Download Algebra II with Trigonometry

Algebra II with Trigonometry
Final Exam Review
Part I
1)
Simplify each of the following.
a)
2)
3
32 x 4 y 6
2
d) (x  1) 3  4
x8
c)

1
2
x y z2
1
6
1
4
x y z

1
3
d)
x y
1 3

1
4
b) (5x)3 = 512
e)
3
2
(x  4)  64
c) 4x = 43x - 6
f)
5
2x 3  64
b)
2x  9  7
c) 3 2x  5  3
d)
3x  2  11  13
b) 34 = 81
c) 641/3 = 4
Write in exponential form.
a) log6 216 = 3
6)
2
3
Write in logarithmic form.
a) 72 = 49
5)
1
3
Solve each of the following equations.
a)
4)

Solve each of the following exponential equations.
a) 3x 35 = 37
3)
1


b) x xyz 4 


b) log 100000 = 5
c) log5 x = 6
Expand each of the following.
a) log6 2x
3
b) log5 8x
c) log
x2y7
z
6
d) log
2x
3y
1
7)
Solve each of the following logarithmic equations.
b) log4 x= –1.5
a) log (5x) = 4
Condense first, then solve:
d) log3 x = 4 log3 2 + log3 5 – log3 4
8)
c) logx 5 = 1/2
e) 2 logm (x + 1) – logm 4 = 0
Identify the transformations (shifts and reflections), domain, range, asymptotes, and x- and yintercepts (if any) for the following functions.
a) f(x) =
x 3 4
d) g(x) = 2
x+1
–3
b) g(x) =  3 x  2  5
e)
f (x) 
x4
x 1
 2
c) f(x) =  
 3
 3
f) g(x) =  
 5
x1
x 4
5
3
9) State the domain of for each function:
a) f(x) = 8  x
b) f(x) = x2 – 3x + 2
d) f(x) =
x
2x 2  7x  3
10) If f(x) = x + 3 and g(x) = 4x – 1, find the following:
a) f(x) – g(x)
b) f(x)  g(x)
c) f(g(x))
d) g(f(x))
11) Find the inverse of the function f(x) = 4x - 2. Is the inverse a function?
12) Verify that f and g are inverses of each other. (Show f(g(x)) = g(f(x)) = x)
f(x) = 6x - 5
1
5
g(x) = x 
6
6
13) You deposit $3000 in an account that pays 2.84% annual interest. Find:
a.) The balance after 2 years when compounded quarterly.
b.) The balance after 2 years when compounded continuously.
c.) Approximately how many years it will take to have a balance of $4000.
14) In 1992, 1219 monk parakeets were observed in the United States. For the next 11 years, about
12% more parakeets were observed each year. Write an exponential model that describes the
situation.
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Algebra 2 Final Exam Review Part II
1)
Perform the division for each of the problems given.
a)
(12x4 - 4x2 - 3)  (2x2 + 1)
b)
(8x3 + 2x2 - 5)  (x + 1)
c)
(6x3 + 11x2 - 4x - 9)  (3x - 2)
d)
(3x4- 5x3 + 15x2 – 4x + 3)  (x2 - x + 4)
2)
What is the remainder when (x3 - 2x2 - 9)  (x + 5)?
3)
Is (x + 4) a factor of (x3 - 12x + 16)?
4)
Find the missing factors for each of the following.
5)
a)
x3 - 5x2 - 12x + 36 = (x + 3)(
b)
2x3 - 13x2 - 13x + 42 = (x - 7)(
)(
)
)(
)
List the possible rational zeros given by the Rational Zero Test for the function
f(x)= 4x3 + 15x2 – 8x – 3
6) How many solutions does the equation 2x3 – 4x2 + 6x = 7 have?
7) Write a polynomial function that has the given zeros and has a leading coefficient
of 1.
a) –6, 4, 2
b) 4, 3i
8) One zero of f(x) = x3 – 2x2 – 9x + 18 is x = 2. Find the other zeros of the function.
9) Factor f(x) = 2x3 + 11x2 + 18x + 9 given that f(-3) = 0.
10) Find all the zeros (real and imaginary) of the following functions:
a)
f(x) = 10x4 – 3x3 – 29x2 + 5x + 12
b)
f(x) = x4 + x3 + 2x2 + 4x – 8
11) Roughly sketch the following polynomial functions. Indicate x- and y-intercepts on your graph
a)
f(x) = – 2(x + 5)(x – 3)(x + 1)2
b)
f(x) = (2x + 6)(x – 1)(x – 1)(x – 2)
12) Roughly sketch the following rational functions. Show the equations of all asymptotes on your
graph.
a)
f (x) 
4
x3
b)
f (x) 
2x  1
4x  2
3
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Algebra 2 Final Exam Review Part III
1)
Multiply or divide each of the following.
a)
2)
3)
16x 3y 4

