Download Algebra II with Trigonometry

Algebra 2 Final Review Guide
To prepare for your final exam, please follow these suggested steps. The more practice
you do, the better prepared you will be for the exam. Be sure to have your calculator
ready the day of the exam with WORKING batteries and more than one sharpened pencil.
Do not expect to borrow materials.
1. Review key vocabulary terms at the beginning of each chapter or use the list at the end
of the chapter in the review. Read the chapter summary at the end of each chapter
several times. Write down any ideas that are unclear.
2. Review the quizzes and tests you have already taken and be sure that you would get 100%
if you took the same test today.
3. Complete the lesson-by-lesson review in the Study Guide and Review section at the
end of each chapter. Complete the practice test at the end of each chapter.
4. Use pages 961-964, 969-970, 972-977 (Ch.7-8, 11-14.2) for extra practice problems
for each section of each chapter. This is the EXTRA PRACTICE section of the back of
your book.
5. Use pages 985-986, 989-992 (Ch.7-8, 11-14.2) for extra practice word problems for
each section of each chapter. This is the MIXED PROBLEM SOLVING section of the
back of your book.
6. For additional practice on areas you may use previous homework assignments. Your
homework should be a solution key to check your work.
7. Use the practice questions in the remainder of this packet for a general review. However,
do not rely solely on these questions.
*REMINDER: Bring all quizzes and tests to the exam for extra credit on the final!! You
must have all of them or no credit. There are 12 and they are listed below:






Quiz square and cube roots
Ch.7 Test
Quiz logs
Ch.8 Test
Packet Sequences and Series
Quiz Statistics






