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

垂直渐近线的零分母项inator, sox=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 havey=1as 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可以通过评估发现的0分母项inator 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 lineis the only vertical asymptote of the graph of.

The vertical asymptote(s) of the graph of a rational function such as可以通过评估发现的0分母项inator.

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, sois an undefined quantity. Consequently, the graph ofhas no-intercept.

The vertical asymptote(s) of the graph of a rational function such as可以通过评估发现的0分母项inatorafter 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:

重写

as

The expression can be reduced by cancellingin 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可以通过评估发现的0分母项inatorafter 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 isand whose product is; through trial and error, we findand 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-intercept(s) of the graph ofare the point(s) at which it intersects the-axis. The-coordinate of each is 0; their-coordinate(s) are those value(s) offor 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, settingin the definition, we see that

由于零denomina,一个未定义的表达式tor. 5 cannot be eliminated similarly. Therefore,is the only-intercept.