24xy5
9x 2 y 2
b)
x2  x  6
2x 2  9x  9

3x 2  6x
c)
x2  4
x 3  27
4x 2  25

2x 3  6x 2  18x
2x 2  x  10
Solve each of the following.
a)
x
2

x 3 x 3
b)
15
6
 4  3
x
x
d)
5
3x  1

x 2 x1
e)
x5
x

5
x 2 x 6
c)
1
3
4


2
x 2 x 3 x  x 6
c)
8
2x
7 x


x 3x  3 x  1
Perform the indicated operation.
a)
4)
3x 2 y
3x
5x

x1 x 2
b)
8x
x 2

x 3 x1
Simplify each of the following.
x
4
3
a)
1
5
x
4
2
x
b)
1
8
2x
3
c)
1
x
x
4
2
5x
2

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Algebra 2 Final Exam Review Part IV
1)
2)
Find the reference angle for each of the following angles
a) 313°
b) –167°
c) 30
d) 213
e)

10
f) 
g)
3
7
h)
4)
5)

6
Find the remaining five trigonometric ratios for  given the following information.
a) sin  = 3/5
3)
6
5
b) cos  = 7/12
c) tan  = 9/5
d) csc  = 13/6
Solve each of the following right triangles. Give side lengths in simplest radical form.
a) C = 90, B = 60, a = 8
b) C = 90, A = 60, a = 18
c) C = 90, B = 45, a = 8
d) C = 90, A = 45, c = 22
Solve each of the following right triangles. Give sides to the nearest tenth.
a) C = 90, B = 31, a = 7
b) C = 90, A = 63, a = 12
c) C = 90, B = 22, a = 11
d) C = 90, A = 16, c = 28
Solve each of the following problems.
a) A ladder that is 6 meters long leans against a house so that its lower end is 1.5 meters from
the building’s base. What angle does the ladder make with the ground?
b) From the top of the lighthouse 180 feet above the ground, the angle of depression to a boat
at sea is 33. Find the boat’s distance from the foot of the lighthouse.
c) A vertical pole 30 ft high casts a shadow 18 ft long. What is the angle of elevation to the
sun?
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6)
Convert each of the following to radians. Express as a fraction times 
a) 40
7)
3
4
d) 440
b)
11
6
c)
24
9
d)
15
7
Find one positive and one negative coterminal angle for each of the following angles.
a) 15
9)
c) –120
Convert each of the following to degrees.
a)
8)
b) 60
b)421
c)
5
12
d)
19 
7
Evaluate each of the following.
a) If tan  = –7/24 in quadrant II, what is cos ?
b) If sec  = 13/5 in quadrant IV, what is sin ?
c) If sin  = –7/11 in quadrant III, what is cot ?
10) Find all values between 0° and 360° that satisfy the expressions.
a) sin = 0.83
b) tan = (–1)
c) cos  = (  3 / 2 )
d) sin = –1
e) cos  = 0
f) tan = –4.6
11) Determine the exact value of each of the following (in simplest radical form).
a) sin 150°
b) sec 225°
c) tan
5
6
d) cot 2
12) Find the requested information for the given triangles.
a) a = 4, b = 6, c = 5, find B
b) C = 16, b = 92, c = 32, find all sides and angles
c) A = 130, a = 20, b = 16, find all sides and angles
d) B = 35, c = 42, a = 25, find b
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