Algebra II
Name___________________
Exam Review:
Work out each problem carefully, show all work. Turn this in on the day of your exam for extra credit.
I.
Justify each equation by using one of the properties of real numbers.
1.
(x + 7) + (-7) = x + (7 + -7)
1.
2. X • 1 = X
2.
3. X + (7 + -7) = x + 0
3.
4. X + 0 = x
4.
5. Xyz = xzy
5.
6. (fg)h = f(gh)
6.
7. (p + q)r = (q + p)r
7.
8. (5 + y)x = x(5 + y)
8.
9. a(x + b) = ax + ab
9.
10. 8 + 0 = 0 + 8
10.
11. V + 0 + w = v + w
11.
II.
Solve each equation or inequality. You must graph all inequalitites.
1.
3.
4(x + 3) ≤ 44
5.
III.
5x + 8 – 12x = 16 – 15x
3(4x – 2) ≥ 7x + 19
2. 3(x + 1) = 2(x + 11)
4. -2(x + 4) + 9 < - 11
6. 12x – 51 = 3(4x + 7)
Solve and graph each compound inequality. Make sure to give a final graph and solution.
1.
3x > - 6 and 2x < 6
2. 6x < -12 or 5x > 5
3.
5.
IV.
5x > 10 or -2x < 10
5x – 4 ≥ 26 or 29 - 3x > 2
4. - 5 < x – 4 < 3
6. – 22 < 3x – 4 < 8
Solve each absolute value equation or inequality. Remember to graph the inequalities.
1.
|𝑥 + 4| = 10
2. 2|3𝑥 − 2| = 14
3.
4|3𝑥 + 4| = 4𝑥 + 8
4. |𝑥 + 3| > 6
5. |𝑥 − 5| ≤ 8
6. |3𝑥 − 4| − 5 ≤ 27
7. 2|4𝑥 − 1| + 6 > 20
V.
Given f(x) = 2x, g(x) = x2 + 4 and h(x) = 3x2, find the following:
1.
F(-3)
2. G(-4)
3.
h(-2)
4. F(g(3))
5.
h(f(2))
6. G(f(-5))
7. f(g(x))
VI.
1.
Give the domain, range, intercepts, and end behavior of the following graphs.
F(x) = 3x – 2
Inverse _____________
Domain:
Range:
Xint:
Yint:
As x -> ∞, y -> ____
As As x -> - ∞, y -> ____
2.
G(x) = - |𝑥| + 3
Domain:
Inverse ___________________
Range:
Xint:
Yint:
As x -> ∞, y -> ____
As As x -> - ∞, y-> ____
H(x) = x3 + 3
Inverse ______________
Domain:
Range:
Xint:
Yint:
As x -> ∞, y -> ____
As As x -> - ∞, y-> ____
VII.
1.
Graph the following equations and inequalities.
Y=x
2. Y = |𝑥|
3.
y = x2
5.
y = |𝑥 − 3| + 1
4. Y = -
2
2
7. 3x – 4y > 12
VIII.
1.
2.
3.
2
3
𝑥+5
6. Y = - 2(x + 1)2 + 2
8. Y < − |𝑥 + 4| − 2
Determine whether each relation is a function and give its domain and range.
Function? _____, Domain:____________, Range:_____________
Function? _____, Domain:____________, Range:_____________
Function? _____, Domain:____________, Range:_____________
IX.
1.
2.
Find the following for each graph:
Y = -5/2 x + 5
Domain:__________
Range: __________
X-int: __________
y-int: __________
end behavior as x —> ∞:__________
end behavior as x —> -∞:__________
two points on inverse graph: __________
1st inverse equation: __________
Is inverse graph a function: __________
Function notation for inverse: __________
Y = (x – 1)(x + 2)(x – 3)
inverse graph
Domain:__________
Range: __________
X-int: __________
y-int: __________
end behavior as x —> ∞:__________
end behavior as x —> -∞:__________
two points on inverse graph: __________
1st inverse equation: __________
Is inverse graph a function: __________
Function notation for inverse: __________
X.
1.
Find the x- and y-intercepts for each line:
-2x + y = 10
2. 4x + 5 = y
3. 3y + 7x = -14
x-int:__________
x-int:__________
x-int:__________
y-int:__________
y-int:__________
y-int:__________
XI.
1.
Simplify Completely. Do not leave negative exponents in your answer.
3
(2a )(5a4) =_____________
2. (-4x-3y5)2 =_____________
3.
4𝑎𝑏 6 𝑐 3
𝑎5 𝑏𝑐 3
=____________
4. (5a3)(-3a-4) =_____________
5. (-3x2)(-4x-2) =_____________
6. (3x2y3)2 =_____________
7. (-4x-4y3)3 =_____________
8.
