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Example Questions
Example Question #21 : Rational Functions
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
?
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.
Example Question #2 : Find Intercepts And Asymptotes
Find the -intercept and asymptote, if possible.
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:
Example Question #3 : Find Intercepts And Asymptotes
Find the -intercepts of the rational function
.
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.
Example Question #4 : Find Intercepts And Asymptotes
Find the vertical asymptotes of the following rational function.
No vertical asymptotes.
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 .
Example Question #5 : Find Intercepts And Asymptotes
Find the y-intercept and asymptote, respectively, of the following function, if possible.
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:
Example Question #6 : Find Intercepts And Asymptotes
What is a vertical asymptote of the following function?
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:
Example Question #7 : Find Intercepts And Asymptotes
What is the -intercept of the following function?
There is no -intercept.
The y-intercept of a function is always found by substituting in .
We can go through this process for our function.
Example Question #8 : Find Intercepts And Asymptotes
Which of these functions has a vertical asymptote of and a slant asymptote of
?
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
Example Question #9 : Find Intercepts And Asymptotes
Find the zeros and asymptotes for
.
Zero: ; Asymptotes:
Zero: ; Asymptote:
Zero: ; Asymptotes:
Zeros: ; Asymptotes:
Zeros: ; Asymptote:
Zero: ; Asymptote:
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.
Example Question #10 : Find Intercepts And Asymptotes
Find the slant and vertical asymptotes for the equation
.
Vertical asymptote: ; Slant asymptote:
Vertical asymptote: ; Slant asymptote:
Vertical asymptote: ; Slant asymptote:
Vertical asymptote: ; Slant asymptote:
Vertical asymptote: ; Slant asymptote:
Vertical asymptote: ; Slant asymptote:
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
.
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