Download 2007Are You Ready for Alg 3 with key

Page 1 of 16
Are You Ready for Algebra 3/Trigonometry?
The Algebra 3/Trig CP course prepares students for Calculus and college science courses. In
order to accomplish this, it is taught at a significantly faster pace than Algebra 2 CP, and
students often find it difficult to adjust. To make the transition easier, spend some time this
summer practicing and reviewing the Algebra 2 topics in this packet! Completion of the packet
is not mandatory, but students who complete it will be better prepared for the Algebra 3/Trig
year.
The packet includes 3 sections - Algebra 1 Topics you are expected to know well,
Algebra 2 Topics you will learn again at a much faster pace, and Common Errors in
Algebra. At the end is a formula sheet and answers to the problems.
If you are unsure how to do any problem, check the Internet for help. Go to google.com
and type “Algebra Tutorial”. Some websites are free, such as
http://www.wtamu.edu/academic/anns/mps/math/mathlab/ and
http://tutorial.math.lamar.edu/sitemap.aspx and www.algebracomplete.com
Section 1 - Algebra 1 Topics you are expected to know well
A. Evaluate using order of operations:
3xy 2  1
1.
when x = -3, y = -2
3x3 y  1
B. Solve equations:
3(3x  2)  6(3  2 x)  2
3.
2.
 4 x  5 when x = 2.4
4.
2  5x  9  3  8  x  2 
C. Linear Equations - Sketch graphs, write linear equations using slope and intercept:
5.
In which quadrant is (5, -4)?
Sketch the graph of the following equations:
6.
x  5
7.
y3
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8.
y  3x  5
10.
Find the slope of (-15, 11) and (8, -4)
11.
Find the intercepts of 4 x  3 y  16
9.
2x  3y  6
x-intercept:
y-intercept:
12.
Rewrite in slope-intercept form:
4 x  3 y  1
slope:
y-intercept:
13.
Write an equation of the line containing (1, 1) and (2, -2).
14.
Write an equation of the line passing through (7, 2) and having a slope of 0.
15.
Write an equation of the line having an undefined slope (no slope) and passing through
the point (-2, 5).
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D. Add, Subtract and Multiply Polynomials:
16.
 2x
2
 3 x  4    x 2  x  1
18. 2 x  4 x 2  3 x  2 
20.
 2 x  5
2
17.  2 x 2  3 x    3 x  2   2  3 x 2  2 x 
19.
 4x  7 3x  2
21.
 x  3  x 2  2 x  3
E. Factoring- Greatest Common Factor, Trinomials, Difference of Squares:
22.
x2 – 49
23.
x2 + 4x – 21
24.
x 2  16 x  64
25.
2 x3  32 x
F. Simplify Using Exponent Rules:
26.
(3)4 (3)2
27.
x 7  x 9
28.
y15
y5
29.
7x 2 y 0
30.
(5m)
31.
y 4 12 x 2

6 x3 xy
32.
5x2
1

3
y 15xy 1
33.
(2 xy 3 )3
0
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Section 2 - Algebra 2 topics you will learn again in Algebra 3, but at a faster pace.
A. Solve Absolute Value Equations:
34.
3x  3  15
35.
2 1  2 x  16
B. Solve and Graph Inequalities on a Number Line:
36.
3x 1  2x 11
37.
3  2x  1  9
38.
3x  5  10
39.
2 x  3  13
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C. Function Operations and Inverses:
Determine whether the relation is a function.
40.
{(0, 7), (2, 6), (2, 11), (8, 6)}
41.
{(2, 3), (5, 7), (8, 0), (3, 7)}
43.
f(x) = x2 – 2x – 4
Find the indicated value of f(x).
42.
f(x) = 3x – 7
f(3)
Let f(x) = 2x – 1 and g(x) = x – 2 Find:
44.
f(x) + g(x)
45.
g(x) – f(x)
46.
f(g(x))
47.
g(f(2))
48.
f(x)  g(x)
Find the inverse of the function (solve for y)
49.
f(x) = 4x + 3
Find the domain and range for each of the following.
50.
f(x) =
x3
51.
f(x) =
5
x4
f  4
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D. Quadratic Equations and Parabolas
52. Solve
1 2
x 8.
2
53.
Solve by completing the square.
x2 + 4x – 3 = 0
54.
Solve by the quadratic formula.
3x2 + 2x – 2 = 0
55.
Determine the vertex and axis of symmetry for the parabola y  x 2  8 x  1
Vertex
Axis of Symmetry
Find all real solutions
56.
x2 – 5x = 0
57.
x 2  3x  10  0
E. Complex (Imaginary) Numbers:
Simplify the following.
58.
9
59.
7
60.
 4
61.
  15
62.
(3i)2
63.
(2i)2
64.
3i2
65.
i4
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Solve the equation.
x2 = 9
66.
67.
x2 = -7
Perform the indicated operation.
68.
(9 + 3i) + ( 7 – i)
69.
(1 – 6i) – (8 + i)
70.
2i(5 + 3i)
71.
(5 + 4i)(2 + i)
F. Radicals and Rational Exponents:
72. Write in radical form and simplify: 9
73. Write in rational exponent form:
74. Simplify, then add like radicals:
5
1
2
43
18  8  4 2
Simplify:
75.
3
27x 3 y 6
16x5
76.
Solve:
77.
4 2x  3  7
78.
79.
x 5  243
80.
3
3x  3 x  4
2x  3  4x  7
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G. Factoring - Trinomials, Factor by Grouping, Sum and Difference of Cubes:
81.
2x2 – 5x – 12
82.
x3 + 5x2 – 9x – 45
83
x3  8
84.
8 x3  27
H. Polynomial Division:
85.
Use long division to divide: (8x2 + 4x + 3)  (2x + 3)
86.
Use synthetic division to divide: (y3 + 2y2 – 5y – 6)  (y + 3)
I. Rational Expressions- Perform the operations and simplify:
x2  6x  8 x2  x  2
 2
87.
x 2  3x
4 x  12 x
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88.
8
2 x  10
 2
x  4 x  16
Simplify the complex fractions
89.
2
3x
3
x 1
2
x 1
1
90.
1 5

