ACP ALGEBRA II FINAL EXAM REVIEW PACKET 2015-2016

Name _____________________________ Per__ Date______

This review packet includes new problems and a list of problems from the textbook. The answers to the odd problems

from the textbook can be found in the back of the textbook and a detailed solution on www.hotmath.com.

Section 6-4 Completing the Square

Vocabulary: Completing the square, Square Root Property

p.310-311 #5-11odd,15-21odd,25-37odd,45

1. Solve the equation by the Square Root Property.

a. 2 14 49 25x x

b. 2100 80 16 9x x

2. Find the value of c that makes each trinomial a perfect square. Then, write the trinomial as a perfect square.

a. 2 12x x c

b. 2 15x x c

3. Solve the equation by completing the square.

a. 2 2 6 0x x b.

22 4 8 0x x

Section 6-6 Analyzing Graphs of Quadratic Functions

Vocabulary: Vertex form

p.325-327 #3-33odd,39-47odd

4. Write the following quadratic function in vertex form, if not already in that form. Then identify the vertex, the

equation of the axis of symmetry, and the direction of the opening.

a. 21

1 23

y x

b. 2 6 1y x x

c. 28 3y x

d. 22 20 35y x x

e. 23 48y x x

5. Write an equation (in vertex form and in 2(y ax bx c ) for the parabola with the following conditions.

a. Vertex is at 3,3 and passes through 5,27 .

b. Vertex is at 4,3 and passes through 3,6 .

c. Vertex is at 3, 2 and passes through 1,8 .

Section 8-3 Circles

Vocabulary: Center of a circle, circle, Distance Formula, general form of an equation of a circle, Midpoint Formula,

standard form of an equation of a circle (center-radius form), tangent

p.426-431 #3-13odd,17-25odd,29-45odd

1. Write the equation for a circle that satisfies the given conditions.

a. Center 10, 6 and radius 9 units.

b. Endpoints of the diameter 5,2 and 3,6 .

c. Center 8, 9 and passes through 21,22 .

d. Center at 8, 7 , and tangent to the y-axis.

e. Center at 5,7 , and tangent to the x-axis.

2. Find the center and radius of the given equation.

d. 22 2 4x y

e. 2 2

4 5 75x y

f. 2 2 14 6 50 0x y x y

g. 2 2 9 8 4 0x y x y

3. Find the center and radius of a circle with the given equation and then graph the circle.

2 2 22 28 196 0x y x y

4. The equation of a circle in the xy-plane is shown below. What is the diameter of the circle? (SAT)

𝑥2 + 𝑦2 − 6𝑥 + 8𝑦 = 144

Section 7-1 Polynomial Functions

Vocabulary: Degree of a polynomial, leading coefficient, polynomial function, polynomial in one variable

p.350-351 #5-11odd,17,19,23-37odd

6. State the degree and the leading coefficient of each polynomial in one variable.

a. 3 23 2 5y y y

b. 4 57 8 6 4b b b

7. Find 2p and 1p for each function

a. 37 4 1p x x x

b. 23 8 5p x x x

8. If 3 22 3 4q x x x and 2 1r x x x , find each value.

a. 2q a

b. 3r a

c. ( 2 )r a

d. 3r a

e. 2q a r a

Section 7-2 Graphing Polynomial Functions

Vocabulary: relative maxima, relative minima, zeros of the polynomial

p.356-358 #5-29odd,41-45odd

1. Using your graphing calculator, sketch the graph of the following polynomials. State the relative maxima(s) and

minima(s), find all zeros. (round answers to the hundredths).

a) 4 3( ) 2 5f x x x

b) 3 2( ) 3 3 4f x x x

c) 5 2( ) 2 2f x x x

d) 4 3 23

( ) 3 44

f x x x x

2. State whether the given graph is an even-degree polynomial or odd-degree polynomial, state the number of real

zeros of the polynomial. Describe the end behavior.

