We're given a rational function and asked to determine the y intercept,
x intercepts, vertical asymptotes, and horizontal asymptotes. When working with rational functions,
it's often helpful to have the numerator and denominator in factored form. So let's start by
factoring both the top and bottom. Looking at the numerator, the factors of two x squared are two x
and x. The second positions must come from the factors of negative five so that the sum
of the inner product and outer product is equal to nine x. If we use a factor of negative five here,
negative five would give us negative ten x. The other factor of five is positive one. So
negative ten x plus one x is equal to negative nine x. So that's factored correctly.
Now for the denominator, we'd have a factor of two x and x here. Now we use the factors of
negative three in the second positions. So the sum of the inner and outer product is equal to
positive five x. If we use a factor of positive three here and negative one here,
we have positive six x plus negative one x is equal to five x.
Now we'll start by finding the y intercept. To find the y intercepts of any function, we set
x equal to zero. And because this is written in function notation, that means we need to evaluate
f of zero. So whether we use the original form or the factored form, it doesn't matter. If we
use the original form, if x is zero, we would have zero minus zero minus five or negative
five over zero plus zero minus three or negative three. This result would be the same if we use the
factored form as well. This simplifies to positive five over three, which means our y intercept is
the point with an x coordinate of zero and a y coordinate of positive five over three.
To find the x intercepts of a function, we set y equal to zero or the function value equal to zero.
But because we have a rational function, as long as there aren't any common factors between the
numerator and denominator, this function will only equal zero when the numerator is equal to
zero. To show why this is true, we'll set f of x equal to zero and then solve for x. So we would
have zero equals in factored form the quantity two x plus one times the quantity x minus five
all over the quantity two x minus one times the quantity x plus three. So
because this is a fraction, if we put zero over one, it's still equal to zero.
Now we can solve by cross multiplying, but notice when we cross multiply,
we would have one times the numerator is equal to zero times the denominator,
which would just give us the equation that the numerator equals zero.
So as long as there aren't any common factors between the numerator and denominator,
the x intercepts can be found by setting the numerator equal to zero. Two x plus one
equals zero when x equals negative one half and x minus five equals zero when x equals five. So
we have two x intercepts, they are the points negative one half comma zero and five comma zero.
Now to determine the vertical asymptotes of a rational function, again, as long as there
aren't any common factors between the numerator and denominator, the values of x that make the
denominator equal to zero give us the equations of the vertical asymptotes. So we can find the
vertical asymptotes by setting the quantity two x minus one times the quantity x plus three equal
to zero. If there are common factors between the numerator and denominator, the zeros of
the common factors give us holes, not x intercepts or vertical asymptotes. So two x minus one equals
zero when x equals one half and x plus three equals zero when x equals negative three. Remember
vertical asymptotes are vertical lines,
so the equations of the vertical asymptotes are x equals one half and x equals negative three.
Now for the last question, we're asked to determine the horizontal asymptotes, which is
really asking us to determine the end behavior or long run behavior of a rational function.
So there's a nice shortcut to determining if the function has horizontal asymptotes,
and if it does, what they are. We can determine the horizontal asymptotes by determining the
degree of the numerator and denominator. Notice the numerator is degree two and the denominator
is also degree two. The rule is if the degree of the numerator and denominator are the same,
the equation of the horizontal asymptote is y equals the ratio of the leading coefficients,
which in this case would be two over two. So if we have y equals two over two,
that's equal to one. So the equation of the horizontal asymptote is y equals one.
Remember it's a horizontal line, so it's in the form of y equals a constant.
And just to review the other possibilities when determining a horizontal asymptote:
if the degree of the numerator is less than the degree of the denominator,
then the horizontal asymptote is y equals zero. If the degrees are equal, as in our case,
the horizontal asymptote is y equals the ratio of the leading coefficients. The third case is
if the degree of the numerator is greater than the degree of the denominator, then there is
no horizontal asymptote. So this comes in very handy when determining horizontal asymptotes.
As a last step, let's go ahead and graph the function and check our work. This might be
a little hard to see, but we do have a y intercept of five over three, an
x intercept of five and negative one half. We have two vertical asymptotes,
one at x equals one half and one at x equals negative three. We have a horizontal asymptote of
y equals one. So this graph does verify that our work is correct. I hope you found this helpful.