Rational Functions - College Algebra

The vertical asymptotes of a rational function are always the zeroes of the polynomial in the denominator. So to solve this problem all we need to do is find the zeroes of .

We can use the quadratic formula, , for a polynomial . In this case, a=4, b=-8, and c=-5. So we must plug these values into the quadratic formula:

And we get the two roots, .

The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced.

First, factor the numerator and denominator.

The numerator is a perfect square trinomial and can be factored as such:

The denominator can be factored as the difference of squares:

Rewrite

as

The expression can be reduced by cancelling in both halves:

Set the denominator equal to 0 and solve:

The only vertical asymptote is therefore the line of the equation .

The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced. The expression is in simplest form, so set the denominator equal to 0 and solve for :

The graph of the line is the only vertical asymptote of the graph of .

The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced.

First, factor the numerator. It is a quadratic trinomial with lead term , so look to "reverse-FOIL" it as

We seek two integers whose sum is and whose product is ; through trial and error, we find and 2, so

Therefore, can be rewritten as

Cancelling , this can be seen to be essentially a polynomial function:

which does not have a vertical asymptote.

The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator.

Set the denominator equal to 0 and solve for :

The only vertical asymptote of the graph is the line of the equation .

The -intercept of the graph of a function is the point at which it intersects the -axis. The -coordinate is 0, so the -coordinate can be found by evaluating . This is done by substitution, as follows:

The value of the denominator here is 0, so is an undefined quantity. Consequently, the graph of has no -intercept.

The -intercept(s) of the graph of are the point(s) at which it intersects the -axis. The -coordinate of each is 0; their -coordinate(s) are those value(s) of for which , so set up, and solve for , the equation:

A fraction is equal to 0 if and only if the numerator is equal to 0, so set

Factor out :

By the Zero Product Property, one of the factors must be equal to 0, so either

or

in which case

.

However, setting in the definition, we see that

, an undefined expression due to the zero denominator. 5 cannot be eliminated similarly. Therefore, is the only -intercept.

To find vertical asymptotes, factor each quadratic and then simplify.

The vertical asymptotes are the zeros of the denominator, so x=3 is a vertical asymptote.

Horizontal asymptotes are found by analyzing the coefficients of the first term in each equation. The line (where a is the coefficient of the first term in the numerator and b is the coefficient of the first term in the denominator) is the horizontal asymptote. So we have y=1 as the horizontal asymptote.

The vertical asymptotes of a rational function are always the zeroes of the polynomial in the denominator. So to solve this problem all we need to do is find the zeroes of .

We can use the quadratic formula, , for a polynomial . In this case, a=4, b=-8, and c=-5. So we must plug these values into the quadratic formula:

And we get the two roots, .

The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced.

First, factor the numerator and denominator.

The numerator is a perfect square trinomial and can be factored as such:

The denominator can be factored as the difference of squares:

Rewrite

as

The expression can be reduced by cancelling in both halves:

Set the denominator equal to 0 and solve:

The only vertical asymptote is therefore the line of the equation .

The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced. The expression is in simplest form, so set the denominator equal to 0 and solve for :

The graph of the line is the only vertical asymptote of the graph of .

The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced.

First, factor the numerator. It is a quadratic trinomial with lead term , so look to "reverse-FOIL" it as

We seek two integers whose sum is and whose product is ; through trial and error, we find and 2, so

Therefore, can be rewritten as

Cancelling , this can be seen to be essentially a polynomial function:

which does not have a vertical asymptote.

The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator.

Set the denominator equal to 0 and solve for :

The only vertical asymptote of the graph is the line of the equation .

The -intercept of the graph of a function is the point at which it intersects the -axis. The -coordinate is 0, so the -coordinate can be found by evaluating . This is done by substitution, as follows:

The value of the denominator here is 0, so is an undefined quantity. Consequently, the graph of has no -intercept.

The -intercept(s) of the graph of are the point(s) at which it intersects the -axis. The -coordinate of each is 0; their -coordinate(s) are those value(s) of for which , so set up, and solve for , the equation:

A fraction is equal to 0 if and only if the numerator is equal to 0, so set

Factor out :

By the Zero Product Property, one of the factors must be equal to 0, so either

or

in which case

.

However, setting in the definition, we see that

, an undefined expression due to the zero denominator. 5 cannot be eliminated similarly. Therefore, is the only -intercept.

To find vertical asymptotes, factor each quadratic and then simplify.

The vertical asymptotes are the zeros of the denominator, so x=3 is a vertical asymptote.

Horizontal asymptotes are found by analyzing the coefficients of the first term in each equation. The line (where a is the coefficient of the first term in the numerator and b is the coefficient of the first term in the denominator) is the horizontal asymptote. So we have y=1 as the horizontal asymptote.

The vertical asymptotes of a rational function are always the zeroes of the polynomial in the denominator. So to solve this problem all we need to do is find the zeroes of .

We can use the quadratic formula, , for a polynomial . In this case, a=4, b=-8, and c=-5. So we must plug these values into the quadratic formula:

And we get the two roots, .

The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced.

First, factor the numerator and denominator.

The numerator is a perfect square trinomial and can be factored as such:

The denominator can be factored as the difference of squares:

Rewrite

as

The expression can be reduced by cancelling in both halves:

Set the denominator equal to 0 and solve:

The only vertical asymptote is therefore the line of the equation .

The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced. The expression is in simplest form, so set the denominator equal to 0 and solve for :

The graph of the line is the only vertical asymptote of the graph of .