Name: ______________________
Class: _________________
1ST. SEM. STUDY GUIDE 2015
Short Answer
Simplify the given expression.
1. 6x + 21y − 26x + 64y
Ê
ˆ Ê 2
ˆ
2
2. ÁÁÁÁ 11x + 3x + 19 ˜˜˜˜ + ÁÁÁÁ 6x − 18x − 8 ˜˜˜˜
Ë
¯ Ë
¯
Ê
ˆ Ê
ˆ
2
2
3. ÁÁÁÁ −2x − 9x + 17 ˜˜˜˜ − ÁÁÁÁ 16x + 17x − 1 ˜˜˜˜
Ë
¯ Ë
¯
3
2
4. −2xy(3xy − 5xy + 7y )
5. Evaluate the given expression if m = 45.
| − 3m |
6. Evaluate the given expression if k = –84.
5 | k + 10 | − | 4k |
Solve the given equation. Check your solution.
7. | m − 8 | =27
1
Date: _________
ID: A
Name: ______________________
ID: A
Solve the given inequality. Graph the solution set on a number line.
8. m − 1 < 9 and m + 2 ≥ 1
9. || p − 1 || < 6
10. Graph the given relation or equation and find the domain and range. Then determine whether the
relation or equation is a function.
y = 2x + 5
2
11. Find the value of f(–9) and g(–2) if f ( x) = −5x − 2 and g ( x) = 3x − 21x.
12. State whether the given equation or function is linear. Write yes or no. Explain your reasoning.
8x + 25y = 5
13. Write the equation 10y = 12x + 0.7 in standard form. Identify A, B, and C.
14. Find the x-intercept and the y-intercept of the graph of the equation 11x + 12y = 8. Then graph the
equation.
15. Find the slope of the line that passes through the pair of points (17, 11) and (21, 19).
16. Write an equation in slope-intercept form for the line that satisfies the following condition.
passes through (6, 11), parallel to the line that passes through (2, 4) and (23, 23)
2
Name: ______________________
ID: A
17. Write an equation in slope-intercept form for the line that satisfies the following condition.
1
passes through (29, 8), perpendicular to the graph of y =
x + 17
13
2
18. Consider the quadratic function f ( x) = −2x + 2x + 2. Find the y-intercept and the equation of the axis of
symmetry.
2
19. Graph the quadratic function f(x) = −2x + 2x + 2.
Determine whether the given function has a maximum or a minimum value. Then, find the maximum or
minimum value of the function.
20.
2
f(x) = x − 2x + 2
Solve the equation by graphing. If exact roots cannot be found, state the consecutive integers between
which the roots are located.
2
21. x + 5x + 4 = 0
2
Write a quadratic equation with the given roots. Write the equation in the form ax + bx + c = 0, where a,
b, and c are integers.
22. –5 and 2
Solve the equation by factoring.
2
23. x + 3x − 28 = 0
3
Name: ______________________
ID: A
2
24. 2x + 3x − 14 = 0
Simplify.
25.
196
26.
245
64
27. (2i)(−3i)(4i)
28. i
7
29. ( 11 + i) + ( 3 − 15i)
30.
( 8 + 10i) (5 − 8i)
31.
3
6 + 7i
32.
6 − 3i
8 − 11i
4
Name: ______________________
20
33.
25x y
ID: A
14
Solve the equation by using the Square Root Property.
2
34. 16x − 48x + 36 = 49
Solve the equation by completing the square.
2
35. x + 2x − 3 = 0
Find the exact solution of the following quadratic equation by using the Quadratic Formula.
2
36. x − 8x = 20
Find the value of the discriminant. Then describe the number and type of roots for the equation.
2
37. −x − 14x + 2 = 0
Write the following quadratic function in vertex form. Then, identify the axis of symmetry.
2
38. y = x + 4x − 6
39. Write an equation for the parabola whose vertex is at ÁÊË 2, 6 ˜ˆ¯ and which passes through ÊÁË 4, − 1 ˜ˆ¯ .
5
Name: ______________________
ID: A
40. Write an equation for the parabola whose vertex is at ÁÊË 3, 3 ˜ˆ¯ and which passes through ÊÁË 5, 27 ˜ˆ¯ .
Graph the quadratic inequality.
2
41. y > x − 3x + 5
Solve the inequality algebraically.
2
42. 2x + 14x < −12
Simplify the given expression. Assume that no variable equals 0.
Ê
−6 11 ˆ Ê
5ˆ
43. ÁÁÁÁ 19x y ˜˜˜˜ ÁÁÁÁ −6xy ˜˜˜˜
Ë
¯Ë
¯
ÊÁ
ÁÁ 32x 18 y 10
44. ÁÁÁÁ
ÁÁ 16x 9 y 20
ÁË
ˆ˜ 2
˜˜
˜˜
˜˜
˜˜
˜¯
Simplify the expression using long division.