2 3x
Solve:
91.
11 1 4
 
3x 3 x 2
93.
1+
1
2x

x 4 x2
2
92.
2
4

3 x  5 x  15
94.
4
12
 2
3x  3 x  1
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Section 3 - Common Errors in Algebra
Many algebra errors come from not mastering the differences between the rules of addition
and multiplication! Test yourself by doing the following problems. Check your answers by
substituting numbers for the variables. There may be more than one correct form of the
answer.
I.
ERRORS INVOLVING PARENTHESES
A. Distributing a negative sign
1. 3  ( x  2) 
2.
3x
x 1


x2 x2
3. .
3x  2 2  x  1


5x
5x
a) 3  x  2
b) 1  x
a)
2x 1
x2
b)
2x 1
x2
a)
1
5
b)
x 1
5x
B. Distributing Left and Right
1. 4( x  2)  2 
a) 8x 16
c) 5  x
c)
b) 4x  4
x4
5x
c) 4x 16
C. DO NOT distribute exponents over addition or subtraction
Does (a  b)2  a 2  b2 ?
Convince yourself:
Does (3  4)2  32  42 ?
D. DO NOT distribute when there is only multiplication
1
1
 1  1 
1
1.    a  (b)  a)  a   b 
b)   ab
c)  ab 
2
2
 2  2 
2
II.
ERRORS INVOLVING FRACTIONS
1. Does
2.
ab a b
  ?
x
x x
1 1
 
a b
Does
a)
1
ab
b)
ba
ab
a)
bx
a
b)
x
ab
a)
1
3x
b)
x
3
x
3.
a
b
1
4.   x 
3
x
x x
  ?
ab a b
d)
ab
2
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5.
1
2
x
a)
1
x2
6.
x 1

x 1
a) 1
7.
( x  1)

( x  1)( x  3)
a)
b)
b) 0
1
x3
b) x  3
50
?
 5 2
8. Which are possible steps for doing on a calculator
a) 50  5  2
III.
1  2x
x
c) 50   5  2 
b) 50  5  2
ERRORS INVOLVING EXPONENTS
x 
2 3
=
a) x 5
b) x 6
2. x 3 x 5 
a) x8
b) x15
1.
3. Does 4 x 2  2 x 2  4 x 2  (2 x)2 ?
4.
5.
9

5x3
a)
7

2x  3
9
b)   x3
5
9 3
x 
5
a) 7  2 x  3
1
2
b) 7  2 x  3
1
2
6. Which are correct steps for doing on a calculator 22 3 ?
b) 2 ^  2  3
a) 2 ^ 2  3
IV.
ERRORS INVOLVING RADICALS
1.
5x 
a) 5 x
b)
5x
c)
2. Does
a 2  b2  a  b ?
Check: Does
32  42  3  4 ?
3. Does
 a  b
Check: Does
3  4
2
 ab ?
2
 3 4?
4.
64

