Advanced Algebra / Trigonometry – 2015 Semester 2 Final Exam Review
Monday
June 15th
Exam Day
Bell Sched.
Tuesday
June 16th
Wednesday
June 17th
Thursday
June 18th
Friday
June 19th
No Exams
8:00 – 9:40
Per. 2/A
Per. 6/B
Advisory
Last Day of
School
8 Period Day
9:50 -11:30
1
5
8
11:40 - 1:20
3/C
7/D
4
1:20 – 3:05
Lunch/
study hall in
cafeteria
Lunch/
study hall in
cafeteria
Lunch/
Make-up Exams
Field Day
1:15 dismissal
SECOND SEMESTER STANDARDS
Unit 5: Polynomials and rational expressions
1. Add, subtract, multiply, and divide polynomials and functions; determine appropriate operation(s) for
given situations
2. Simplify rational expressions, including those involving addition, subtraction, multiplication, and division
3. Solve equations involving rational expressions; model problem situations using rational functions
4. Compose two or more functions using symbolic representations, graphs, or tables
Unit 6: Exponential functions and logarithms
5. Model and answer questions about exponential growth and decay situations using graphical, tabular,
and algebraic representations. Move between representations with and without technology.
6. Understand the inverse relationship between exponents and logarithms and use this relationship to
solve one-step problems involving logarithms and exponents.
7. Manipulate logarithmic expressions using properties of logarithms including addition, subtraction,
exponent and common identities.
8. Solve exponential equations using logarithms, including natural and common logs as appropriate.
9. Apply logarithms to situations involving continuous compounding (e), natural events (earthquakes),
science and measurement (pH, sound).
Unit 7: Function transformations
10. Given a parent function and another function in the same family, describe the transformation(s) that
occurred and graph the transformed function.
11. Given a parent function a description of transformation(s), write the transformed function rule and
graph the transformed function.
Unit 8: Trigonometry
12. Solve right triangles by selecting and applying the appropriate tool.
13. Derive and apply Law of Sines and Law of Cosines to solve non-right triangles.
14. Measure any angle using degrees and radians, including negative and angles over 360 degrees
15. Derive the unit circle, finding the exact length of sides of special right triangles 45-45-90 and 30-60-90.
16. Graph sin(x) and cos(x) with any transformation.
1
Complete practice questions on a separate sheet.
UNIT 5: Polynomials and Rational Expressions
1) Factor using GCF: −6𝑦 10 − 4𝑥𝑦10 − 8𝑥 2 𝑦 8 + 10𝑦 7
2) Rewrite in standard polynomial form (expand): (𝑥 – 3)3
Simplify. Remember to check for invalid (extraneous) solutions.
3) (6𝑥 3 + 8𝑥 2 − 18𝑥 − 20) ÷ (2𝑥 + 4)
4)
𝑥−6
𝑥 2 +2𝑥−24
÷
𝑥+1
5)
𝑥−4
45𝑥 2
𝑥−9
∙
𝑥 2 −5𝑥−36
5𝑥−1
6)
3𝑥 3 +12𝑥 2
𝑥 2 −3𝑥+2
−
3
2𝑥−4
Solve. Remember to check for invalid (extraneous) solutions.
7)
9)
1
𝑘
1
8)
𝑘 2 +𝑘
1
𝑟+3
=
𝑟+4
𝑟−2
+
6
𝑟−2
The speed of a freight train is 20 mph slower than the speed of a passenger train. The passenger train travels
440 miles in the same time that the freight train travels 280 miles. Find the speed of the each train (recall:
𝑡𝑖𝑚𝑒 =
10)
=5+
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
).
𝑠𝑝𝑒𝑒𝑑
𝑓(𝑥) = 𝑥 2 − 𝑥 and 𝑔(𝑥) = 25 − 2𝑥 2 and ℎ(𝑥) = √𝑥 2 − 10𝑥 + 9 and 𝑘(𝑥) = 4𝑥 − 1
a. (𝑓° 𝑔)(5) =
b. (ℎ° 𝑘)(1) =
c. (𝑘° 𝑔)(𝑥) =
d. (𝑔° ℎ)(2𝑥) =
UNIT 6: Exponential functions and logarithms
1. If you deposit $1,250 into an account paying 3.5% interest compounded weekly, to the nearest cent, how much
money will be in the account after 8 years?