45.
2
(9x − 41x − 6) ÷ (x − 4)
Simplify the expression using synthetic division.
46.
3
2
(3x − 35x + 128x − 140) ÷ (x − 5)
6
Name: ______________________
ID: A
4
3
2
47. Find p ( −3 ) and p(5) for the function p ( x) = 4x + 8x − 2x + 13x + 10.
For the given graph,
a. describe the end behavior,
b. determine whether it represents an odd-degree or even-degree polynomial function, and
c. state the number of real zeros.
48.
5
4
3
2
49. Graph the function f ( x) = 3x + 8x − 3x − 10x + 12 by making a table of values.
For the given function, determine consecutive values of x between which each real zero is located.
4
3
2
50. f ( x) = −2x − 4x − 2x + 3x + 8
Estimate the x-coordinates at which the relative maxima and relative minima occur for the function.
3
2
51. f ( x) = 8x + 2x − 8
7
Name: ______________________
ID: A
Factor the polynomial completely.
4
2
52. 5x y − 10x y
3
2
2
53. 30x − 50x + 27x − 45
2
54. 4x − 13x + 9
4
2
55. Use synthetic substitution to find g ( 2 ) and g (–7) for the function g ( x) = 5x − 3x + 6x − 4.
Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some of the
factors may not be binomials.
3
2
56. 16x − 144x − 81x + 729; x − 9
Solve the given equation. State the number and type of roots.
2
57. x + 2x − 8 = 0
58.
9x − 9 + 5 = 10
59. 2
9
n − 11 = 1
16
8
Name: ______________________
ID: A
3
2
60. Find all of the zeros of the function f(x) = x − 15x + 73x − 111.
61. List all of the possible rational zeros of the following function.
6
5
4
3
2
f ( x) = 2x − 10x − 23x + 80x + 28x − 20x + 9
4
3
2
62. Find all the rational zeros of the function f ( x) = 78x + 143x − 39x − 26x.
63. Find (f − g) ( x) for the following functions.
f ( x) = 12x + 15
2
g ( x) = −20x + 2x + 30
64. Find ÊÁË f ⋅ g ˆ˜¯ ( x) for the following functions.
2
f ( x) = 3x − 4x − 5
g ( x) = 11x − 3
ÁÊÁ f ˜ˆ˜
65. Find ÁÁÁÁ ˜˜˜˜ ( x) for the following functions.
ÁË g ˜¯
3
2
f ( x) = 20x − 4x + 10x − 13
2
g ( x) = −12x − 7
È
˘
66. Find ÍÍÎ g û h ˙˙˚ ( x) and [h û g](x).
g ( x) = 3x
h ( x) = −6x − 5
9
Name: ______________________
ID: A
Find the inverse of the given relation.
ÔÏ
Ô¸
67. ÔÌ ÁÊË 1, − 5 ˜ˆ¯ , ÁÊË 12, − 7 ˜ˆ¯ , ÁÊË 9, − 9 ˜ˆ¯ , ÁÊË 16, − 13 ˜ˆ¯ Ô˝
Ó
˛
Find the inverse of the given function.
68. f ( x) =
7x − 3
16
69. Determine whether each pair of functions are inverse functions.
11x + 4
1) f ( x) =
2) f(x) = x − 8
4
g(x) = x + 8
9x − 6
g ( x) =
11
70. Graph the given function. State its domain and range.
y = −12 5 − 6x + 6
71. Graph the inequality y <
4x − 9 + 8.
72. Use a calculator to approximate the value of
73. Simplify
5
72 x y
12
4
.
10
(954)
2
to three decimal places.
Name: ______________________
74. What is
4116 divided by
ID: A
7?
Simplify.
75.
162 +
Ê
76. ÁÁÁÁ 5 +
Ë
77.
78.
32 −
ˆÊ
5 ˜˜˜˜ ÁÁÁÁ 7 −
¯Ë
50
ˆ
2 ˜˜˜˜
¯
11
6−
5
3+
5
4−
5
Write the given expression in radical form.
Ê 17 ˆ
79. ÁÁÁÁ x ˜˜˜˜
Ë
¯
9
7
Write the given radical using rational exponents.
80.
2
5
6a b
9
11
Name: ______________________
ID: A
Simplify each expression.
ÊÁ 3
ÁÁ
Á 4
81. ÁÁÁÁ y
ÁÁ
ÁÁ
Ë
82.
b
b
83.
4
ˆ˜
˜˜
˜˜
˜˜
˜˜
˜˜
˜¯
7
8
2
3
1
7
49
Solve the given inequality.