a) b)

c) d)

3. If the function f has five distinct zeros, which of the following could represent the complete graph of f in the

xy-plane? (SAT)

4. In the xy-plane, the graph of function f has x-intercepts at -3, -1, and 1. Which of the following could define f ? (SAT)

Section 7-4 The Remainder and Factor Theorems

Vocabulary: Depressed Polynomial, Factor Theorem, factor, zero

p.368-369 #7-9odd,21-29odd

1) Given a polynomial and one factor, find the remaining factors of the polynomial. Some factors may not be

binomials.

a) 3 27 26 72; 4x x x x

b) 4 3 211 9 18; 1x x x x x

c) 3 23 4 5 2;3 1x x x x

d) 3 24 12 3; 3x x x x

e) 3 26 5 3 2;3 2x x x x

Section 7-5 Roots and Zeros and 7-6Ration Zero Theorem

Vocabulary: Fundamental Theorem of Algebra, rational zeros, Rational Zero Theorem,

p.375-376 #5-11odd,13-35odd; pages 380-381 #5-33odd

1. Solve each equation. State the number and types of roots.

a) 5 7 0x

b) 23 10 0x

c) 4 3 22 23 60x x x x

d) 3 2 3 3 0x x x

2. State the all of the possible zeros of each function. Use your calculator to find one zero, then factor or use

synthetic division to find the remaining zeros. (answers should be exact value – no decimals)

a) 4 3 2( ) 3 5 2 7 5f x x x x x

b) 4 3 2( ) 20 16 11 12 3f x x x x x

c) 3 2( ) 3 6 5 8f x x x x

d) 3 2( ) 16 79 114f x x x x

e) 3 2( ) 2 2 34 30f x x x x

f) 3 2( ) 10 17 7 2f x x x x

g) 4 3 2( ) 6 29 40 7 12f x x x x x

3. Write a polynomial function of least degree with integral coefficients that has the given zeros.

a) 3,1,2

b) 5, 3,3,5

c) 3, 5,1

d) 2, 2,0,4

Section 2.7Graphing inequalities

Page 98 #5-23odds

1. Graph each inequality

a) 5 3 15x y b) 3 2 10x y c) 2x y

Section 3-3Solving Systems of Inequalities by Graphing

Pages 125-126 #5-9odd, 13-29odd

1. Solve each system of linear inequalities by graphing

a) 3

2

2

xy

y x

b) 3 3

2 4

x y

x y

c)

1

1

x

y

d)

2

4

x

x

e)

3 4

2 3 6

y x

x y

Section 3-4 Linear Programming

Vocabulary: constraints, feasible region, bounded, unbounded, vertices, maximize, minimize

Pages 132-133 #3-37odd

1. Graph the following constraints. Determine the vertices of the critical region, then test these vertices to determine

the point that maximizes AND minimizes the system.

0

0

8

14

5 50

( , ) 5 4

x

y

y

x y

x y

f x y x y

2. In the x-y plane, if a point with coordinates (a,b) lies in the solution set of the system of inequalities above, what is the

maximum possible value of b? (SAT)

𝑦 ≤ −15𝑥 + 3000

𝑦 ≤ 5𝑥

Section 5-8 Radical Equations

Vocabulary: Radical equation, extraneous solution

p.266 #4-30odd

1. Solve and check each equation.

a) 5 20 4 3x

b) 10 9 2 5x

c) 2 5 6 1 5x

d) 1 2 7x x

e) 2 1 5x x

Section 9-1 Multiplying and Dividing Rational Expressions

Vocabulary: Complex fraction, rational expression

p.472-478 #5-43odd

1. Simplify

f.

32

2 2

3

9

x y

x y

g. 2 2

x y

x y

h.

2 3 5

5 4 7 3

5 23

3 42

x y x y

a b a b

i.

2 2

2 2

5 10 75 2 10 28

4 24 28 7 10

x x x x

x x x x

j.

