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Algebra 2
Review- Final Exam
Name:____________________________________
Date:_____________________________________
Topics:
 Factoring and Rational Expressions
 Functions
o Linear
o Quadratic
o Absolute Value
o Radical
o Log
o Exponential
o Piecewise
 Trig
o Basics
o Identities
o Graphs & Equations



Sequences and Series
Statistics
o Designs
o Graphs
o MCT
o MS
o ND
o CLT
Probability
o Basics, Addition, Multiplication rules
o PDF (including normal distributions)
o Counting
o Binomial
Helpful tips:
Be prepared to:
1. solve any equation in more than one way (algebraically, graphically, and/or algorithmically).
2. explain your reasoning using complete sentences
3. check your answers to see if they make sense (ex: plane’s velocity in flight = 4mph, etc.)
4. use given statements/facts to draw conclusions (logical reasoning)
5. derive the quadratic formula
6. verify trig id’s using multiple approaches
7. analyze the solution to a problem for veracity
8. explain calculator processes
Factoring
Perfect Square Trinomial
To factor a perfect square trinomial follow the steps below:
1) Create two empty binomials as indicated on the right
(
2) Take the square root of the first term of the given trinomial
3) Take the square root of the last term of the given trinomial
4) Take the result of step 2 and put in the 1st position in each binomial
5) Take the result of step 3 and put in the 2nd position in each binomial
6) The signs in the binomials should be the same as the middle term of the binomial
Example 1:
factor : 4 x 2  12 x  9
Answer:
)(
)
(2x - 3)2
1
Difference of Two Squares a 2  b 2
To factor a difference of two squares follow the steps below:
1) Create two empty binomials as indicated on the right
(
)(
)
st
2) Take the square root of the first term of the given binomial and put in the 1 position of each
binomial
3) Take the square root of the last term of the given binomial and put in the 2nd position of each
binomial
4) Make one binomial a sum and the other binomial a difference
Example 2:
9 x 2 y 2  16
Answer:
(3xy - 4) (3xy + 4)
a 2  b2
Sum of Two Squares
The sum of squares is not factorable
Sum of Two Cubes
To factor a sum of two cubes follow the steps below:
1) Create an empty binomial and an empty trinomial as indicated on the right (
)(
2) Take the cube root of the first term of the given expression and put it in the 1st position in the
binomial. Square it and put it in the first position of the trinomial.
**note- ignore all signs until the last step**
3) Take the cube root of the last term of the given expression and
a. put it in the 2nd position in the binomial
b. square it and put it in the last position of the trinomial
4) Find the product of the terms in the binomial and put it in the middle position of the trinomial
5) Arrange the signs as follows ( + )( − + )
Example 3:
8 x 3  27
Answer:
)
(2x + 3) (4x2 - 6x + 9)
Difference of Two Cubes
The only difference between a difference of two cubes and a sum of two cubes is the sign arrangement.
Arrange the signs as follows ( − )( + + )
Example 4:
64 z 6  x 3
Answer:
(4z2 - x) (16z + 4xz + x2)
Answer:
(x+7) (x2 + 2)
Factoring by Grouping
Steps:
1) Find a convenient point in the polynomial to partition
2) Factor within each group
3) Factor across the groups
Example 5:
( x3 + 7x2 ) + ( 2x + 14)
x2( x + 7) + 2(x + 7)
2
Factoring a Trinomial with Leading Coefficient not = 1 in the form ax2+ bx + c
Steps:
1) Multiply the a term by the c term
2) Find the factors of (ac) which will add to the b term
3) Rewrite the b term as the sum of two x terms with coefficients being the factors of (ac)
4) Group the first two terms and the last two terms each in a set of parentheses
5) Factor out greatest common factor from each group
Example 6:
2x3 + x2 + 8x + 4
(2 x3 + x2 ) + (8x + 4)
x2( 2x + 1) + 4(2x + 1)
Answer:
(2x + 1) (x2 + 4)
Multiplying Rational Expressions:
Steps:
 Factor each numerator and denominator completely
 Cancel any like factor in any numerator with any like factor in any denominator
 Multiply the remaining expressions in each numerator
 Multiply the remaining expressions in each denominator
 Reduce if possible
y2  y y2  4

y  2 y 2  3y

y ( y  1) ( y  2)( y  2)


y2
y ( y  3)
Answer:
y( y  1)( y  2)
y( y  3)
Dividing Rational Expressions:
Steps:
 Multiply the first fraction by the reciprocal of the second (KCF)
 Continue using the rules for Multiplying Rational Expressions:
b2
1