Quiz 13.1-13.3
Quiz Unit Circle
Ch.13 Test
Quiz 13.4-13.5, modeling
Quiz Trig Identities
Test Trig part II
1
Algebra II with Trigonometry
Final Exam Review – 2015
CHAPTER 7 – RATIONAL EXPONENTS
AND
RADICAL FUNCTIONS
 Properties of rational exponents
 Properties of radicals
 Simplifying expressions with rational exponents
 Simplifying expressions with radicals
 Solving radical equations
 Domain and range
 Function operations - addition, subtraction, multiplication, division, composition
 Inverse functions
1. Simplify each of the following using exponent rules (positive exponents only).
-
a)
2
1
6x y
3
1
-
4
1
(16a b c )
e)
-3 ö
æ
ç 2x 8 ÷
ç -4 ÷
ç
÷
è 3y 9 ø
(2x 2y 3 ) -3
d)
x
2 5
y
xy -1 3
3
-6 0 2
b)
4
3
1
c)
(4x y z 4 ) -3
f)
-5 ö
æ
ç 3x 2 ÷
ç -2 ÷
ç 9 ÷
è y ø
3
4
6
-3
2
2. State the domain and range of f(x) = 8 - x .
3. Simplify each radical expression.
a)
3
40a 7b 3c 15
b)
4
c)
5 45 + 6 80
d)
4 24 - 3 40 + 150
e)
3
32x 2 · 4x 4
f)
æ 3 - 3 2 ö æ 1 - 23 6 ö
è
øè
ø
3
54x 2
h)
4
g)
3
3
2x
256x 6y 8z 10
3
3
16a 5
3b
3
4. Solve each of the following equations.
a)
9x 3 5 = 72
c)
(4x + 1) - 5 = -2
1
3
d)
3
e)
(x + 4) 2 = -64
8x + 12 = 6
b)
f)
3
3
3x + 2 - x - 9 = 0
2x + 43 - 4 = x
4
CHAPTER 8 – EXPONENTIAL
AND
LOGARITHMIC FUNCTIONS
 Converting between logarithmic and exponential forms
 Evaluating logarithmic expressions
 Solving logarithmic and exponential equations
1. Write in logarithmic form.
a) 72 = 49
b) 34 = 81
c) 641/3 = 4
2. Write in exponential form.
a) log6 216 = 3
b) log 100000 = 5
3. Solve each of the following exponential equations.
a) 3x · 35 = 37
b) (5x)3 = 512
4. Solve each exponential equation.
a) 82+x = 2
b)
c) log5 x = 6
c)
4x = 43x - 6
2x = 13
5
c)
3x+2 + 1 = 15
d)
8(3x) - 1 = 22
5. Solve each of the following logarithmic equations.
a) log (5x) = 4
b) log4 x= -1.5
c)
logx 5 = 1/2
e)
logm (x + 1)2 - logm 4 = 0
d)
log 4 (x2 – 17) = 3
6
CHAPTER 11 – SEQUENCES
AND
SERIES
 Arithmetic and Geometric sequences
 Writing rules (explicit formulas) for sequences
 Write recursive formulas for sequences
 Find sums of series and infinite geometric series
 Binomial Expansion
1. Find the sum of the series
5
å(i - 5)
i=2
Write a rule for the nth term of the arithmetic sequence.
2. 4, 7, 10, 13, …
3. d = 6, 𝑎7 = 31
4. CARS Chris buys a $20,000 car. He makes a $4,400 down payment and then pays a
$325 monthly payment. Write a rule for the total amount of money paid on the car after
n months.
Write a rule for the nth term of the geometric sequence.
5. 512, 64, 8, 1, …
6. r = 3, 𝑎5 = 162
7
7. Find the sum of the series
9
å8(2)
i-1
i=1
8. Find the sum of the infinite geometric series, if it exists:
i-1
æ 5ö
å3çè 8 ÷ø
i=1
¥
9. Write the first five terms of the sequence. 𝑎1 = 4, 𝑎𝑛 = 𝑎𝑛−1 + 9
10. Write a recursive rule for the sequence. 7, 13, 19, 25, 31,…
8
CHAPTER 12 – STATISTICS
 Find measures of central tendency and dispersion
 Use normal distributions
 Select and draw conclusions from samples
1. The list shows the average price of a gallon of gasoline each year from 1994 to
2004. Find the mean, median, mode, range, and standard deviation of the prices.
$1.04, $1.13, $1.13, $1.26, $1.13, $.97, $1.30, $1.47, $1.14, $1.47, $1.59
2. The heights of 1800 teenagers are normally distributed with a mean of 66 inches
and a standard deviation of 2 inches.
(a) About how many teens are between 62 and 70 inches?
(b) What is the probability that a teenager selected at random has a height
greater than 68 inches?
9
3. The cholesterol level for adult males of a specific racial group is normally
distributed with a mean of 158.3 and a standard deviation of 6.6. How many of the
900 men in a study have cholesterol between 145.1 and 171.5.
4. The time a fire department takes to arrive at the scene of an emergency is
normally distributed with a mean of 6 minutes and a standard deviation of 1 minute.
What is the probability that the fire department takes at most 8 minutes to arrive
at the scene of an emergency?
5. A town council wants to know if residents support having an off-leash area for dogs
in the town park. Eighty dog owners are surveyed at the park. Identify the type of
sample described. Then tell if the sample is biased. Explain.
6. If you find 100 students, half of which have part-time jobs, and compare their
grade-point averages, is that a survey, observational study, or experiment?
10
CHAPTER 13 - TRIGONOMETRY





Angles
SOHCAHTOA
Laws of sines and cosines
Converting degrees and radians
Evaluating trig functions
1. Given the triangle below, find each trig function.
sin  = __________
tan  = __________
cot  = __________
cos  = __________
sec  = __________
csc  = __________
2. Given csc  =
16
12