2

2

2

8 8

8 8

26 31

x

x

x

x x

2. Under what conditions is 2

5

12 28

xk x

x x

undefined?

Section 9-2 Adding and Subtracting Rational Expressions

Vocabulary: Least common multiple

p.479-484 #5-11odd,15-39odd

1. Find the LCM of each set of polynomials

a. 212 , 6y y

b. 3 4 2 228 , 16 , a c b b c

c. 2 22 , 4x x x

d. 2 22 3, 2 5 3t t t t

2. Simplify

e. 2

1711

a

b

f. 3 2 1

4 5 2m m m

g. 7 4

8 8y y

h. 2

8 3

2 4

y

y y

i. 2

2

4 3 6

m

m m

j. 2 2

1 5

9 20 10 25h h h h

Section 9-6 Solving Rational Equations

Vocabulary: Lowest common denominator, rational equation

p.505-511 #5-27odd (change any inequality sign to an equal sign and you can still solve the problem)

1.Solve and check the solution(s).

a. 4

27 14s

b. 4 5

2 8

g g

g g

c. 2

2 1

36 6 6

a

a a a

d. 2

3 1 7

5 6 2 3

x

x x x x

2) What is one possible solution to the equation 24

𝑥+1−

12

𝑥−1= 1 ? (SAT)

Section 10-1 Exponential Functions

Vocabulary: Exponential decay, exponential equation, exponential function, exponential growth

p.523-530 #3-9odd,13-17odd,21-31odd,39-55odd (change any inequality sign to an equal sign and you can still solve the

problem)

1. Determine whether the function is exponential growth or decay.

k. 8x

f x

l. 2

53

x

f x

2. Solve

a. 41

819

n

n

b. 5 66 1,296n

3. Simplify each expression.

a. 3 25 5

b. 5 10 54 4

c. 10

27

4. In planning maintenance for a city’s infrastructure, a civil engineer estimates that, starting from the present, the

population of the city will decrease by 10 percent every 20 years. If the present population of the city is 50,000, which

of the following expressions represents the engineer’s estimate of the population of the city t years from now? (SAT)

A) 50,000(0.1)20𝑡

B) 50,000(0.1)𝑡

20

C) 50,000(0.9)20𝑡

D) 50,000(0.9)𝑡

20

5. Sketch the graph of the given function. Then, state the function’s domain and range.

1.2 3x

y

Section 10-2 Logarithms and Logarithmic Functions

Vocabulary: Exponential form, logarithm, logarithmic equation, logarithmic expression, logarithmic form, logarithmic

function

p.531-538 #5-17odd,21-41odd,47-59odd (change any inequality sign to an equal sign and you can still solve the

problem)

6. Write the given equation in logarithmic form. 5 1

959,049

7. Write the given equation in exponential form. 9

1log 4

6,561

8. Evaluate the logarithmic expression.

d. 12log 144

e. 2

1log

8

f. 8log 32,768

g. 7

5log 5

9. Solve and check your solution(s).

a. 3log 6x

b. 2

10log 1 1x

c. 25

3log

2n

d. 2 2log 4 10 log 1y y

e. log 121 2b

Section 10-3 Properties of Logarithms

Vocabulary: Product property of logarithms, quotient property of logarithms, power property of logarithms, extraneous

solution.

p.544-545 #5-31odd

10. Use 3log 28 3.0331 and 3log 4 1.2619 to approximate the value of each expression.

a. 3log 36

b. 3log 7

c. 3log 256

11. Solve and check each equation.

h. 3 3 32log log 4 log 25x

i. 2 2log log ( 2) 3x x

j. 3 3 3log ( 3) log (4 1) log 5x x

k. 5 5log ( 3) log (2 1) 2x x

l. 4 42log ( 1) log (11 )x x

m. 6 6log (2 5) 1 log (7 10)x x

n. 2 2 24log log 5 log 405x

o. 2 23log 2log 5 2x x

p. ln ln3 12x x

q. 2ln( 12) ln ln8x x

Section 10-5 Base e and Natural Logarithms

Vocabulary: Natural base e, natural logarithm.