2
b 9 3b
b2
3b


(b  3)(b  3) 1
Answer:
 (b  2)
(b  3)
Adding and Subtracting Rational Expressions
Steps:
 Factor all denominators
 Find the least common denominator among all fractions (if none already exists)
 Multiply each denominator and denominator by an appropriate factor to make it equivalent to the LCD
 Combine all numerators (make sure the signs are placed appropriately), simplify and put over LCD
 Reduce if possible
 Combine each set of rational expressions and simplify.
Examples
x
2
x
2
x( x  3)  2( x  3)
x 2  3x  2 x  6






x 2  x  6 x 2  5x  6
( x  3)( x  2) ( x  3)( x  2)
( x  3)( x  2)( x  3)
( x  3)( x  2)( x  3)
x 2  5x  6

( x  3)( x  2)( x  3)
Answer:
( x  6)( x  1)
( x  3)( x  2)( x  3)
3
Functions
1.
Which graph of a relation is also a function?
(a)
2.
(b)
(c)
Determine the Domain and Range of:
f ( x)  3x  4
a.
(d)
b.
g ( x)  x 2  9
3.
Find the x and y intercepts for the following linear equations:
a. x  3 y  7
b. 3x  4 y  12
4.
Write and graph the equation of the line given the following information:
a. m  3, and passes through (3,2)
b. passes through (5,1) and (2,0)
5.
Write & graph the equation of the line that is parallel to y  3x  2 and passes through (4,1).
6.
Write and graph the equation of the line that is perpendicular to y  3x  2 and passes through its x
intercept.
7.
8.
9.
If the following graph is y = f(x), what is the value of f(1)?
(a) -1
(b) -2
(c) 1
Given f(x) = 4x – 7 and g(x) = 2x – x2, evaluate f(2) + g(-1)
Which function is not one to one?
(a)
10.
(d) 2
(b)
(c)
(d)
(c)
(d)
Which function is not onto?
(a)
(b)
4
11.
Given f ( x)  3x  4; g ( x)  x 2  9 , find ( f  g )( x) and ( g  f )( x) .
12.
Find the inverse of the following and state the domain.
a.
f(x) = 5x + 2
4
f ( x) 
b.
x3
Perform the four basic operations on f ( x)  3x  4; g ( x)  x 2  9 and determine the domain of the
result.
13.
14.
a.
Complete the following transformations on graph paper. Label your images.
b.
rx axis (2,1)
D2 [ f ( x)  x 2  1]
c.
R0,90 [ g ( x)  2 x  1]
d.
T2,3 [h( x)  2 x 2  4 x  2]
Review- Absolute Values, Inequalities, Piecewise and Variation
Graph each of the following:
1.
2 x  y  1
2.
2 y  2  2x  4
y
f ( x)  1 x  2  3
3.
y
x
y
x
x
For #4-12, solve and check:
4.
x 1  4
5.
3 y  5
6.
2  3d  d  4
7.
2m  1  2
8.
2x  5  x  1
9.
x  3  3(2 x  1)
10.
x  7x  6
11.
2  3d  d  4
12.
2x  3y  1
y
Graph the following:
2 x 2 if x  1
13.
f ( x)  
2 x  1 if x  1
x
5
14.
1
 x  2 if x  2
g ( x)   2
 x 2 if x  2

y
x
15.