4𝑎8
9.
12𝑥 5 𝑦 3
10.
(6𝑥 3 )0
=_____________
2𝑎4
=_____________
4𝑥 −1
=_____________
3𝑥𝑦 −2
11. (
2𝑥 4
3
−3
=_____________
)
12. (-4m2n3)(2mn)
13. (2x3y7)-2
=_____________
=_____________
14. (h8k6)0 =_____________
15.
16.
𝑟 2𝑠4𝑡 6
𝑟 3 𝑠 4 𝑡 −6
−16𝑥 3 𝑦
4𝑦
=_____________
=_____________
17. (s4t)2 (st) =_____________
18. (x3y4)3(x4y2)2 =_____________
19.
(3ab2)2 (4a3b)3 =_____________
20. (-2x-1y3) (4x3y-2)2 =____________
21. (-3x2 + 7x + 23) + (-8x2 – 5x + 13)
22. (-3x2 + 7x + 23) – (-8x2 – 5x + 13)
23. 5a2b(4a – 3b)
24. -4x5(-3x4 – x3 + x + 7)
25. (2x – 3)(4x + 7)
26. (3X – 5)(-2X – 1)
27. (2X – 3)(X2 + 4X + 7)
28. (2X + 5)(2X – 5)
29. (3X – 7)(3X + 7)
30. (5X – 2)2
31. (X + 7)2
XII.
1.
Factor Completely.
4
4x y + 3x2y2
2. X2 – 11x + 18
3. x3 – 8
4. 4x2 + 9
5. 3x2 + 4x + 1
6. X2 + 4x – 12
7. 4x2 – 17x + 15
8. 81a2 – 4b2
9. 3x2 + 8x + 4
10. Y4 – 81
11. x2 – 3x – 40
12. 2x2 – 128
13. 4ab – 2b – 2a + 1
14. 3x2 + 9x – 30
15. xy – 3y + 2x – 6
16. 5y3 – 30y2 – 45y
17. 4x2 + 16xy – 10y2
18. 5x3 + 40
19. 125 – y3
20. 8x4 - 24x3 + 18x2
XIII. Solve each quadratic equation by factoring.
1.
X2 – 4x – 12 = 0
2. Y2 – 16y + 64 = 0
3.
x2 + 25 = 10x
4. 9x = 10x2
5. 7y2 = 4y
6. c2 = 2c + 99
7. 5w2 – 35w + 60 = 0
8. 3x2 +24x + 45 = 0
9. 15x2 + 19x = -6
10. 4k2 + 6 = 11k
11. 36y2 = 25
12. 12m3 – 8m2 = 15m
13. 6a3 = 5a2 + 6a
14. 9 = 64x2
XIV.
Solve using the method described:
Square Root:
1.
3.
X2 + 64 = 0
2. 4x2 + 1 = 0
2x2 + 5 = - 31
4. 5x2 – 15 = 0
Complete the Square:
1.
X2 + 10x – 1 = 0
2. 3x2 + 4x = 2x2 + 3
Quadratic Formula:
1.
X2 + 4x = 3
2. 2x2 – 12 = 5x
3.
4x2 + 4x = 7
4. X2 = 4x – 1
Evaluate the discriminant for each equation. Determine the number and what kind of solutions.
1.
X2 + 5x + 8 = 0
2. X2 – 5x + 4 = 0
3.
-9x2 = 4 – 12x
4. 6x2 = x + 2
XV.
Simplify.
1. √25
16.
2
5𝑖
2. √50
3. √−18
17. (-3 – 7i) + (1 + 2i) 18. (-3 – 5i)(1 + 2i)
20. (8 - √−1)(−3 + √−25)
21. -5(1 + 3i) + 3i(3 – 2i)
4. √−40
5. √
19. (9 + √−4) − (6 − √−16)
2
5
XVI.
Graph each function. Identify the vertex, axis of symmetry, the maximum or minimum value, the
domain, range and zeros of each function.
1.
f(x) = (x  3)2 – 2
2. F(x) = -3x2 + 18x - 27
Vertex: ____________
Vertex: ____________
AOS: __________
AOS: __________
Max/Min: __________
Max/Min: __________
Domain: __________
Domain: __________
Range: __________
Range: __________
Zeros _____________
Zeros _____________
Y – int __________
Y – int __________
3. y = 2x2 + x - 4
Vertex: ____________
AOS: __________
Max/Min: __________
Domain: __________
Range: __________
Zeros _____________
Y – int __________
Write a quadratic function to model each graph.
4.
_______________________
5.
Write equation : y = 3x2 + 18x + 32, in vertex form. ____________________________