p.557-558 #5-51odd

12. Write an equivalent exponential or logarithmic equation.

a. 3 45xe

b. 5 0.2xe

c. ln(4 ) 9.6x

d. ln 8x

e. ln0.0002 x

13. Evaluate each expression

a. 3ln e

b. ln 42e

c. ln ye

d. ln 2xe

14. Solve and check

a. 8 50xe

b. ln(5 3) 3.6x

c. 32 5 2xe

d. ln( 3) 5 2x

e. 16 3 21xe

10-6 Exponential Growth and Decay

Vocabulary: exponential growt,. exponential decay, formulas for exponential growth, formulas for exponential decay

p.563-564 #4-17 odd

15. Solve

a. Able Industries bought a scanner/fax machine for $825. It is expected to depreciate at a rate of 18% per year.

What will the value of the scanner/fax machine in 31/2 years?

b. For a certain strain of bacteria, k is 0.872 when t is measured in days. How long will it take 9 bacteria to

increase to 738 bacteria?

c. Radium-226 decomposes radioactively. It’s half-life, the time it takes for half of the sample to decompose, is

1800 years. Find the constant k in the decay formula for this compound.

d. The population of a city 10 years ago was 45,600. Since then, the population has increased at a steady rate

each year. If the population is currently 64,800, find the annual rate of growth for this city.

e. In 10 years, the mass of a 200-gram sample of an element is reduced to 100 grams. This period is called half-

life. Find the constant k for this element.

f. Assume $100 is deposited in a savings account. The interest rate is 6% compounded continuously. When will

the money be double the original amount? If the money is to be doubled, the final amount will $200.

Section 13-1 Right Triangle Trigonometry

Vocabulary: Angle of depression, angle of elevation, cosecant, cosine, cotangent, secant, sine, SOH-CAH-TOA, special

angles, solving a right triangle, tangent, trigonometric functions, trigonometry

p.701-708 #5-25odd,29-43odd,49

1. Find the values of the six trigonometric functions for the angle .

15 17

2. Find the values of the six trigonometric functions for the angle .

2 5

16

3. Solve ABC by using the measurements below. Round measures of sides to the nearest tenth and measures of

angles to the nearest degree. (note: picture is not scaled)

A

c b

B C

a

a. 90 , 40 , and 10B A a

b. 90 , c 14, and 13B a

4. Solve PQR by using the measurements below. Round measures of sides to the nearest tenth and measures of

angles to the nearest degree. (note: picture is not scaled)

P

r q

Q R

p

a. 90 , 80 , and 15Q R r

b. 90 , 19, and 31Q p q

5. Solve each of the following:

a. In a tourist bus near the base of the Eiffel Tower at Paris, a passenger estimates the angle of elevation to

the top of the tower to be 60 . If the height of the Eiffel Tower is about 984 feet, what is the distance

from the bus to the base of the tower?

b. A man standing 20m from a tower estimates the angles of elevation to the top and bottom of a flagpole on

the tower as 58 and 55 . Calculate the height of the flagpole.

c. An engineer estimates the angle of elevation to the top of the building to be 50 . After moving 1.5m

further away, the angle of elevation was 40 . How high is the top of the building?

d. A kite at a height of 75meters from the ground is attached to a string inclined at 60 to the horizontal.

Find the length of the string to the nearest meter.

6. If 4

tan5

, find the value of cos .

Section 13-2 Angles and Angle Measure

Vocabulary: Co-terminal angles, degree measure, initial side, radian measure, standard position, terminal side, unit circle

p.709-715 #5-55odd

7. Draw an angle with the given measure in standard position.

a. 140

b. 240

c. 640

d.

