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# Rational Functions

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```RATIONAL
FUNCTIONS
A rational function is a function of the form:
where p and q
are polynomials
What would the domain of a rational
function be?
WeвЂ™d need to make sure the
denominator п‚№ 0
5x
2
3пЂ« x
пЃ»x пѓЋ пѓ‚ : x п‚№
Find the domain.
xпЂ­3
x пЂ­1
x пЂ« 5x пЂ« 4
2
пЃ»x пѓЋ пѓ‚ : x п‚№
пЂ­ 3пЃЅ
пЂ­ 2 , x п‚№ 2пЃЅ
If you canвЂ™t see it in your
head, set the denominator = 0
and factor to find вЂњillegalвЂќ
values.
пЃ»x пѓЋ пѓ‚ : x п‚№
пЂ­ 4 , x п‚№ пЂ­ 1пЃЅ
The graph of
1
x
2
looks like this:
If you choose x values close to 0, the graph gets
close to the asymptote, but never touches it.
Since x п‚№ 0, the graph approaches 0 but never crosses or
touches 0. A vertical line drawn at x = 0 is called a vertical
asymptote. It is a sketching aid to figure out the graph of
a rational function. There will be a vertical asymptote at x
values that make the denominator = 0
LetвЂ™s consider the graph f пЂЁ x пЂ© пЂЅ
1
x
We recognize this function as the reciprocal function
from our вЂњlibraryвЂќ of functions.
Can you see the vertical asymptote?
LetвЂ™s see why the graph looks
like it does near 0 by putting in
some numbers close to 0.
The closer to 0 you get
1
пѓ¦ 1 пѓ¶
fпѓ§
пЂЅ 10
пѓ·пЂЅ
for x (from positive
1
пѓЁ 10 пѓё
direction), the larger the
10
function value will be
Try some negatives
пѓ¦ 1 пѓ¶
fпѓ§
пѓ·пЂЅ
пѓЁ 100 пѓё
1
1
100
пЂЅ 100
1
пѓ¦ 1 пѓ¶
f пѓ§пЂ­
пЂЅ
пЂЅ пЂ­ 10
пѓ·
пѓЁ 10 пѓё пЂ­ 1
10
1 пѓ¶
1
пѓ¦
f пѓ§пЂ­
пЂЅ
пЂЅ пЂ­ 100
пѓ·
пѓЁ 100 пѓё пЂ­ 1
100
Does the function f пЂЁ x пЂ© пЂЅ
1
have an x intercept? 0 п‚№ 1
x
x
There is NOT a value that you can plug in for x that
would make the function = 0. The graph approaches
but never crosses the horizontal line y = 0. This is
called a horizontal asymptote.
A graph will NEVER cross a
vertical asymptote because the
x value is вЂњillegalвЂќ (would make
the denominator 0)
A graph may cross a horizontal
asymptote near the middle of
the graph but will approach it
when you move to the far right
or left
Graph Q пЂЁ x пЂ© пЂЅ 3 пЂ«
1
x
пЂЅ
1
x
пЂ«3
vertical translation,
moved up 3
This is just the reciprocal function transformed. We can
trade the terms places to make it easier to see this.
Q пЂЁx пЂ© пЂЅ 3 пЂ«
1
x
1
x
The vertical asymptote
remains the same because in
either function, x в‰ 0
The horizontal asymptote
will move up 3 like the graph
does.
VERTICAL ASYMPTOTES
Finding Asymptotes
There will be a vertical asymptote at any
вЂњillegalвЂќ x value, so anywhere that would make
the denominator = 0
x пЂ« 2x пЂ« 5
2
LetвЂ™s set the bottom = 0
and factor and solve to
find where the vertical
asymptote(s) should be.
So there are vertical
asymptotes at x = 4
and x = -1.
HORIZONTAL ASYMPTOTES
We compare the degrees of the polynomial in the
numerator and the polynomial in the denominator to tell
1<2
degree of top = 1
If the degree of the numerator is
less than the degree of the
1
2x пЂ« 5
denominator, (remember
the x axis isdegree
a
is the highest
horizontal
asymptote.
power onThis
any is
x
x пЂ­ 3x пЂ« 4
term) the
along
the xline
axis
y=
is0.
a horizontal
asymptote.
degree of bottom = 2
HORIZONTAL ASYMPTOTES
is the number in front of
the highest powered x
term.
degree of top = 2
If the degree of the numerator is
equal to the degree of the
denominator, then there is a
horizontal asymptote at:
y = leading coefficient of top
2x пЂ« 4x пЂ« 5
2
1 x2 пЂ­ 3x пЂ« 4
degree of bottom = 2
horizontal asymptote at:
yпЂЅ
2
1
пЂЅ2
OBLIQUE ASYMPTOTES
If the degree of the numerator is
greater than the degree of the
denominator, then there is not a
horizontal asymptote, but an
degree of top = 3
oblique one. The equation is
found by doing long division and
3
2
x пЂ« 2 x пЂ­ 3x пЂ« 5
the quotient is the equation of
2
the oblique asymptote ignoring
x пЂ­ 3x пЂ« 4
the remainder.
degree of bottom = 2
x пЂ« 5 пЂ« a remainder
x пЂ­ 3x пЂ­ 4 x пЂ« 2 x пЂ­ 3x пЂ« 5
2
3
2
Oblique asymptote
at y = x + 5
SUMMARY OF HOW TO FIND ASYMPTOTES
Vertical Asymptotes are the values that are NOT in the
domain. To find them, set the denominator = 0 and solve.
To determine horizontal or oblique asymptotes, compare
the degrees of the numerator and denominator.
1. If the degree of the top < the bottom, horizontal
asymptote along the x axis (y = 0)
2. If the degree of the top = bottom, horizontal asymptote
of bottom
3. If the degree of the top > the bottom, oblique
asymptote found by long division.
Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
St StephenвЂ™s School вЂ“ Carramar
www.ststephens.wa.edu.au
```
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