2. To the nearest cent, how much money would you need to deposit into an account paying 5.25% annual interest,
compounded continuously, in order to have $15,000 in the account after 10 years?
3. Change each logarithmic form to an equivalent exponential form or vice versa:
1
log 4 64 = −3
ln 50 ≈ 3.912
53 = 125
15 = √225
4. Evaluate each log. Write your answers in exact form – no rounding.
1
log 9 9 ∙ log 7 (49)
5. For each equation find the value of x.
log 4 (log 2 (log 3 81))
log 3 𝑥 = 5
log 3 81 = x
2
6. Condense each expression, and write it as a single logarithm. Then simplify, if possible.
a) log5x – log52
b) log82 + log832
c) log95 + log9y – log94
d) 3logbz + logby – 4logbz
7. Completely expand each expression as the sum or difference or logarithms.
a) log3(15q)
 3a 2
c) log 9 
 7
b) log 8  64 
 y



8. Solve each equation for 𝑥; round your answer, if necessary, to two decimal places
2ln(4𝑥) = 15
log(1 − 𝑥) = −1
ln𝑥 − ln(𝑥 − 3) = ln2
9. If money is invested in an account that compounds interest continuously at an annual rate of 5%, the
amount in the account, A, is determined by the formula 𝐴 = 𝑃𝑒 0.05𝑡 where P is the principal initially
invested and t is the time in years. To the nearest tenth of a year, determine the amount of time
required to double an investment.
10. Katya is a ranger in a nature reserve in Siberia, Russia, where she studies the changes in the reserve's
bear population size over time. She found that the number of bears, B, in the reserve at t years since
the beginning of the study can be modeled by the following function:
𝐵(𝑡) = 5000 ∙ 2−0.05𝑡
How long will it take the bear population to get to 2000? Round your answer to 2 decimal places.
11. The magnitude of an earthquake is measured on a logarithmic scale called the Richter scale. The
magnitude M is given by 𝑀 = log10 𝑋, where 𝑋 represents the amplitude of the seismic wave caused
by ground motion. A) The Chilean Earthquake of 1960 and registered 9.5 on the Richter scale. The
explosion at the Chernobyl nuclear power plant in 1986 registered 3.87 on the Richter scale. How
many times greater was the Chilean earthquake than the Chernobyl explosion?
B) The Loma Prieta earthquake in San Francisco in 1989 registered 8.3 on the Richter scale. In the
same year, an earthquake in South America was four times stronger. Find the difference in the Richter
scale magnitudes of the earthquakes.
UNIT 7: Function transformations
1. The function 𝒈(𝒙) is the function 𝒇(𝒙) = 𝟐(𝒙 − 𝟏)𝟐 + 𝟑 transformed by a horizontal shift of 3 units to
the right and
a vertical shift of 2 units down. Which of the following best represents 𝒈(𝒙)?
𝐴. 𝑔(𝑥) = 2(𝑥 + 1)2 + 6
C.
𝑔(𝑥) = 2(𝑥 − 4)2 + 1
𝐵. 𝑔(𝑥) = 2(𝑥 − 3)2 − 2
D. 𝑔(𝑥) = 2(𝑥 + 2)2 + 3
2. The function 𝑔(𝑥) is the function 𝑓(𝑥) = 2(𝑥 − 1)2 + 3 transformed by a horizontal shift of 2 units to
the left and a
vertical shift of 2 units down and reflected over the x-axis. Which of the following
best represents 𝑔(𝑥)?
A. 𝑔(𝑥) = −2(𝑥 − 3)2 + 1
C. 𝑔(𝑥) = 2(𝑥 + 1)2 + 1
B. 𝑔(𝑥) = −(𝑥 + 2)2 − 2
D. 𝑔(𝑥) = −2(𝑥 + 1)2 + 1
3
Function
Transformation
Description
3.