8. Rewrite the radian measure in degrees.

a. 9

b. 6

c. 19

12

9. Rewrite the degree measure in radians.

a. 90

b. 400

c. 225

Section 13-3 Trigonometric Functions of General Angles

Vocabulary: Quadrantal angles, reference angle

p.717-724 #5,11-23odd,33-43odd,47-51odd

10. Find the exact value of the given trigonometric function.

a. cos300

b. sec240

c. tan 210

d. cot180

e. sin315

f. csc150

11. Find the exact value of the remaining five trigonometric functions of .

a. Suppose is an angle in standard position whose terminal side is in Quadrant IV and cot 2 .

b. Suppose is an angle in standard position whose terminal side is in Quadrant I and 60

tan11

.

c. Suppose is an angle in standard position whose terminal side is in Quadrant II and tan 3 .

d. Suppose is an angle in standard position whose terminal side is in Quadrant III and 10

csc3

.

12. Find the exact values of the six trigonometric functions of if the terminal side of in standard position

contains the given point.

a. Point 4, 3

b. Point 5, 10

13. Which of the following is equal to sin (𝜋

6)? (SAT)

A) −cos (𝜋

6) B) −sin (

𝜋

6) C) cos (

𝜋

3) D) 𝑠𝑖𝑛 (

𝜋

3)

TO BE REVIEWED FOR ACCURACY AND

COMPLETENESS FOR 2016 FINAL

Answer Key for Final Exam Review Packet

ACP ALG II

1a. 14 14i 1b. 10 4i

1c. 24

1d. 112

1e. 5 2 5 2

3 3 3

ii

1f. 18 21 18 21

85 85 85

ii

1g. 81 42 81 42

185 185 185

ii

1h. i

1i. 1

1j. 7i

1k. 6 12100x y i x

1l. 100 6

2a. 3x i

2b. 7x i

3a. 4 ; 5m n

3b. 15 ; 8m n

4a. down, maximum

4b. up, minimum

5a. 26 0.5 200 10 1200 40 5I x x x x x

5b. $4.00 per mug

5c. 4 $1280f

6a. minimum 1,1

6b. maximum 2, 48

6c. 1 9

maximum ,2 2

6d. maximum 2,5

7ia. .

3y-int 0, 4 ; AOS:

2

3 25Vertex: , ; x-ints 4,0 1,0

2 4

x

7ib.

X Y

-1 0

0 -4

3

2

25

4

3 -4

4 0

7ic.

7iia.

y-int 0, 1 ; AOS: 2 ; Vertex: 2,3

x-ints 2 3,0 2 3,0

x

7iib.

X Y

0 -1

1 2

2 3

3 2

4 -1

7iic.

7iiia.

1 1 5

y-int 0,2 ; AOS: ; Vertex: ,2 2 2

1 5 1 5x-ints ,0 ,0

2 2 2 2

x

7iiib.

X Y

-1 -2

0 2

1

2

5

2

1 2

2 -2

7iiic.

8. 9 8f x x x

9. 4 seconds

10a. 2,0 and 1,0

10b. between -2 and -1 ; between 4 and 5

11a. 1,0 4,0

11b. 6,0 ; 3,0

11c. 3,0 ; between -3 and -2

12a. 7,4x

12b. 5x

12c. 5

0,3

x

12d. 9

, 12

x

12e. 4

,53

x

13a. 2 4 3 0x x

13b. 2 3 10 0x x

13c. 22 15 7 0x x

13d. 215 2 8 0x x

14a. 2, 12x

14b. 7 1

,10 10

x

15a. 2

36 ; 6c x

15b.

2225 15

; 4 2

c x

16a. 1 5x i

16b. 2x i

16c. 5 4 2x

16d. 1 3x i

17. 2 4b ac

18.

