Download Unit 6B Review

Name: ______________________
Class: _________________
Date: _________
ID: A
Unit 6B Review
Multiple Choice
Identify the choice that best completes the statement or answers the question.
Sketch the asymptotes and graph the function.
____
1. y =
−4
x+2
−3
a.
b.
c.
d.
____
2. Write an equation for the translation of y =
a. y =
4
x−6
+ 7 b. y =
4
x+7
4
x
that has the asymptotes x = 7 and y = 6.
+ 6 c. y =
1
4
x−7
+ 6 d. y =
4
x+6
+7
____
3. This graph of a function is a translation of y =
a. y =
____
2
2
2
2
− 5 b. y =
− 3 c. y =
+ 3 d. y =
+5
x−3
x−5
x−5
x−3
4. This graph of a function is a translation of y =
a. y =
2
. What is an equation for the function?
x
4
. What is an equation for the function?
x
4
4
4
4
+ 4 b. y =
− 4 c. y =
− 3 d. y =
+3
x+3
x+3
x+4
x+4
Find any points of discontinuity for the rational function.
____
5. y =
(x + 3)(x − 5)(x + 7)
(x + 1)(x + 4)
a. x = 1, x = 4 b. x = –1, x = –4 c. x = 3, x = –5, x = 7 d. x = –3, x = 5, x = –7
2
x+8
____
6. y =
____
7. What are the points of discontinuity? Are they all removable?
2
x − 9x + 14
a. x = –2, x = –7 b. x = 2, x = –7 c. x = –8 d. x = 2, x = 7
y =
(x − 7)(x − 3)
2
x − 10x + 21
a. x = 1, x = –8, x = –2; yes b. x = 7, x = 3; yes c. x = –7, x = –3; no d. x = –1, x = 8, x = 2; no
____
8. What are the points of discontinuity? Are they all removable?
y =
(x − 5)
2
x − 6x + 5
a. x = –1, x = –5, x = 8; yes b. x = 6, x = –7, x = –8; no c. x = –1, x = –5; no d. x = 1, x = 5; no
____
(x − 3)(x − 1)
9. Describe the vertical asymptote(s) and hole(s) for the graph of y =
.
(x − 1)(x − 5)
a. asymptote: x = 5 and hole: x = 1 b. asymptote: x = –5 and hole: x = –1 c. asymptote: x = –3 and
hole: x = 5 d. asymptote: x = 5 and hole: x = –1
x−1
____ 10. Describe the vertical asymptote(s) and hole(s) for the graph of y =
.
2
x + 6x + 8
a. asymptotes: x = –4, –2 and hole: x = 1 b. asymptote: x = 1 and no holes c. asymptote: x = 1 and
holes: x = –4, –2 d. asymptotes: x = –4, –2 and no holes
6
____ 11. Find the horizontal asymptote of the graph of y =
a. y = 3 b. y =
3
7
3x − 7x + 9
2
7x + 7x + 9
.
c. y = 0 d. no horizontal asymptote
3
____ 12. Find the horizontal asymptote of the graph of y =
−2x + 3x + 2
3
2x + 6x + 2
a. y = 1 b. y = −1 c. no horizontal asymptote d. y = 0
.
Simplify the rational expression. State any restrictions on the variable.
2
____ 13.
t − 4t − 32
t−8
a. t − 4; t ≠ −8 b. t + 4; t ≠ 8 c. −t − 4; t ≠ 8 d. −t + 4; t ≠ −8
3
2
____ 14.
k −k−2
2
k − 4k − 5
−(k − 2)
−(k − 2)
k−2
a.
; k ≠ 5 b.
; k ≠ −1, k ≠ 5 c.
; k ≠ −1, k ≠ −5 d.
k−5
k−5
k−5
k−2
; k ≠ −1, k ≠ 5
k−5
What is the product in simplest form? State any restrictions on the variable.
____ 15.
y
2
2
y−3
⋅
y −y−6
2
y + 1y
2
a.
y + 2y
2
, y ≠ 3, − 1 b.
y+1
y+2
, y ≠ 3, − 1
y+1
y + 2y
y+1
, y ≠ 3, 0, − 1 c.
y+2
y+1
, y ≠ 3, 0, − 1 d.
Simplify the trigonometric expression.
____ 16.
1
1
+
1 + sin θ
1 − sin θ
2
a. 2 cos θ b. 2 sec 2 θ c. 2 csc 2 θ d. 2 cot 2 θ
____ 17. secθ cosθ
a. cotθ b. tanθ c. sinθ d. 1
____ 18.
sin 2 θ
1 − cos θ
a. 1 + cos θ b. sin θ c.
