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Graphs of Rational Functions

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Graphs of Rational Functions
Rational Function
пЃ®
A function of the form
f ( x) пЂЅ
p( x)
q( x)
where p(x) & q(x) are
polynomials and q(x)в‰ 0.
Hyperbola
f ( x) пЂЅ
1
x
A type of rational function.
пЃ® Has 1 vertical asymptote and
1 horizontal asymptote.
пЃ® Has 2 parts called branches.
(blue parts) They are
symmetrical.
We’ll discuss 2 different forms.
x=0
y=0
Hyperbola
yпЂЅ
a
пЂ«k
пЃ®
One form:
пЃ®
Has 2 asymptotes: x=h (vert.) and y=k (horiz.)
пЃ®
Graph 2 points on either side of the vertical
asymptote.
пЃ®
Draw the branches.
xпЂ­h
yпЂЅ
3
пЂ«2
Ex: Graph
x пЂ­1
State the domain & range.
Vertical Asymptote: x=1
Horizontal Asymptote: y=2
x
y
-5 1.5
-2
1
2
5
4
3
Left of
vert.
asymp.
Right of
vert.
asymp.
Domain: (-в€ћ, 1) U (1, в€ћ)
Range: (-в€ћ, 2) U (2, в€ћ)
Hyperbola (continued)
ax пЂ« b
пЃ®
Second form:
пЃ®
Vertical asymptote: Set the denominator equal to
0 and solve for x.
пЃ®
Horizontal asymptote:
yпЂЅ
cx пЂ« d
y пЂЅ
a
c
пЃ®
Graph 2 points on either side of the vertical
asymptote. Draw the 2 branches.
yпЂЅ
xпЂ­2
Ex: Graph
3x пЂ« 3
State domain & range.
Vertical asymptote:
3x+3=0 (set denominator =0)
3x=-3
x= -1
x
Horizontal Asymptote:
-3
y пЂЅ
a
c
y пЂЅ
1
3
y
.83
-2 1.33
0
-.67
2
0
Domain:
(-в€ћ, -1)U(-1, в€ћ)
Range:
(-в€ћ, 1/3)U(1/3, в€ћ)
We have graphed rational functions
where x was to the first power only.
What if x is not to the first power?
Such as:
f ( x) пЂЅ
x
x пЂ«1
2
Steps to graph when x is not to the 1st power
1.
2.
3.
4.
Find y-intercepts. (let x=0 and solve)
Find the x-intercepts. (Set numerator = 0 and solve)
Find vertical asymptote(s). (set denom=0 and solve)
Find horizontal asymptote. 3 cases:
a.
If degree of top < degree of bottom, y=0
lead. coeff. of top
y
пЂЅ
b.
If degrees are =, lead. coeff. of bottom
c.
If degree of top > degree of bottom, no horiz.
asymp, but there will be a slant asymptote.
5. Make a T-chart: choose x-values on either side &
between all vertical asymptotes.
6.Graph asymptotes, pts., and connect with curves.
7.Check solutions on calculator.
Ex: Graph. State domain & range.
yпЂЅ
x
x пЂ«1
2
y-intercepts: 0
2.
x-intercepts: x=0
3.
vert. asymp.: x2+1=0
x2= -1
No vert asymp
x пЂЅ пЂ­1
1.
(No real solns.)
3.
horiz. asymp:
1<2
(deg. of top < deg. of bottom)
y=0
x
y
-2
-.4
-1
-.5
0
0
1
.5
2
.4
Domain: (-в€ћ,в€ћ)
Range:
пѓ©пЂ­1 1пѓ№
пѓЄ 2 , 2пѓє
пѓ«
пѓ»
Ex: Graph, then state the domain and range.
yпЂЅ
1.
2.
2.
3.
3x
2
x пЂ­4
2
y-intercepts: 0
4. x y
x-intercepts:
3x2=0
4
4
On right of x=2
x2=0
asymp.
3
5.4
x=0
1
-1
Between the 2
Vert asymp:
0
0
asymp.
2
x -4=0
-1
-1
2
x =4
-3
5.4
On left of x=-2
x=2 & x=-2
-4
4
asymp.
Horiz asymp:
(degrees are =) y=3/1 or y=3
Domain: (- в€ћ, -2) U (-2, 2) U (2, в€ћ)
Range: (- в€ћ, 0] U (3, в€ћ)
Ex: Graph,
x state
пЂ­ 3 x пЂ­the
4
then
yпЂЅ
xпЂ­2
2
domain & range.
y-intercepts: -4/-2 = 2
2.
X-intercepts:
4. x
y
x2-3x-4=0
(x-4)(x+1)=0
-1
0
Left of x=2
x-4=0 x+1=0
0
2
asymp.
x=4 x=-1
1
6
2.
Vert asymp:
3
-4
Right of
x-2=0
x=2
x=2
asymp.
4
0
3.
Horiz asymp: 2>1
(deg. of top > deg. of bottom)
no horizontal asymptotes, but there is a slant!
1.
Slant asymptotes
x пЂ­ 3x пЂ­ 4
2
yпЂЅ
xпЂ­2
Do synthetic division (if possible); if not, do long
division!
пЃ® The resulting polynomial (ignoring the remainder)
is the equation of the slant asymptote.
In our example:
Ignore the remainder,
2 1 -3 -4
пЃ®
1
2
-2
-1
-6
use what is left for the
equation of the slant
asymptote: y=x-1
Domain: all real #’s except 2
Range: all real #’s
2
Ex: Graph, then state
x пЂ«the
x пЂ­ 6domain & range.
yпЂЅ
1.
2.
3.
4.
5.
6.
7.
8.
y-intercepts:
X-intercepts:
Vert asymp:
Horiz asymp:
Slant:
Domain
Range
Graph
x пЂ«1
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