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9.3 – Rational Function and Their Graphs

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9.3 – Rational Function and
Their Graphs
Review: STEPS for GRAPHING
HOLES
Discontinuous part of the graph where the line jumps over.
___________________________________________
Represented by a little open circle.
___________________________________________
y пЂЅ
(x пЂ­ 3)
( x пЂ­ 3 )( x пЂ« 5 )
Hole @ x = 3
EX _________________________________________
y пЂЅ
EX
x( x пЂ­ 2)
Hole @ x = 2
No hole at x = 0
x (x пЂ­ 2)
_________________________________________
2
Review: STEPS for GRAPHING
VERTICAL ASYMPTOTES
Discontinuous part of the graph where the line cannot cross over.
___________________________________________
Represented by a dotted line called an asymptote.
___________________________________________
y пЂЅ
(x пЂ« 5)
(x пЂ­ 2)
VA @ x = 2
EX _________________________________________
y пЂЅ
EX
x
Hole @ x =0
x ( x пЂ­ 2 )( x пЂ« 5 )
VA @ x = 2, -5
_________________________________________
Review: STEPS for GRAPHING
HORIZONTAL ASYMPTOTES
n = degree of numerator
d = degree of denominator
5x пЂ« 7
2
y пЂЅ
No HA
(
x
пЂ­
2
)
_______________________________________________
Case 1 n > d
y пЂЅ
xпЂ«3
HA @ y = 0
3
x
пЂ­
1
_______________________________________________
Case 2 n < d
4x пЂ« 1
2
Case 1 n = d y пЂЅ
HA is the ratio of coefficients
HA @ y = 4 / 5
5 ( x пЂ­ 2 )( x пЂ« 2 )
_______________________________________________
Finding holes and asymptotes
VA: x=-1, -5
HA: y=0 (power of the denominator
is greater than the numerator)
Holes: none
VA: none (graph is the same as
y=x-1 once the (x-2)s cancel
HA: none (degree of the numerator
is greater than the denominator)
Hole: x=2
Let’s try some
Find the vertical, horizontal asymptotes and any holes
VA: x=3
HA: none (power of the numerator is
greater than the denominator)
Holes: x=2
VA: x=-5,0 ( cancel the (x-3)s
HA: y=0 (degree of the denominator
is greater than the numerator)
Hole: x=3
GRAPHING
y = x / (x – 3)
1) HOLES?
no holes since nothing cancels
2) VERTICAL ASYMPTOTES?
Yes ! VA @ x =3
3) HORIZONTAL ASYMPTOTES?
Yes ! HA @ y =1
4)
T-CHART
X
Y = x/(x – 3)
4
Y=4
2
Y = -2
0
Y=0
5
Y = 5/2
GRAPHING
1) HOLES?
hole @ x = -1
2) VERTICAL ASYMPTOTES?
None!
3) HORIZONTAL ASYMPTOTES?
None!
4) The graph -
What cancels?
Graph the function
y=x with a hole
at x=-1
GRAPHING
y пЂЅ
1) HOLES?
x( x пЂ­ 2)
x ( x пЂ« 2 )( x пЂ­ 5 )
y пЂЅ
(x пЂ­ 2)
( x пЂ« 2 )( x пЂ­ 5 )
hole @ x = 0
2) VERTICAL ASYMPTOTES?
Yes ! VA @ x =-2 , 5
3) HORIZONTAL ASYMPTOTES?
Yes ! HA @ y =0 (Power of the denominator
is greater than the numerator)
4)
T-CHART
X
y пЂЅ
(x пЂ­ 2)
( x пЂ« 2 )( x пЂ­ 5 )
6
Y = 1/2
-3
Y = -5/8
1
Y = 1/12
2
3
Y=0
Y = -1 / 10
WAIT –
What
about the
Horizontal
Asymptote
here?
Remember,
Horizontal
Asymptotes only
describe the ends of
the function (left and
right). What happens
in the middle is �fair
game’.
T-CHART
X
To find out what the graph looks like between the
vertical asymptotes, go to a T Chart and plug in
values close to the asymptotes.
y пЂЅ
(x пЂ­ 2)
( x пЂ« 2 )( x пЂ­ 5 )
Left
-1
Y = 1/2
Right
4
Y = -1/3
Middle
2
Y=0
Let’s try one:
Sketch the Graph
1) HOLES?
none
2) VERTICAL ASYMPTOTES?
Yes ! VA @ x = 1
3) HORIZONTAL ASYMPTOTES?
Yes ! HA @ y =0 (Power of the denominator
is greater than the numerator)
4)
T-CHART
X
0
Y=0
-1
Y = 1/4
-2
Y = .22
2
3
Y=-2
Y = -3/4
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