Secondary III 6-2 HW
Name: ______________________
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must be made through “File inf
Evaluate: Homework
and
Practice
CorrectionKey=NL-A;CA-A
Asymptotes of Rational Functions
Book 8.2
t0OMJO
State
the
domain
using
interval
notation.
For
anynotation,
x-value excluded
from the
domain, state
State the domain using an inequality, set
and interval
notation.
For any
t)JOUT
whether the graph has a vertical asymptote or a “hole” at that x-value. Use a graphing calculator
t&YUSB
excluded
tox-value
check your
answer.from the domain, state whether the graph has a vertical asymptote
or a “hole” at that x-value. Use a graphing calculator to check your answer.
Sketch the graph of the
given
rational
function. (I
2x 3
x+5
x2 +
(x) = __
2.
ƒ
1. ƒ(x) = _
2
of the numerator is 1 xmore
x+1
- 4xthan
+ 3the degree of the
find the graph’s slant asymptote by dividing the nu
denominator.) Also state the function’s domain an
inequalities, set notation, and interval notation. (I
indicates that the function has maximum or minim
a graphing calculator to find those values to the n
when
the range.)
Find any holes, asymptotes, and intercepts and state the
end determining
behavior.
3.
f ( x) =
7.
x −1
2
x + x−6 x-1
ƒ(x) = _
x+1
Divide the numerator by the denominator to write the function in the form ƒ(x) = quotient +
and determine the function’s end behavior. Then, using a graphing calculator to examine the f
graph, state the range using an inequality, set notation, and interval notation.
3x + 1
ƒ(x) = _
x made
- 2 through “File info”
O NOT EDIT--Changes must be
4.
3.
x
ƒ(x) = __
(x - 2)(x + 3)
Sketch the graph of the given rational function. Also state the function’s domain and range
rrectionKey=NL-A;CA-A
using interval notation. Find any x and y intercepts, state the end behavior, and behavior
around the asymptotes.
8. f (x) =
x +1
(x − 1)2 (x + 2)
Sketch the graph of the given rational function. (If the degree
of the numerator is 1 more than the degree of the denominator,
find the graph’s slant asymptote by dividing the numerator by the
denominator.)
Also state the function’s domain and range using
Domain:
inequalities, set notation, and interval notation. (If your sketch
8.
Range:
indicates that the function has maximum or minimum values, use
X–intercept:
a graphing
calculator to find those values to the nearest hundredth
when determining
the range.)
Y–intercept:
hton Mifflin Harcourt Publishing Company
7.
x-1
)=_
ƒ(xVAsymptote:
x+1
x 2 - 5x + 6
_
(
)
5.
ƒ
x
=
Hole:
x-1
increasing:
decreasing:
EndBehavior:
y
4
x-1
ƒ(x) = __
(x - 2)(x + 3)
2
x
-4
-2
6.
4x - 1
ƒ(x) = _
2
x +x-2
AsymptotesBehavior:
0
2
-2
2
9.
(x + 1)(x - 1)
ƒ(x) = __
x+2
y
x
Name: ______________________
-6 -4 - 2 0
2
Secondary III 6-2 HW
9. f (x) =
x 2 + 2x − 3
x2 + x − 2
-2
-4
x-1
y
8. ƒ(x) = __
4
(x - 2)(x + 3)
Domain:
2
Range:
X–intercept:
-4 -2 0
-2
Y–intercept:
VAsymptote:
-4
Hole:
increasing:
decreasing:
EndBehavior:
AsymptotesBehavior:
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=NL-A;CA-A
© Houghton Mifflin Harcourt Publishing Company
Module 8
Domain:
Range:
X–intercept:
Y–intercept:
VAsymptote:
Hole:
Increasing:
Decreasing:
EndBehavior:
-8
x
2
4
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-3x(x - 2)
__
(x + 1)(x - 1)
10.
ƒ(x) =9. ƒ(x) = __
(x - 2)(x + x2+
) 2
-6
y
-6
-6
-4
-4
-2
-2
y
0 x
0 -2 2
-2
-4
-4
Lesson 2
456
-6
-6
-8
AsymptotesBehavior:
-3x(x - 2)
10. ƒ(x) = __
y
x
2
x -x+ 6
⎩
⎭
⎫
⎧
B. The function’s domain is ⎨xǀx⎧≠ -2 and x ≠ -3⎬. ⎫
A. The
function’s
domain⎩ is ⎨xǀx ≠ -2 and x ≠
⎭ 3⎬.