2
a)
32
b)
8
or 4
2
5.
32

2
a)
16
b)
4 2
or 2 2
2
5
x
Page 12 of 16
6. Which are correct calculator steps for
a)
3 2 
4 2
b)
c)  3^ 2  4 ^ 2  ^1  2
V.
32  42 ?
3 2  4 2


d)  3^ 2  4 ^ 2 ^ (1  2)
ERRORS INVOLVING DIVIDING OUT COMMON FACTORS
A. When Simplifying Fractions
1.
a  bx

a
a) 1  bx
b
b) 1    x
a
2.
a  ax

a
a) a  x
b) 1  x
3.
24 3

2
a) 2  2 3
b) 1  2 3
c) 1  4 3
4.
12  2 3

6
a) 2 
3
3
c) 2  2 3
5.
4x

x4
a)
x
1 4
b) -1
c) Can't simplify
6.
8x

2x  4
a)
4x
x2
b) -1
c) Can't simplify
3
3
b) 12 
c) 1 
bx
a
Page 13 of 16
Answer Key – Are you ready for Algebra 3/Trig
1.
37
 .227
163
2. 14.6
14
3
5
4. x 
18
3. x 
5. IV
6.
14. y  2
15. x  2
16. x 2  2 x  3
39. x  5 or x  8
17. 4 x 2  4 x  2
40. no
18. 8 x3  6 x 2  4 x
41. yes
19. 12 x 2  13 x  14
20. 4 x 2  20 x  25
21. x3  x 2  9 x  9
22.  x  7  x  7 
42.
43.
44.
45.
23.
24.
25.
7.
38. 5 3  x  5
13. y  3x  4


 x  7  x  3
2
 x  8
2 x  x  4 x  4
8.
10. m 
11.
12.
15
23
x  int : 4
y  int : 16 3
slope : 4 3
y  int : 1 3

50.
51.
52.
53.
x
y10
7 x2
1
x  3
All Re al , x  4
x  4
x  2  7
34. x  4 or x  6
57.
35. x  9 2 or x  7 2
58. 3i
36. x  2
59. i 7
2  28 1  7

6
3
55. Vertex  4, 17  ; x  4
56. x  x  5  0; x  0,5




 x  5 x  2   0
x  5, 2
60. 2i

61. i 15
62. 9i 2  9
37. 2  x  4


49. x  4 y  3; y   x  3 4
54. x 



48. 2 x 2  5 x  2
2



-16
20
3x  3
 x 1
2 y3
x2
xy 4
32.
3
1
33.
8x3 y 9


47. f  2  3; g  f  2   1
31.
9.

46. 2 x  5
26. 729
27.
28.
29.
30.




63. 4

Page 14 of 16
64.
65.
66.
67.
68.
69.
70.
71.
72.
–3
1
x  3i
x  i 7
16  2i
7  7i
10i  6i 2  6  10i
10  13i  4i 2  6  13i
9 3
3
73. 4 5
74. 3 2  2 2  4 2  2
75. 3xy 2
76. 4x 2 x
77.
78.
79.
80.
81.
82.
83.
84.
1
2
3x  x  4 so x  2
x3
2x  3  4x  7 so x  2
 2x  3 x  4
2 x  1 so x 
 x  3 x  3 x  5
 x  2  x2  2x  4
 2 x  3  4 x 2  6 x  9 
15
2x  3
2
86. y  y  2
4  x  4
87.
x 1
6 x  22
88.
 x  4  x  4 
4
89.
3 x  10
x4
90.
2
91. LCD : 3 x 2
 x 12 x  1  0; x  12, 1
85. 4 x  4 
92. Cross Multiply; x  1
93. LCD :  x  2 x  2
 x 1 x  3  0; x  1,3
94. Cross Multiply; x  10
Page 15 of 16
Answers to Common Errors in Algebra
I.
A.
1. c
2. b 3. a
B.
1. a
C.
NO!!!
D.
1. b or c or d
II.
1. Yes, No
2. b
3. b
4. b
5. b
6. a
7. a
8. b or c
III.
1. b
2. a
3. NO!!
4. a
5. b
6. b
IV.
1. c
2. NO!!
3. Yes
4. b
5. b
6. b or d
V. A. 1. b or c
2. b
3. b
4. a
5. c
6. a
Page 16 of 16
ALGEBRA II
FORMULA SHEET
y 2  y1
x 2  x1
Slope of a Line
m
Slope Intercept Form of a Line
y  mx  b
Point Slope Formula
(y  y1 )  m(x  x1 )
Vertex Form of a Quadratic
y  a(x  h)  k
Quadratic Formula
b  b  4ac
x
2a
Discriminant
b 2  4ac
x-coordinate of the Vertex of a Parabola
x
Pythagorean Theorem
a 2  b 2  c2
2
2
Distance Formula
b
2a
d  (x2  x1 )  (y2  y1 )
2
Midpoint Formula
x2  x1 , y2  y1 
 2
2 
Direct Variation
y  kx or
Inverse Variation
y
Joint Variation
z  kxy
y
k
x
k
or xy  k
x
Difference of Two Cubes
a3  b3  (a  b)(a 2  ab  b2 )
Sum of Two Cubes
a3  b3  (a  b)(a 2  ab  b2 )
2