2

f ( x )  2 x
1
 x2 1
2
y
if x  3
if  3  x  2
if x  2
x
16.
If y varies inversely as x, and y is 2 when x is 9, find y when x is 18.
17.
If y varies directly as x, and y is 3 when x is 9, find y when x is 4.
18.
If y varies jointly as x and z, and y is 3 when x is 9 and z is 2, find y when x is 4 and z is 6.
Quadratics
1.
Graph the equation: f ( x)  3x 2  6 x  1
2.
Using the three methods discussed in class, find the roots of g ( x)  6 x 2  19 x  7
3.
Write f ( x)  2 x 2  6 x  9 in standard form and state the coordinates of the vertex.
4.
Determine the nature of the roots of: h( x) 
5.
Write the quadratic equation in standard form that has zeroes at x= 2 and x= -9
6.
What are the sum and product of the roots of g ( x)  3x 2  6 x  9
7.
Write the quadratic equation in standard form that has roots whose sum and product are 3 and -7,
respectively.
8.
Derive: the quadratic formula, the sum of the roots, the product of the roots.
1 2
x  6x  3
2
6
9.
Using the three methods discussed in class, find the roots of g ( x)  2 x 2  4 x  2
10.
Write f ( x)  4 x 2  6 x  9 in standard form and state the coordinates of the vertex.
11.
A farmer has 200 yards or fencing available to create a rectangular enclosure for his livestock. Using a
stream as one side of the enclosure, he uses the 200 yards of fencing create three sides to enclose the
rectangular area. What is the maximum area he can enclose under such conditions? What are the
dimensions of the enclosure? Justify your answers by showing all work. Include a diagram of the
situation.
12.
Eli Manning has been selected to be the first NFL quarterback to travel to Mars. While there, he throws
a football from an elevation of 6 feet at a speed of 132 feet per second. Assuming the acceleration due
to gravity on Mars is 33 feet per second squared (fictitious):
write the equation that models this situation.
determine the maximum height of the football
find the time it takes to reach that height
determine the height when time equals 1.25 sec
find the time(s) it would take to reach a height of 13 feet
a)
b)
c)
d)
e)
13.
14.
Determine the solution of the following quadratic/linear system using the algebraic method.
2x  y  7
y  2 x 2  5x  7
Determine the solution of the following quadratic/linear system using any method.
y  2x  3
y  x 2  2x  1
15.
State the rules for:
rx  axis , ry  axis , rorigin , Tm ,n , Dm
16.
Given g ( x)  3( x  1) 2  4 write equations, in standard form, for:
T1, 3 g ( x), rx  axis g ( x), ry  axis g ( x), rorigin g ( x), D1 g ( x)
8
17.
Solve and graph:
2x  7x  4  0
2
Exponential Functions
1.
The bear population at Yellowstone National Park has been declining since 2005. The rate of decrease
has been determined to be 30% per year. If the initial population was 4000 bears, what is the current
bear population (2009)?
2.
What interest rate compounded monthly is required for a $2,500 investment to triple in 10 years?
3.
Jessica made an investment in an account that compounds continuously at a rate of 5.75%. Determine
how much time, to the nearest year, is required for her investment to double.
4.
What interest rate (to the nearest hundredth of a percent) compounded annually is required for an $8,000
investment to grow to $13,000 in 9 months? Is this a reasonable rate?
7
5.
There is a population of 3,000 gophers living on various golf courses in the tri-state area. If the number
of gophers is increasing at an average rate of 9.2% per year, predict the population after six years.
6.
a.
Solve for x.
3x1  27 x3
7. Simplify: log
i2
12
b.
3x  5 x1
a2
b 2
3
8. Simplify: log b
c.
( x  1)6  729
9. The value of log 4 8  log 4 2 is:
10. The population of Katonah was 11,020 in 2005 and was growing at a rate of 1.1% per year. Assuming that
the population of Katonah continues to grow at the same rate, find Katonah’s projected population in the year
2010.
a) 12,294
b) 13,438
c) 28,583
d) 11, 640
11. In January 1995, the 6,230 residents of Floodplains begin leaving at a rate of 60% per year. If the at rate
stays constant, when will there be less than 100 residents?
a) by 1996
b) by 2000
c) by 2008
d) by 2004
12. A sum of money is invested at a certain annual interest rate, compounded continuously. After 10.2 years,
the investment has doubled. At what interest rate was the money invested?
a) 5.7%
b) 8.3%
c) 6.8%
d) 2.9%
13. Mike invests $2,000 at 4.5% interest compounded continuously. What is the value of his investment after 6
years?
a) $3,297.44
b) $2,619.93
c) $2,866.66
d) $29,759.46
14. In reference to #7, how much money would Mike lose if he chose to compound his interest daily over the
same period of time?
15. Suppose the value of a laptop computer depreciates at a rate of 12.3% a year. Determine the value, to the
nearest cent, of a laptop computer four years after it has been purchased for $3,650.
16. Andre made an investment in an account that compounds continuously at a rate of 5.75%. Determine how
much time, to the nearest year, is required for his investment to triple.
17. What interest rate (to the nearest hundredth of a percent) compounded annually is required for an $18,000
investment to grow to $49,500 in 6 years?
8
18. When Grandma was planning for her retirement, she found an account that earned 4.6% interest
compounded monthly. If she made her investment back on January 1, 1984, in what year will the single deposit
double in her account?
19. There is a population of 3,000 crickets living in Mathematical Meadow. If the number of crickets is
increasing at an average rate of 9.2% per year, predict the population after six years.
20. The number of rabbits living in Mathematical Meadow doubles every month. If there are 80 rabbits present
initially:
(a)
(b)
(c)
Express the number of rabbits as a function of the time t.
Using your answer from part (a), find how many rabbits are present after 1 year?
Using your answer from part (a), find, to the nearest month, when will there be 10,000 rabbits?
Trigonometry
Find the exact value of each expression:
1.
cos 270º
2.
sin 90º
7
and sin   0 .
8
Find the values of the three trigonometric functions for angle  in standard position if a point with the
coordinates (-3, -5) lies on its terminal side.
Without finding , find the exact value of tan  if cos   
3.
4.
Given the following triangle find the measure of angle  exactly
5.

