucial to immediately identify the domain of this function
The denominator cannot be zero, since it is undefined to divide numbers by this value. After simplification, the equation is:
The domain is
Since it's not possible to substitute
Therefore, the correct answer is:
What is the 
-intercept of the following function?
What is the
Tap to see back →
The y-intercept of a function is always found by substituting in 
.
We can go through this process for our function.
The y-intercept of a function is always found by substituting in
We can go through this process for our function.
What is a vertical asymptote of the following function?
What is a vertical asymptote of the following function?
Tap to see back →
To find the vertical asymptote of a function, we set the denominator equal to 
.
With our function, we complete this process.
The denominator is 
, so we begin:
To find the vertical asymptote of a function, we set the denominator equal to
With our function, we complete this process.
The denominator is
Find the zeros and asymptotes for
.
Find the zeros and asymptotes for
Tap to see back →
To find the information we're looking for, we should factor this equation:
This means that it simplifies to 
.
When the equation is in the form of a fraction, to find the zero of the function we need to set the numerator equal to zero and solve for the variable.
To find the asymptote of an equation with a fraction we need to set the denominator of the fraction equal to zero and solve for the variable.
Therefore our equation has a zero at -3 and an asymptote at -2.
To find the information we're looking for, we should factor this equation:
This means that it simplifies to
When the equation is in the form of a fraction, to find the zero of the function we need to set the numerator equal to zero and solve for the variable.
To find the asymptote of an equation with a fraction we need to set the denominator of the fraction equal to zero and solve for the variable.
Therefore our equation has a zero at -3 and an asymptote at -2.
Find the slant and vertical asymptotes for the equation
.
Find the slant and vertical asymptotes for the equation
Tap to see back →
To find the vertical asymptote, just set the denominator equal to 0:
To find the slant asymptote, divide the numerator by the denominator, but ignore any remainder. You can use long division or synthetic division.
The slant asymptote is
.
To find the vertical asymptote, just set the denominator equal to 0:
To find the slant asymptote, divide the numerator by the denominator, but ignore any remainder. You can use long division or synthetic division.
The slant asymptote is
Find the slant asymptote for
.
Find the slant asymptote for
Tap to see back →
By factoring the numerator, we see that this equation is equivalent to
.
That means that we can simplify this equation to 
.
That means that 
isn't the slant asymptote, but the equation itself.
is definitely an asymptote, but a vertical asymptote, not a slant asymptote.
By factoring the numerator, we see that this equation is equivalent to
That means that we can simplify this equation to
That means that
Which of these functions has a vertical asymptote of 
and a slant asymptote of 
?
Which of these functions has a vertical asymptote of
Tap to see back →
In order for the vertical asymptote to be 
, we need the denominator to be 
. This gives us three choices of numerators:
If the slant asymptote is 
, we will be able to divide our numerator by 
and get 
with a remainder.
Dividing the first one gives us 
with no remainder.
Dividing the last one gives us 
with a remainder.
The middle numerator would give us what we were after, 
with a remainder of -17.
The answer is
In order for the vertical asymptote to be
If the slant asymptote is
Dividing the first one gives us
Dividing the last one gives us
The middle numerator would give us what we were after,
The answer is
Find the y-intercept of 
, if any.
Find the y-intercept of
Tap to see back →
Be careful not to confuse this equation with the linear slope-intercept form. The y-intercept of an equation is the y-value when the x-value is zero.
Substitute the value of 
into the equation.
Simplify the equation.
The y-intercept is: 
Be careful not to confuse this equation with the linear slope-intercept form. The y-intercept of an equation is the y-value when the x-value is zero.
Substitute the value of
Simplify the equation.
The y-intercept is:
Find the horizontal asymptote of the function:
Find the horizontal asymptote of the function:
Tap to see back →
To find the horizontal asymptote, take the leading term of the numerator and the denominator and divide. In this case:
To find the horizontal asymptote, take the leading term of the numerator and the denominator and divide. In this case:
Determine the asymptotes, if any: 
Determine the asymptotes, if any:
Tap to see back →
Factorize both the numerator and denominator.
Notice that one of the binomials will cancel.
The domain of this equation cannot include 
.
The simplified equation is:
Since the 
term canceled, the 
term will have a hole instead of an asymptote.
Set the denominator equal to zero.
Subtract one from both sides.
There will be an asymptote at only: 
The answer is: 
Factorize both the numerator and denominator.
Notice that one of the binomials will cancel.
The domain of this equation cannot include
The simplified equation is:
Since the
Set the denominator equal to zero.
Subtract one from both sides.
There will be an asymptote at only:
The answer is:
Where is an asymptote located, if any? 
Where is an asymptote located, if any?
Tap to see back →
Factor the numerator and denominator.
Rewrite the equation.
Notice that the 
will cancel. This means that the root of 
will be a hole instead of an asymptote.
Set the denominator equal to zero and solve for x.
An asymptote is located at: 
The answer is: 
Factor the numerator and denominator.
Rewrite the equation.
Notice that the
Set the denominator equal to zero and solve for x.
An asymptote is located at:
The answer is:
Find the vertical and horizontal asymptotes of the function
Find the vertical and horizontal asymptotes of the function
Tap to see back →
The function
is already in simplified form.
To find the vertical asymptotes, we set the denominator equal to and solve for .
yields the vertical asymptotes
To find the horizontal asymptote, we examine the largest degree of between the numerator and denominator
Note that
Because the largest degree of in the numerator is less than the largest degree of in the denominator, or
we find the horizontal symptote to be
The function
is already in simplified form.
To find the vertical asymptotes, we set the denominator equal to
yields the vertical asymptotes
To find the horizontal asymptote, we examine the largest degree of
Note that
Because the largest degree of
we find the horizontal symptote to be
Suppose the function below has an oblique (i.e. slant asymptote) at .
If we are given , what can we say about the relation between and and between and ?
Suppose the function below has an oblique (i.e. slant asymptote) at
If we are given
Tap to see back →
We can only have an oblique asymptote if the degree of the numerator is one more than the degree of the denominator. This stipulates that must equal .
The slope of the asymptote is determined by the ratio of the leading terms, which means the ratio of to must be 3 to 1. The actual numbers are not important.
Finally, since the value of is at least three, we know there is no intercept to our oblique asymptote.
We can only have an oblique asymptote if the degree of the numerator is one more than the degree of the denominator. This stipulates that
The slope of the asymptote is determined by the ratio of the leading terms, which means the ratio of
Finally, since the value of
Find the -intercept and asymptote, if possible.
Find the
Tap to see back →
To find the y-intercept of , simply substitute and solve for .
The y-intercept is 1.
The numerator, , can be simplified by factoring it into two binomials.
There is a removable discontinuity at , but there are no asymptotes at since the terms can be canceled.
The correct answer is:
To find the y-intercept of
The y-intercept is 1.
The numerator,
There is a removable discontinuity at
The correct answer is:
Find the -intercepts of the rational function
.
Find the
Tap to see back →
The -intercept(s) is/are the root(s) of the numerator of the rational functions.
In this case, the numerator is .
Using the quadratic formula,
the roots are .
Thus, are the -intercepts.
The
In this case, the numerator is
Using the quadratic formula,
the roots are
Thus,