what's up guys in this video what I want
to do is go over three different
examples of rational expressions that
don't have a vertical asymptote that a
lot of students always think or confuse
that they do actually have a vertical
asymptote so let's just kind of go
through a quick little review of why we
have vertical asymptotes and then what
are these kind of mistakes that you know
students usually make when they look at
an expression and think oh there's going
to be a vertical asymptote here so just
a real quick like you know if I had like
one over X remember we can't divide by
zero right so X in this case cannot
equal zero that is the value like zero
is the value that makes my denominator
equal to zero so therefore I'm going to
have an undefined value and what that's
going to do is that's actually going to
create a vertical asymptote right so
this graph is going to look something
like this it approaches a0 from left
hand and right
now sometimes my denominator gets a
little bit more crazy right so if I have
like one over you know an X let's say a
minus one all that simply does is just
shift the graph over one unit to the
right right because now X cannot equal
now it's not X cannot equal zero because
you can plug in zero right zero minus
one is negative one one divided by
negative 1 Works however now X cannot
equal one so what happens here is my
vertical asymptote gets shifted over but
it still is existing here right it's
just X can now not equal a one so at x
equals one is we now have this new
vertical asymptote where the function is
going to be undefined and that's the
main thing I want you to take away from
this
when we're trying to find the vertical
asymptotes one thing we're looking for
is when the function is not going to be
defined however a function cannot be
defined and it not be a vertical
asymptote and if you're wondering what
I'm talking about I'm good that's
exactly what I'm going to explain in
this video but let's go and take a look
at the first example and a very common
one where students a lot of times think
there's a vertical asymptote just
because it is a
rational expression
all right and then me expression like
this so here I have y equals 3x squared
plus 6X plus 9 all divided by 3. all
right and I think a lot of times once we
you know kind of get depth and like kind
of get deep into teaching with rational
expressions seeing fractions students
just kind of get so happy and say
you know since I have something in the
denominator like that's going to be my
vertical asymptote right so let's say
the vertical asymptote is going to be an
x equals A3 and no right no please do
not do this okay the important thing
guys is the vertical asymptote is like
for what values of X right make the
denominator equal to zero we just have a
number in the denominator here right
which is three now again like what's
happening with this three is there's a
couple ways we could write this you
could rewrite dividing by 3 is the same
thing as multiplying by a 3X all right
we're sorry multiplying by one-third so
plus a 6X Plus 9. or you could also
distribute this 3 into each and every
one of these terms
right because just by applying division
here into each and every one of these
terms you can actually simplify this now
I don't care what which way you really
want to do that but what I want you to
understand is when you do like simplify
this expression this rational expression
what you're actually going to get is an
x squared plus a 2X
plus 3 which guess what is a quadratic
right so and hopefully you recognize
like a quadratic equation is going to
have that Parabola approach now I don't
know exactly what this graph is going to
look like it doesn't look like it's
going to be anything fun but it's going
to look something like this where
there's no vertical asymptote right
there's no discontinuities no matter
what value you plug in for X it's always
going to be defined
when we're looking for vertical
asymptotes we're looking for values that
when you plug them
when we're looking for vertical systems
we're looking for values that make the
denominator equal to zero this
denominator is never going to be zero so
whenever you have just a number in the
denominator you don't have to worry
about your vertical asymptotes all right
now let's go and take a look at a
another example here
and in this example I have y is equal to
a x squared minus 6X minus 7. all over a
to X Plus 2.
okay
um now in this case you can definitely
say all right we have some numbers in
the denomer
um we have some values in the
denominator that uh that we have some
numbers in the denominator that actually
could make sorry we have some values for
X that can make my denominator equal to
zero so typically what I always tell
students to do is like well
to find the vertical asymptotes just set
your denominator equal to zero right and
then go ahead and solve so you
subtracted two on both sides
2x is equal to a negative 2 divided by 2
divided by 2 x equals a negative 1. you
say vertical asymptote is at x equals
negative one and guess what it's not now
how do we know this well one thing you
should always do before you find the
vertical asymptotes is simplify right
that's what we did up here we simplified
this rational expression by either
dividing by 3 or multiplying by the
one-third right
and that's exactly what we want to do in
this example we want to factor this like
we have a quadratic and what we simply
want to do then is typically when you
have a quadratic look to factor it
um so I can Factor this as a product of
two binomials now what I'm looking at
here is I could probably Factor this as
to x minus say 7 times an X plus one
right what two numbers multiply to give
you negative seven add to give you
negative 6. I think that is going to
work out and then over here I could
factor out a 2 and I'd be left with a X
Plus 1.
now What's Happening Here what I want
you to see is the X Plus ones divide out
so the simplified version of this is
again we have this x minus 7 divided by
2 right so you could write it like that
or you could write it like this doesn't
really matter but what I want you to see
is this X Plus 1 actually divided out
now what that means is that is now a
what we call a removable discontinuity
not a non-rulable discontinuity so when
you can get something divide out it
doesn't mean it goes away like it's
still a part of the answer like negative
when x equals negative 1 is still makes
this function not defined
however the it is not going to when you
look at this like graphically it's not
an asymptote anymore what this is is
actually what we call a hole so our hole
is going to be is going to take your
removal discontinuity and then set it
equal to zero which in this case is x
equal to negative one so what we
actually have here is we have a rational
expression we're at negative I'm sorry a
linear equation we're at x equals
negative one we actually have a whole so
you know again I'm not trying to graph
this correctly I'm just trying to show
you what the graph would look like but
there's no vertical asymptotes here it's
actually going to be in the linear
equation with a hole at x equals
negative one
all right let's go and take a look at
one more example of a rational function
that does not have vertical asymptotes
so in this case we have y equals let's
say a 4x squared plus a 7X
plus 3 divided by a 4x squared plus a
16. okay
um this one looks uh pretty interesting
here in this case now
one thing I'm always going to recommend
like we could go ahead and try to factor
this one out you know if you want to
um you know kind of look into like if
anything's going to factor out in the
numerator denominator but I kind of
already noticed something that is
um always going to be an issue like in
this case and again like I'm just trying
to show why it does not have a vertical
asymptote or what would be a possibility
here or why it doesn't have a vertical
vertical asymptote remember vertical
asymptotes are the values that make your
denominator equal to zero right so if
we're trying to find these values that
are making equal zero one you got to
have a variable in the denominator right
number two you got to make sure it's not
a whole and number three you got to make
sure there's actually values that are
real numbers so in this case like this
might seem line let's say you're like
hey you know what I don't really want to
be factoring something out
um but I'm not sure if this has a
vertical asymptote because sometimes you
can do the math in your head sometimes
you maybe need to work it out
but another possibility I want you to
look at is what about when you cannot
what about when there's no real numbers
that make your denominator equal to zero
so my point in this case is let's go
ahead and solve let's set your
denominator equals zero so 4x squared
equals a negative 16.
divide by 4 on both sides
x squared equals a negative 4. now when
you introduce the square root
hopefully recognize remember when you
introduce the square root
of a negative number that's going to
like that's not going to be a part of
your real number system right so X is
going to equal a plus or minus I
times not two that's going to be plus or
minus 2i
right so now we need to use the complex
number system to be able to take the
even root of a negative number right so
again like if you're thinking about this
as a graphical approach
like we can't graph these on the real
number line a plus or minus a 2i so a
lot of times what we just say is no
vertical asymptotes right because we
have to have a real number for us to
have a vertical asymptote so there you
go ladies and gentlemen that is three
examples for vertical asym or for
rational expressions that rational
equations with no vertical asymptotes I
hope it was helpful and if it was I look
forward to seeing you in the next video