3
6.
7.
8.
6
8
and tan   0 .
9
Find the values of the three trigonometric functions for angle  in standard position if a point with
the coordinates (5, -4) lies on its terminal side.
Without finding , find the exact value of cos  if sin   
Find the length of the missing side and the exact value of the three trigonometric functions of the angle 
in each figure:
a)

b)
c)
3
5

2
7
12
d)


8
e)
13
9
7

11
9
9.
Find the exact value of each expression without using a calculator:
cos 210
sin 150
a.
b.
cos 315
c.
10.
Two adjacent apartment buildings in Geometry Garden Estates share a triangular courtyard. They plan
to install a new gate to close the courtyard that forms an angle of 1048 with one building and an angle
of 4820 with the second building, whose length is 527 feet.
a.
Find, to the nearest tenth, the area of the courtyard.
b.
Find, to the nearest tenth, the length of this new gate.
11.
A lamppost tilts toward the sun at a 2 angle from the vertical and casts a 25 foot shadow. The angle
from the tip of the shadow to the top of the lamppost is 45. Find the length of the lamppost to the
nearest tenth of a foot.
2
45
25 ft
12.
A derrick at the edge of a dock has an arm 25 meters long that makes a 122 angle with the floor of the
dock. The arm is to be braced with a cable 40 meters long from the end of the arm back to the dock. To
the nearest tenth of a meter, how far from the edge of the dock will the cable be fastened?
40 m
25 m
122
13.
Using the picture seen to the right, and rounding to the nearest tenth of a
meter, find the height of the tree.
110
23
120 m
14.
Solve ABC if c = 49, b = 40, and A = 53 (round each answer to the nearest tenth)
15.
Solve ABC, to the nearest tenth, if A = 50, b =12, and c = 14 & find the area.
10
16.
Two forces act upon a body at rest. The first force, 35N, is separated from the second force, 52N, by 63
degrees. Determine the resultant force to the nearest Newton.
17.
The resultant force acting on a body at rest is 90 pounds. If one of the component forces is 70 pounds
and the other is 110 pounds, find the angle separating the resultant force from the larger of the
component forces to the nearest ten minutes.
18.
Find each of the following exactly:
 
sec 
a.
b.
6
19.
c.