7
11
, find the following.
cos  = __________
sec  = __________
cot  = __________
3. Evaluate the six trigonometric ratios for 𝜃 if the terminal ray of 𝜃 contains the
point P(-6, -8).
4. Evaluate the five remaining trigonometric ratios for 𝜃 given that cot 𝜃 =
lies in Quadrant III.
8
15
and 𝜃
11
5. Find the missing value. Round angles to whole degree and sides to tenths.
A)
B)
11
x
9
41
63
x
`
C)
D)
25
5
15
x
x
6. Convert each of the following.
a)
–120
28
b)
440
c)
24
9
d)
15
7
7. Find the reference angle for the given angle:
A) 156
B)
23p
9
C) -172
8. Find one positive and one negative coterminal angle for each of the following
angles.
5
19 
a) 15
b) 421
c)
d)
12
7
12
9. Draw an angle with the given measure in standard position.
A) 207
D)
-
7p
10
B)
E)
-320
C)
9p
6
F) 505
19p
4
10. Evaluate each of the following providing EXACT answers. Strategy: use unit circle.
A) tan 135
D) sin
G) sin
J) tan
M) cot
4p
3
-11p
4
17p
6
5p
2
B) sec -315
2p
E) cos
3
13p
H) cos
K) cot
6
5p
N) sec
4
9p
2
C) sin 240
p
F) tan
4
10p
I) tan
L) sec
3
17p
O) csc
3
5p
2
13
11. Solve each of the following special right triangles. Give side lengths in simplest
radical form.
a) C = 90, B = 60, a = 8
b) C = 90, B = 45, a = 8
12. 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
13. 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?
14
14. Find the requested information for the given triangles. Draw diagrams.
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
15
CHAPTER 14 - TRIG (Cont.)
 Model real world situations
 Find exact values of trig functions using trig identities
 Simplify trig identities
 Verify trig identities
1. A Ferris wheel in China has a diameter of approximately 520 feet. The height of a
compartment h is a function of time t. It takes about 30 seconds to make one
complete revolution. Assume a person gets on the wheel at the bottom of the
wheel. Write a function and sketch a graph for this situation.
2. In a city, the average high temperature for each month is shown in the table.
(a) Sketch a graph
(b) Describe the period of the function
(c) Find a sine regression equation to model the data.
3
3. Find the exact value for csc 𝜃 , 𝑖𝑓 cos 𝜃 = 5 𝑎𝑛𝑑 0° < 𝜃 < 90°
16
4. Find the exact value for sec 𝜃 , 𝑖𝑓 tan 𝜃 = −1 𝑎𝑛𝑑 270° < 𝜃 < 360°
1
5. Find the exact value for tan 𝜃 , 𝑖𝑓 sin 𝜃 = 2 𝑎𝑛𝑑 0° < 𝜃 < 90°
6. Simplify 𝑠𝑒𝑐𝜃 tan2 𝜃 + 𝑠𝑒𝑐𝜃
7. Simplify 𝑐𝑜𝑡𝜃𝑠𝑒𝑐𝜃
8. Simplify sinθ(1 + cot 2 𝜃)
9. Verify cos2 𝜃 + tan2 𝜃 cos2 𝜃 = 1
17
10. Verify that 1 + sec 2 𝜃 sin2 𝜃 = sec 2 𝜃
11. Verify that (𝑠𝑖𝑛𝜃 − 1)(𝑡𝑎𝑛𝜃 + 𝑠𝑒𝑐𝜃) = −𝑐𝑜𝑠𝜃
18
GRAPHING REVIEW
 Graph a given function
 Identify domain and range
 Identify function shifts
 Identify equations of asymptotes
 Describe right and left end behaviors of functions
 Find vertices, axes of symmetry, end points, turning points, etc for a given function
1. Graph
a) f ( x)  3 cos( 2 )
1
3
b) g(x) = - sin q + 4
2 2
c) h(x) = 6 tanq
2. Sketch the graph and complete the following information for:
f(x) = x - 2 + 3
Horizontal Shift _____________
Vertical Shift _____________
Domain _____________
Range _____________
Table of Values
19
3. Sketch the graph and complete the following information for:
f(x) = -3 x + 4 - 1
Horizontal Shift _____________
Vertical Shift _____________
Domain _____________
Range _____________
Table of Values
4. Sketch the graph and complete the following information for:
Equation of Asymptote __________
x-intercept _____________
y-intercept _____________
Domain _____________
Range _____________
Table of Values
20
5. Sketch the graph and complete the following information for:
æ1ö
f(x) = ç ÷
è2ø
x +1
+2
Equation of Asymptote __________
x-intercept _____________
y-intercept _____________
Domain _____________
Range _____________
Table of Values
6. Sketch the graph and complete the following information for:
f(x) = log2 (x + 3) - 1
Equation of Asymptote __________
x-intercept _____________
y-intercept _____________
Domain _____________
Range _____________
Table of Values
21
Match each function below to the correct graph.
æ1ö
f(x) = ç ÷
è3ø
x +2
-1
_____
f(x) = 3 x + 2 - 3 _____
(A)
(B)
æ1ö
f(x) = ç ÷
è3ø
x +1
-1
_____
f(x) = -3 x + 2 + 3 _____
(C)
_____
f(x) = log3 (x + 4) - 1 _____
(D)
(E)
(F)
(G)
(H)
(I)
(J)
(K)
(L)
22