1 − sin θ
1 + sin θ
d.
cos θ
cos θ
Use the unit circle to find the inverse function value in degrees.
ÊÁ
ˆ
ÁÁ 3 ˜˜˜
Á
˜˜
____ 19. sin ÁÁÁ
˜
ÁÁ 2 ˜˜˜
Ë
¯
a. 30° b. 60° c. 240°
−1
d. 150°
−1
____ 20. tan
3
a. 120° b. 90°
c. 60°
d. 30°
4
ÁÊÁ
˜ˆ
ÁÁ 3 ˜˜˜
Á
˜˜
____ 21. cos ÁÁ
ÁÁ 2 ˜˜˜
Ë
¯
a. 60° b. 30°
−1
c. 240° d. 150°
For a standard-position angle determined by the point (x, y), what are the values of the
trigonometric functions?
____ 22. For the point (16, 12), find sinθ and cosθ.
a.
b.
16
20
12
sin θ =
20
cos θ =
cos θ =
20
12
12
sin θ =
20
c.
cos θ =
sin θ =
d.
cos θ =
c.
csc θ =
d.
csc θ =
c.
tan θ =
d.
tan θ =
20
16
16
12
16
12
16
sin θ =
20
____ 23. For the point (9, 12), find cscθ and secθ.
a.
csc θ =
sec θ =
b.
csc θ =
15
12
15
9
15
12
12
sec θ =
15
9
15
12
sec θ =
15
9
12
9
sec θ =
15
____ 24. For the point (4, 3), find tanθ and cotθ.
a.
tan θ =
cot θ =
b.
tan θ =
cot θ =
3
4
4
3
5
4
4
3
5
4
5
3
cot θ =
5
4
3
4
cot θ =
5
Short Answer
25. Verify the basic identity. What is the domain of validity?
cot θ = cos θ csc θ
.
26. Verify the Pythagorean Identity.
2
2
1 + cot θ = csc θ
.
Verify the identity. Justify each step.
27. tan θ + cot θ =
1
sin θ cos θ
.
28.
sec θ
sec θ
−
= 2 csc θ
csc θ − cot θ
csc θ + cot θ
6
ID: A
Unit 6B Review
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
C
C
B
D
B
D
B
D
A
D
D
B
B
D
B
B
D
A
B
C
B
A
A
A
SHORT ANSWER
25.
cot θ =
cos θ
sin θ
=
cos θ ⋅
1
sin θ
= cos θ csc θ
The domain of validity is {x | x ≠ n⋅π, where n = any integer}
1
ID: A
26.
2
cos θ
2
1 + cot = 1 +
Tangent Identity
2
sin θ
2
2
=
sin θ
=
sin θ + cos θ
2
sin θ
+
cos θ
2
sin θ
2
=
Find a common denominator.
2
2
sin θ
1
Add.
Pythagorean Theorem
2
sin θ
2
= csc θ
Reciprocal Identity.
27.
tan θ + cot θ
=
sin θ
cos θ
+
cos θ
sin θ
Tangent Identity and Cotangent Identity
=
sin 2 θ
cos 2 θ
+
sin θ cos θ
sin θ cos θ
Write the fractions with a common
denominator.
=
sin 2 θ + cos 2 θ
sin θ cos θ
Add.
=
1
sin θ cos θ
Pythagorean Identity
2
ID: A
28.
sec θ
sec θ
−
csc θ − cot θ
csc θ + cot θ
=
sec θ(csc θ + cot θ)
( csc θ − cot θ) (csc θ + cot θ)
−
sec θ(csc θ − cot θ)
Write the fractions with
( csc θ − cot θ) (csc θ + cot θ) a common
denominator.
=
sec θ csc θ + sec θ cot θ
sec θ csc θ − sec θ cot θ
−
csc 2 θ − cot 2 θ
csc 2 θ − cot 2 θ
Distributive Property
=
2 sec θ cot θ
csc 2 θ − cot 2 θ
Subtract.
=
2 sec θ
cos θ
sin θ
θ−
cot 2
csc 2
2
=
csc 2
csc 2
2
=
θ
1
cos θ
cos θ sin θ
θ−
2
=
Cotangent Identity
cot 2
Reciprocal Identity
θ
1
sin θ
θ−
cot 2
Simplify.
θ
1
sin θ
Pythagorean Identity
1
= 2 csc θ
Reciprocal Identity
3