Parent
One:
𝒇(𝒙) = 𝒙𝟐
Table
Parent
Function
(𝒙, 𝒇(𝒙))
Graph
(Graph transformed points)
Transformed
Function
(−2, 4)
Transformed
Function
𝑔(𝑥) =– 3𝒙𝟐 + 12
(−1, 1)
Two:
(0, 0)
(1, 1)
(2, 4)
Three:
4. The functions 𝒇(𝒙) and 𝒈(𝒙) are defined in the function tables below. The function 𝒈(𝒙) is a
transformation of 𝒇(𝒙). Which of the following best describes 𝒈(𝒙) in terms of 𝒇(𝒙)?
A
B
C
D
E
𝒙
-2
0
1
2
5
𝒇(𝒙)
6
3
4
1
0
A
B
C
D
E
𝒙
1
3
4
5
8
𝒈(𝒙)
4
1
2
-1
-2
A. 𝒈(𝒙) = 𝒇(𝒙 − 𝟑) − 𝟐
C. 𝒈(𝒙) = 𝒇(𝒙 − 𝟑) + 𝟐
B. 𝒈(𝒙) = 𝒇(𝒙 + 𝟑) − 𝟐
D. 𝒈(𝒙) = 𝒇(𝒙 + 𝟑) + 𝟐
5. 𝑔(𝑥), the dashed graph, is a transformation of 𝑓(𝑥) = 2𝑥 .
Which of the following function rules best describes 𝑔(𝑥)?
A. 𝑔(𝑥) = −2𝑥+5 − 2
B. 𝑔(𝑥) = −2𝑥−5 − 2
C. 𝑔(𝑥) = 2−𝑥−5 − 2
D. 𝑔(𝑥) = 2−𝑥+5 − 2
4
UNIT 8: Trigonometry
1. Solve for 𝑥. Give your answer in simplified radical form.
b.
a.
b.
2. A cross-section of the cornea of the eye, a circular arc, is shown in the
figure. Find the arc radius 𝑅 given the chord 𝐶 = 11.8 millimeters and
the central angle 𝜃 = 98.9°. Round your answer, if necessary, to one
decimal place.
3. Two lookout posts, A and B (10.0 miles apart), are established along a coast to watch for illegal ships coming
within the 3-mile limit. If post A reports a ship S at angle 𝐵𝐴𝑆 = 37° and post B reports the same ship at angle
𝐴𝐵𝑆 = 20°, how far is the ship from post A? Round your answer, if necessary, to two decimal places.
4. A five meter long ladder leans against a wall, with the top of the ladder being four meters above the ground.
What is the approximate angle that the ladder makes with the ground? Round your answer, if necessary, to one
decimal place.
̅̅̅̅ . Round to the nearest tenth.
5. Find side length 𝐵𝐶
6. A baseball diamond is a square with the bases set at 90° angles. If the bases are 90 feet apart, how far is it from
home plate to second base? Give your answer in simplified radical form.
7. If we take the ray pointing along the positive x-axis (the initial side) and rotate it counterclockwise by
in which quadrant will the terminal side lie? Justify your answer with a sketch or other reasoning.
13𝜋
3
radians,
8. Find the angle depicted with an arrow in degrees AND radians.
5
9. Find the EXACT values of the following using a unit circle. (YOU MUST BE ABLE TO DERIVE THE UNIT CIRCLE)
cos(−135°) = _______
9π
cos( 4 ) = _______
sin (90°)= _______
sin(
10. 𝑦 = 2 sin(𝑥) − 1
7π
)
4
= ______
cos(240°) = _______
π
cos(− 6 ) = _______
Transformation type(s): ______________________________________________
Attribute(s) affected: ________________________________________________
Parent 𝒚 = 𝐬𝐢𝐧(𝒙)
Transformation
Minimum:
Minimum:
Maximum:
Maximum:
Midline:
Midline:
Amplitude:
Amplitude:
Period:
Pattern of critical points:
Period:
Pattern of critical points:
11.
𝑦 = −2 cos(𝑥)
Transformation type(s): ______________________________________________
Attribute(s) affected: ________________________________________________
Parent 𝒚 = 𝐜𝐨𝐬(𝒙)
Transformation
Minimum:
Minimum:
Maximum:
Maximum:
Midline:
Midline:
Amplitude:
Amplitude:
Period:
Pattern of critical points:
Period:
Pattern of critical points:
12.
𝑦 = − sin(𝑥) − 3
6