1 Proportionality
Quantities x and y are inversely proportional only if growing x by the factor k (
k
>
1
k greater than 1 ) means shrinking y by the factor
1
k
.
1 over k , .
|
Inverse Variation (Lesson 8-1)
Are ℓ and w inversely proportional?
A
=
ℓ
w
eh equals script l w
P
=
2
ℓ
+
2
w
p equals 2 script l plus , 2 to the w
- for a constant area—yes
- for a constant perimeter—no
|
The Reciprocal Function Family (Lesson 8-2)
A
(
l
)
=
5
l
eh , open l close , equals , 5 over l is a stretch of the graph of
A
(
l
)
=
1
l
eh , open l close , equals , 1 over l by a factor of 5.
|
2 Function
A rational function may have no asymptotes, one horizontal or oblique asymptote, and any number of vertical asymptotes.
3 Equivalence
f
(
x
)
=
x
+
a
x
2
−
a
2
,
x
≠
±
a
,
f , open x close , equals . fraction x plus eh , over x squared , minus , eh squared end fraction . comma x not equal to plus minus eh comma and
g
(
x
)
=
1
x
−
a
,
x
≠
±
a
,
g , open x close , equals . fraction 1 , over x minus eh end fraction . comma x not equal to plus minus eh comma are equivalent. |
Rational Functions and Their Graphs (Lesson 8-3)
Asymptotes: For
y
=
2
x
2
x
2
−
9
y equals . fraction 2 , x squared , over x squared , minus 9 end fraction
horizontal:
y
=
2
y equals 2
Vertical:
x
=
±
3
x equals plus minus 3
For
y
=
2
x
3
+
6
x
2
x
2
+
1
y equals . fraction 2 , x cubed , plus 6 , x squared , over x squared , plus 1 end fraction
oblique:
y
=
2
x
+
6
.
y equals 2 x plus 6 .
y
=
x
4
+
5
x
2
+
1
y equals . fraction x to the fourth , plus 5 , over x squared , plus 1 end fraction has no asymptotes.
Image Long Description
|
Solving Equations Involving Rational Expressions (Lessons 8-4, 8-5, and 8-6)
2
x
2
x
2
−
9
=
x
−
6
x
−
3
+
18
x
2
−
9
2
x
2
x
2
−
9
=
(
x
−
6
)
(
x
+
3
)
(
x
−
3
)
(
x
+
3
)
+
18
x
2
−
9
2
x
2
=
x
2
−
3
x
−
18
+
18
x
2
+
3
x
=
0
x
(
x
+
3
)
=
0
x
=
0
✓
or
x
=
−
3
✗
table with 6 rows and 3 columns , row1 column 1 , fraction 2 , x squared , over x squared , minus 9 end fraction , column 2 equals , column 3 fraction x minus 6 , over x minus 3 end fraction . plus . fraction 18 , over x squared , minus 9 end fraction , row2 column 1 , fraction 2 , x squared , over x squared , minus 9 end fraction , column 2 equals , column 3 fraction open , x minus 6 , close . open , x plus 3 , close , over open , x minus 3 , close . open , x plus 3 , close end fraction . plus . fraction 18 , over x squared , minus 9 end fraction , row3 column 1 , 2 , x squared , column 2 equals , column 3 x squared , minus 3 x minus 18 plus 18 , row4 column 1 , x squared , plus 3 x , column 2 equals , column 3 0 , row5 column 1 , x open x plus 3 close , column 2 equals , column 3 0 , row6 column 1 , x , column 2 equals , column 3 0 check mark , or , x equals negative 3 ballot x , end table
|