,
2
phase shift = 
b. period = 8,
phase shift = -,
vertical translation = 12
vertical translation = -2
Write a cosine function with each given period, phase shift, and vertical translation:
a. period = ,
phase shift = 
b. period = 4,
phase shift =

,
4

,
8
vertical translation = -1
vertical translation = 5
21.
State the amplitude, period, phase shift, and vertical translation for each function:
a.
y  2 cos 0.5x  3
A=
P=
PS =
VT =
b.
y
2
3
cos
x
3
7
A=
P=
PS =
VT =
22.
csc(300)
Write a sine function with each given period, phase shift, and vertical translation:
a. period = 2,
20.
 5 
cot  
 4 
y  cos x     4
   x  
y

x
11
23.
y  2 sin 4x  2 
   x  
y
24.
x 
y  2 cos     1
2 2

x

x

x

x
   x  
y
25.
x 
y  2 sec   
2 2
   x  
y
26.


y  csc  2 x  
2

   x  
y
12
27.
y  cot 2 x
   x  
y

x
SHOW ALL WORK!
USE THE SUM OR DIFFERENCE IDENTITIES TO
USE THE HALF-ANGLE IDENTITIES TO FIND THE
FIND THE EXACT VALUES OF EACH
TRIGONOMETRIC EXPRESSION:
EXACT VALUES OF EACH TRIGONOMETRIC
EXPRESSION:
28. cos 74 cos 44  sin 74 sin 44
29.
tan 110  tan 50
1  tan 110 tan 50
30. cos 345 
32. cos 112.5 
33. sin165
34. tan 105 
31. sin195
Verify Each Identity
35.
1
1

 2 sec 2 x
1  sin x 1  sin x
36.
cos x  sin x
 1  tan x
cos x
37.
1  cos 2 x
 2 csc 2 x  1
2
sin x
38.
ln tan x   ln cot x
39.
2 tan x  sin 2 x
 sin 2 x
2 tan x
40.
cos 4 x  sin 4 x  cos 2 x
2 sin x cos 3 x  2 sin 3 x cos x  sin 2 x
42.
1  sin x
cos x

cos x
1  sin x
41.
13
43.
1  tan 2 x
1  tan 2 x
45.
 1  cot x 


 csc x 
47.
1  sin x
cos x

cos x
1  sin x
49.
cos(u  v)
 tan u  cot v
cos u sin v
51.
tan m  tan n 
53.
cos 2 x 
cot x  tan x 
44.
4 cos 2 x  2
sin 2 x
2
 1  sin 2 x
46.
ln sec x  tan x   ln sec x  tan x
48.
2 sin 2  cos 2   cos 4   1  sin 4 
50.
tan 3t  tan t
2 tan t

1  tan 3t tan t 1  tan 2 t
52.
tan( x  y ) 
log(cos x  sin x)  log(cos x  sin x)  log cos 2 x
54.
tan y  sin y
y
 cos 2
2 tan y
2
55.
1  sin x
 (sec x  tan x) 2
1  sin x
56.
sec 4 s  tan 2 s  tan 4 s  sec 2 s
57.
tan x  cot x
1

sec x  csc x cos x  sin x
58.
tan x  cot x  (sec x  csc x)(sin x  cos x)
59.
tan x  sin x
x
 sin 2
2 tan x
2
60.
cos 3   sin 3  2  sin 2

