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Rational Functions

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RATIONAL
FUNCTIONS
A rational function is a function of the form:
R пЂЁx пЂ© пЂЅ
pпЂЁx пЂ©
qпЂЁx пЂ©
where p and q
are polynomials
What would the domain of a rational
function be?
R пЂЁx пЂ© пЂЅ
We’d need to make sure the
denominator п‚№ 0
R пЂЁx пЂ© пЂЅ
H пЂЁx пЂ© пЂЅ
F пЂЁx пЂ© пЂЅ
5x
2
3пЂ« x
пЃ»x пѓЋ пѓ‚ : x п‚№
Find the domain.
xпЂ­3
пЂЁ x пЂ« 2 пЂ©пЂЁ x пЂ­ 2 пЂ©
x пЂ­1
x пЂ« 5x пЂ« 4
2
пЂЁ x пЂ« 4 пЂ©пЂЁ x пЂ« 1 пЂ© пЂЅ 0
пЃ»x пѓЋ пѓ‚ : x п‚№
pпЂЁx пЂ©
qпЂЁx пЂ©
пЂ­ 3пЃЅ
пЂ­ 2 , x п‚№ 2пЃЅ
If you can’t see it in your
head, set the denominator = 0
and factor to find “illegal”
values.
пЃ»x пѓЋ пѓ‚ : x п‚№
пЂ­ 4 , x п‚№ пЂ­ 1пЃЅ
The graph of
f пЂЁx пЂ© пЂЅ
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
Let’s consider the graph f  x  
1
x
We recognize this function as the reciprocal function
from our “library” of functions.
Can you see the vertical asymptote?
Let’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 “illegal” (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«
f пЂЁx пЂ© пЂЅ
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
“illegal” x value, so anywhere that would make
the denominator = 0
x пЂ« 2x пЂ« 5
2
R пЂЁx пЂ© пЂЅ 2
пЂЁ x xпЂ­ 4пЂ­пЂ©пЂЁ3xxпЂ«пЂ­1 пЂ©4пЂЅ 0
Let’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
us about horizontal asymptotes.
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
R пЂЁx пЂ© пЂЅ 2
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
The leading coefficient
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
R пЂЁx пЂ© пЂЅ
2x пЂ« 4x пЂ« 5
2
1 x2 пЂ­ 3x пЂ« 4
degree of bottom = 2
horizontal asymptote at:
yпЂЅ
2
1
пЂЅ2
leading coefficient of bottom
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
R пЂЁx пЂ© пЂЅ
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
at y = leading coefficient of top over leading coefficient
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
Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au
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