84. 8 +
x + 4 > 18
Sketch the graph of the given function. Then state the function’s domain and range.
85. y = –1.2(3) x
12
ID: A
1ST. SEM. STUDY GUIDE 2015
Answer Section
SHORT ANSWER
1. −20x + 85y
2
2. 17x − 15x + 11
2
3. −18x − 26x + 18
4.
5.
6.
7.
2
4
2
2
−6x y + 10x y − 14xy
135
34
{35, –19}
ÔÏ
Ô¸
8. ÔÌ m | − 1 ≤ m < 10 Ô˝
Ó
˛
3
9. The solution set is {p | –5 < p < 7}.
10.
The domain and the range are all real numbers.
The equation represents a function.
11. f(–9) = 43
g(–2) = 54
12. Yes, the equation is linear. It is in the form Ax + By = C.
13. 120x − 100y = −7 where A = 120, B = −100, and C = −7.
1
ID: A
8
.
11
2
The y-intercept is .
3
14. The x-intercept is
15. 2
19
39
x+
21
7
17. y = −13x + 385
18. The y-intercept is + 2.
16. y =
The equation of the axis of symmetry is x =
1
.
2
19.
20. The function has a minimum value. The minimum value of the function is 1.
2
ID: A
21.
ÏÔ
¸Ô
The solution set is ÌÔ −4, − 1 Ô˝ .
Ó
˛
2
22. x + 3x − 10 = 0
ÔÏ
Ô¸
23. ÌÔ −7, 4 ˝Ô
Ó
˛
7
24. {− , 2}
2
25. 14
26.
27.
28.
29.
30.
31.
32.
7
5
8
24i
−i
14 − 14i
120 − 14i
18 21
−
i
85 85
81
42
+
i
185 185
10
7
33. 5x y
1 13
34. {− ,
}
4 4
ÏÔ
¸Ô
35. ÔÌ −3, 1 Ô˝
Ó
˛
ÔÏ
Ô¸
36. ÔÌ −2, 10 Ô˝
Ó
˛
37. The discriminant is 204. Because the discriminant is greater than 0 and is not a perfect square, the two
roots are real and irrational.
2
38. The vertex form of the function is y = ( x + 2 ) − 10.
The equation of the axis of symmetry is x = −2.
2
39. y = −1.75 ( x − 2 ) + 6
2
40. y = 6 ( x − 3 ) + 3
3
ID: A
41.
42. {x |−6 < x < −1 }
43.
44.
−114y
x
4x
y
16
5
18
20
45. quotient 9x − 5 and remainder –26
2
46. quotient 3x − 20x + 28 and remainder 0
47. 61; 3,525
48. The end behavior of the graph is f ( x) → +∞ as x → +∞ and f ( x) → −∞ as x → −∞.
It is an odd-degree polynomial function.
The function has five real zeros.
49.
50. There are zeros between x = 1 and x = 2, x = –1 and x = –2.
51. The relative maximum is at x = −0.17, and the relative minimum is at x = 0.
ˆ
2 Ê 2
52. 5x y ÁÁÁÁ x − 2y ˜˜˜˜
Ë
¯
4
ID: A
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
2
(10x + 9)(3x − 5)
(4x − 9)(x − 1)
76, 11,812
(4x −9)(4x + 9)
The equation has two real roots, 2 and –4.
34
9
7
n=
9
3, 6 – i, 6 + i
1
3
9
±1, ±3, ±9, ± , ± , ±
2
2
2
1
1
, 0, − , −2
2
3
2
63. 20x + 10x − 15
3
2
3
2
64. 33x − 53x − 43x + 15
65.
66.
67.
68.
69.
70.
20x − 4x + 10x − 13
2
2
,x ≠−
7
12
−12x − 7
[g û h](x) = −18x − 15
[h û g](x) = −18x − 5
ÔÏ ÊÁ
Ô¸
ÌÔ Ë −5, 1 ˆ˜¯ , ÊÁË −7, 12 ˆ˜¯ , ÊÁË −9, 9 ˆ˜¯ , ÊÁË −13, 16 ˆ˜¯ ˝Ô
Ó
˛
−1
16x + 3
f ( x) =
7
Only 2 is an inverse function.
The domain is x ≤
5
and the range is y ≤ 6.
6
5
ID: A
71.
72. 30.887
73. 6 x y
2
6
74. 14
3
75. 8
2
76. 35 − 5
77.
78.
79.
2x
6
2 +7
11 +
31
5 −
10
55
17 + 7 5
11
7
x
1
2
153
5
2
80. 6 a b
81. y
82. b
9
2
21
32
11
21
83.
7
84. x > 96
6
ID: A
85.
The domain is all real numbers and the range is all negative numbers.
7