cos   sin 
2
61.
1  cos 5 x cos 3x  sin 5 x sin 3x  2 sin 2 x
62.
cos 2  cot 2   cot 2   cos 2 
sin( m  n)
cos m cos n
cot x  cot y
cot x cot y  1
FIND ALL SOLUTIONS OF EACH EQUATION FOR THE GIVEN INTERVAL:
63.
4 cos2 x  1, 0  x  360
64.
2 cos x  1  0, 0  x  
65.
sin 2x  sin x  0, 0  x  2
66.
cos2 x  sin2 x  sin x, 0  x  360
67.
sin 2x  3 sin x, 0  x  2
68.
2 sin2 x  cos x  1, 0  x  360
69.
2 sin x  tan x  0, 0  x  2
70.
2 sin2 x  5 cos x  1  0, 0  x  360
71.
cos 2x  3 cos x  1  0, 0  x  180
72.
3 cos 2x  5 cos x  1, 0  x  360
73.
9 tan 2 x  3  0, 0  x  
14
Sequences and Series
Let a1, a2, a3, …,an, … be an arithmetic sequence. Find the indicated quantities.
1.
a1 = -5, d = 4, a2 = ?, a3 = ?, a4 = ?
2.
a1 = -3, d = 3, a2 = ?, a3 = ?, a4 = ?
3.
a1 = -3, d = 5, a15 = ?, S11 = ?
4.
a1 = 1, a2 = 5, S21 = ?
5.
a1 = 7, a2 = 5, a15 = ?
6.
a1 = 1/3, a2 = ½ , a11 = ?, S11 = ?
7.
a1 = 3, a20 = 117, d = ?, a101 = ?
8.
a1 = -12, a40 = 22, S40 = ?
9.
Find g(1) + g(2) + g(3) + … + g(51) if
g(t) = 5 - t
10.
Find the sum of all the even integers
between 21 and 135.
12.
an = n + 3
14.
1
an = 1  
n

Write the first four terms of the given sequences
11.
an = n – 2
13.
n 1
an =
n 1
15.
n+1
an = (-2)
16.
n
n1

 1
an =
2
n
Write the first five terms in each sequence
1
1 
1  n 
3  10 
17.
an = (-1)n+1n2
18.
an =
19.
an = (- ½ )n-1
20.
a1 = 7, an = an-1 – 4, n > 2
21.
a1 = 4, an = ¼ an-1, n > 2
Find the general term of the sequence for which the first four terms are given:
22.
4, 5, 6, 7, …
23.
3, 6, 9, 12, …
24.
1 1 1 1
, , , ,...
2 3 4 5
25.
1, -1, 1, -1, …
26.
–2, 4, -8, 16, ….
27.
1, -3, 5, -7, ….
28.
x,
x2 x3 x4
, , ,...
2 2 2
29.
x, -x3, x5, -x7, ….
15
Write each series in expanded form and then find the sum
(1) k 1

k
k 1
30.
31.
6
34.
 (1) k 1 k
1

k
k 1 10
(2 k 1 )

k
k 1
32.
35.
k
4
33.
36.
 (1)
k
k 1
3
4
k 1
1
 

k 1  3 
5
3
5
(1) k 1 k
x

k
k 1
5
1 k 1
x

k
k 1
37.
Write each series using summation notation with the summing index k starting at k=1
1 1
1
1
 2  3  ...  n
2 2
2
2
38.
12 + 22 + 32 + 42
41.
1 – 4 + 9 – 16 + 25 – 36 + 49 – 64
39.
40.
1
1
1
1
 2  ...  2
2
2
3
n
Find the indicated quantities in each geometric sequence
42.
a1 = -6, r = - ½, a2 = ?, a3 = ?, a4 = ?
43.
44.
a1 = 100, a6 = 1, r = ?
45.
2
, a2 = ?, a3 = ?, a4 = ?
3
a1 = 5, r = -2, S10 = ?
46.
a1 = 81, r =
1
, a10 = ?
3
47.
a1 = 3, a7 = 2,187, r = 3, S7 = ?
Find the sum of each infinite geometric series that has a sum
1
48.
3 + 1 + + ….
49.
3
50.
2–
1
1
+ +…
2
8
51.
a1 = 12, r =
2+4+8+…
5+
5
5
+ +…
2
4
16