Secondary
III 6-2
HW
Name: ______________________
⎩
⎭
⎫
⎧
C. The function’s range is ⎨yǀy ≠⎧ 0⎬.
⎫
⎩
⎭
xǀx
≠
-2
and
x
≠
-3
B. Thex 2function’s
domain
is
⎨
⎬.
−1
f ( x) =
⎩
⎭
⎫
⎧
+2
11. D. The xfunction’s
< y ⎫< +∞⎬.
range is ⎨yǀ-∞
⎧
⎩is ⎨yǀy ≠ 0⎬.
⎭
C. The
function’s
range
x
2
-3x
(
)
y
__
(
)
10.
ƒ
x
=
⎩
⎭
E. The
has
vertical
asymptotes
at
x
=
-2
and
x
=
3.
x
(x function’s
- 2)(x + 2graph
)
Domain:
0
⎫ -6 -4 -2
⎧
2
yǀ-∞
<
y
<
+∞
.
D.
The
function’s
range
is
⎨
⎬
F.
The
function’s
graph
has
a
vertical
asymptote
at
x
=
-3
and
a
“hole”
at
x
=
2.
-2
Range:
⎩
⎭
© Houghton
Harcourt Publishing Company
© Houghton Mifflin Harcourt
PublishingMifflin
Company
© Houghton Mifflin Harcourt Publishing Company
X–intercept:
G.
graph
hashas
a horizontal
asymptote ataty x==
0. -2 and-x4 = 3.
E. The
Thefunction’s
function’s
graph
vertical asymptotes
Y–intercept:
H.
hashas
a horizontal
1. -3 and-a6 “hole” at x = 2.
VAsymptote:
F. The
Thefunction’s
function’sgraph
graph
a verticalasymptote
asymptoteataty =
x=
Hole:
G. The function’s graph has a horizontal asymptote at y = 0.
increasing:
H.O.T. Focus on Higher Order Thinking
decreasing:
x+a
H. The
function’s graph
hasvalue(s)
a horizontal
asymptote
1.
(x) = _________
18. Draw
Conclusions
For what
of
a does
the graphatofy ƒ=
EndBehavior:
AsymptotesBehavior:
x 2 + 4x + 3
have a “hole”? Explain. Then, for each value of a, state the domain and the range of
ƒ(x) using
notation.
H.O.T.
Focusinterval
on Higher
Order Thinking
x+a
x + 4x + 3
18. Draw Conclusions For what value(s) of a does the graph of ƒ(x) = _________
2
have a “hole”? Explain. Then, for each value of a, state the domain and the range of
ƒ(x) using interval notation.
Module 8
Lesson 2
457
4x - 1
19. Critique Reasoning A student claims that the functions ƒ(x) = ______
and
4x + 2
4x
+
2
______
g(x) = 2
have different domains but identical ranges. Which part of the
2
4x - 1
student’s claim is correct, and which is false? Explain.
4x - 1
19. Critique Reasoning A student claims that the functions ƒ(x) = ______
and
4x + 2
4x + 2
______
g(x) = 2
have different domains but identical ranges. Which part of the
2
4x - 1
student’s claim is correct, and which is false? Explain.
Module 8
461
Lesson 2
Secondary III 6-2 HW
Name: ______________________
Selected Answers:
1.
VA:
x = −1
3. X–intercept:(1,0)
⎛
1⎞
Y–intercept: ⎜ 0, ⎟ ⎝ 6⎠
VAsymptote: x = −3, x = 2 HAsymptote: y = 0 EndBehavior: lim f (x) = 0 , lim f (x) = 0 AsymptotesBehavior: lim− f (x) = −∞ , lim+ f (x) = ∞ , lim− f (x) = −∞ , lim+ f (x) = ∞ x→−∞
x→∞
x→−3
x→−3
x→2
10. Domain: (−∞,−2) U (−2,∞) Range: (−∞,−3) U (−3,∞) X–intercept:(0,0)
Y–intercept:(0,0)
VAsymptote: x = −2 HAsymptote: y = −3 Hole: ⎜ 2,− ⎟ 2⎠
⎝
increasing: ∅ decreasing: (−∞,−2) U (−2,∞) EndBehavior: lim f (x) = −3 , lim f (x) = −3 AsymptotesBehavior: lim− f (x) = −∞ , lim+ f (x) = ∞ ⎛
3⎞
x→−∞
x→∞
x→−2
x→−2
x→2