All Precalculus Resources
Example Question #1 : Rational Functions
Determine the equation(s) of the vertical asymptote(s) of the following rational equation:
There are no vertical asymptotes for this equation.
We recall that a vertical asymptote of a rational function is found whenever the factors of the denominator equal 0, provided that there are no cancelling factors in the numerator. We have to factor both the numerator and the denominator.
cancels, so there is a hole in our graph at .
The remaining factor in the denominator can be set equal to 0 to find the asymptote:
Example Question #2 : Rational Functions
Identify the equation(s) of the hole or holes (also called removable discontinuities) in the following rational equation:
There are no holes/removable discontinuities in this equation.
In order to have a hole/removable discontinuity, the numerator and the denominator must share a factor - and, that factor must not appear a second time in the denominator. If it does appear a second time in the denominator, then the zero for that factor will still create a vertical asymptote, not a hole.
To find out what will happen with this equation, we factor the top and bottom completely:
Let's compare factors now:
The removable discontinuity must be at .
Example Question #3 : Rational Functions
Determine the equations of all asymptotes of the following rational equation:
Vertical asymptote: None
First, in order to find vertical asymptotes, we factor the top and bottom and look for factors in the denominator which do not cancel with factors in the numerator.
cancels, leaving only in the denominator.
Therefore, there is a vertical asymptote at .
This leaves us to look for the horizontal/slant asymptote. We know that it will be a slant asymptote because the numerator is of a higher degree than the denominator. To find a slant asymptote, we divide the factors which did not cancel and consider the quotient.
We can disregard the remainder, so there is no need to do fully long devision here. We are left with a slant asymptote of .
Example Question #1 : Polynomial Functions
Find the vertical asymptote of the function
Vertical asymptotes occur when the function values increase without bound as the input approaches a certain value. In other words for some value of x, which is the line describing the vertical asymptote.
Now, anytime a denominator is 0, the value of a fraction or rational function is undefined. This is where .
The denominator in this function is
So, solve for x in
Example Question #5 : Rational Functions
Find the oblique, or slant asymptote of
When the degree of the numerator is greater than the degree of the denominator by a factor of 1, then a function will have an oblique asymptote. The numerator increases faster than the numerator by a factor of x, so there will be a line with a slope that the graph will approach but not intersect.
The asymptote can be found by performing polynomial or synthetic division, and ignoring the remainder.
Synthetic division of gives
3_| 2 4 3
2 10 |_33
The quotient witout the remainder is the asymptote
Example Question #2 : Polynomial Functions
Determine the vertical asymptotes of the following equation:
There are no vertical asymptotes
First, start by isolating x and y on seperate sides to get the equation into the form of y=x. Do this by adding 4x to both sides.
Next, factor out a y on the right side
Divide both sides by to get in the for of y=x
Now that the equation is in the y=x form, we can set the denominator equal to 0 and solve for x
Example Question #3 : Polynomial Functions
Find the horizontal asymptote of
A horizontal asymptote occurs when y approaches a certain value as x goes to infinity or negative infinity. This happens in a rational expression when the degrees of the numerator and denominator are the same. The independent variable of both terms increases or decreases at the same rate, while the terms themselves change according to their leading coefficients. This is to say that there is a line defined by the ratio of leading coefficients that the graph will approach but not inetrsect as x increases or decreases indefinely.
The leading coeffiecint of the numerator is 3, while the leading coefficient of the denominator is 2. So the asymptote is the line
Example Question #8 : Polynomial Functions
What is the horizontal asymptote for the function below?
Since the leading terms of the numerator and the denominator have the same degree, the ratio will move towards a constant. The ratio of the leading coefficients is 1/1, so the function will taper off to y = 1 in both the positive and negative directions for x.
Example Question #8 : Rational Functions
Given the equation , determine the horizontal asymptote.
Change the equation into the y=x form.
When the degree of the exponent in the numerator is higher than the degree of the exponent in the denominator, there is no horizontal asymptote. Thus the answer is no horizontal asymptote.
Example Question #9 : Rational Functions
At what value(s) of are the vertical asymptote(s) of the function located?
Asymptotes can occur where the function is undefined, or when the denominator is equal to . Simplfying our function gives us,
Now solving for when our denominator is zero gives us,
Therefore, there is a vertical asymptote at