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in this lesson we're going to talk about
how to find the X and the y-intercepts
of a rational function
let's begin by finding
the y-intercept first
in order to find the y-intercept
we need to set x equal to zero and we
need to solve for y
let's go ahead and do that
so we're going to have 8
minus 2 times 0 over 5.
minus 3 times 0.
now zero times anything is zero so we
simply get eight over five as our answer
so that's the y-intercept now we could
write the answer as an ordered pair
we can write it as zero comma eight over
five
so that's all you need to do in order to
find the y-intercept
now let's get the x-intercept
in order to find the x-intercept
y equal to zero and then solve for x
so we're going to have zero is equal to
eight minus two x over five minus 3x
now what I'm going to do is I'm going to
rewrite zero as a fraction 0 is
equivalent to 0 over 1.
now we can cross multiply so I have 1
times 8 minus 2x which is just 8 minus
2X
and that equals zero times five minus 3x
now zero times anything is simply zero
next we can add 2x to both sides so we
get 8 is equal to 2X
and then to isolate X we could divide
both sides by two
so X is eight over two
which is 4.
so the x-intercept is going to be 4
comma zero as an ordered pair
so that's how we could find the X and
the y-intercepts of a rational function
so remember in order to find the
x-intercept
y equal to zero in order to find
the y-intercept set x equal to zero and
solve for y
for the x-intercept after you set y
equal to zero then you can solve for x
now let's try another problem
let's say we have y is equal to 8
divided by x minus 3.
go ahead and find the X and the
y-intercepts for this particular
rational function
so let's start with the y-intercept
so let's set x equal to zero
this is going to be eight over zero
minus three
zero minus three is negative 3. so we
simply get negative eight over three
so we can write our answer as zero comma
negative eight over three
so that's the y-intercept
for this particular problem
now let's focus on
the x-intercept
so this time we're going to set y equal
to zero and we're going to solve for x
so zero is equal to eight over x minus
three
let's put this over one and cross
multiply
one times eight is eight zero times x
minus three is zero eight does not equal
zero
so when you get an unequal expression
like this it means that there is no
x-intercept there's none
so that's it for this problem
now let's move on to the next one
let's say we have y
is equal to x squared minus 7x Plus 12.
and this is going to be divided by
3x squared minus 48.
so go ahead and find the X and Y
intercepts for that particular rational
function
now before we start this problem
we should attempt to factor every
possible expression that we can
so let's go ahead and do that it'll make
the math a lot easier
so how can we Factor this trinomial
where the leading coefficient is one
what we need to do is find two numbers
that multiply to 12 but add to the
middle coefficient negative seven
four times three is twelve
and the same is true from negative 4
Negative 3. they multiply to positive
12. but they add to a negative seven so
to factor the numerator it's going to be
x minus 4.
times x minus 3.
now for the denominator the first thing
we need to do is take out the GCF which
is 3.
3x squared divided by 3 is x squared
negative 48 divided by 3 is negative 16.
now we have a difference of two squares
so to factor it it's going to be X plus
4 and x minus 4.
you take the square root of x squared
that'll give you X and then the square
root of 16 will give you 4.
1 will be positive the other will be
negative
so notice that we have a common factor
that we could cancel out which is x
minus 4.
so we're left with x minus 3 divided by
three
X Plus 4.
so now let's go ahead and find
the y-intercept
so we're going to set x equal to zero
and we're going to solve for y
so this is 0 minus 3 over 3
times 0 Plus 4.
zero plus four is four
we could cancel a three
and so we're left with negative one over
four
so we can write the y-intercept as zero
comma negative one over four
now let's focus on the x-intercept
so this time we're going to set y
equal to zero and we're going to find X
so zero is equal to x minus three
over three times X Plus 4.
just like before we're going to put this
over one and cross multiply
so we have one times x minus 3 which is
that and that's equal to zero times the
stuff on the bottom which will be zero
add in 3 to both sides we get that X is
equal to 3.
so we can write the x-intercept as three
comma zero
and so that's it for this problem so now
you know how to find the X and Y
intercepts of any rational function
pretty much you just need to follow the
procedure that you've learned in this
video and that should give you the
answer