2 4

2

b b ac

a

19a. discriminant = 204 ; 2 real irrational roots

19b. discrim = 27 ; 2 complex imaginary roots

19c. discriminant = 0 ; 1 real rational root

19d. discriminant = 81 ; 2 real rational roots

20a. 2,10x

20b. 11x

20c. 14

12

x

20d. 2 3x i

21. see page 316

22. see page 317

23a. 2 5x i

23b. 3 41 3 41

8 8 8x

23c. 5 97 5 97

4 4 4x

23d. 4x

23e. 4

0,7

x

23f. 3 3 1

2 2 2

ix i

23g. 5

3x

23h. 2 2 2 2

3 3 3

ix i

24a.

211 2

3

Vertex: 1,2 ; AOS: 1 ; Opens up

y x

x

24b.

23 8

Vertex: 3, 8 ; AOS: 3 ; Opens up

y x

x

24c.

2 28 0 3 8 3

Vertex: 0,3 ; AOS: 0 ; Opens down

y x x

x

24d.

22 5 15

Vertex: 5, 15 ; AOS: 5 ; Opens down

y x

x

24e.

23 8 192

Vertex: 8,192 ; AOS: 8 ; Opens down

y x

x

25a. 2 26 3 3 6 36 57f x x x x

25b. 2 23 4 3 3 24 51f x x x x

25c. 2 25 5 41

3 2 152 2 2

f x x x x

26a. 1

second4

26b. 31 feet

26c. 1.64 seconds

27a.

27b.

27c.

27d.

28a. degree 3 ; leading coefficient 3

28b. degree 5 ; leading coefficient 8

29a. 2 49 ; 1 2p p

29b. 2 25 ; 1 16p p

30a. 3 22 16 12 4q a a a

30b. 23 3 3 3r a a a

30c. 3 22 3 108 54 8q a a a

30d. 23 5 5r a a a

30e. 3 22 2 2 3q a r a a a a

31a. 222 7 19f g x x x

31b. 3 23 24f g x x x x

31c. 3 211 103 85 44f g x x x x

31d.

3 220 4 10 13

5 2

2|

5

f x x xx

g x

D x x

31e.

3

4

| 4,5,8,10

f xx

g x

D x x

32.

18 15

18 30

g h x x

h g x x

33.

2

2

64 9

8 72

f h x x

h f x x

34.

2,7

5,4 , 6,3 , 3, 5

f g x

g f x

35a. 2 28f g

35b. 1 26h g

35c. 1 175

2 16f h

35d. 1 28f g h

35e 4 448f g

35f. 7 46g h

36. 1 3,1 , 5, 4 , 2,4f x

37. 1 16 208 16 208

9 9 9

xf x x

38a.

Since ,

then and are not inverse functions.

f g x g f x x

f x g x

38b.

Since ,

then and are inverse functions.

f g x g f x x

f x g x

39a. 1 31

4f x x

40a. 2 2

10 6 81x y

40b. 2 2

1 4 20x y

40c. 2 2

8 9 1130x y

40d. 2 2

8 7 64x y

40e. 2 2

5 7 49x y

41a. Center 0, 2 ; radius 2

41b. Center 4,5 ; radius 5 3

41c. Center 7, 3 ; radius 2 2

41d. 9 129

Center ,4 ; radius2 2

42a. Center 11,14 ; radius 11

43a. 43x y

43b. 1

x y

43c.

2 2

3

70

23

a y

bx

43d.

5 3

2 1

x

x

43e.

2

1 1 26 31

8

x x x

x

44. 2,14x

45a. 212y

45b. 3 4 2112a b c

45c. 2 2x x x

45d. 2 3 1 1t t t

46a. 187 2

11

b a

b

46b. 3

20m

46c. 3

8y

46d. 5 16

2 2

y

y y

46e. 5 4

3 2 2

m

m m

46f.

2

4 15

4 5

h

h h

47a. 4

13s

47b. 14g

47c. 6 extraneous, so NO SOLUTIONa

47d. 2, 3 are extraneous, but 7 worksx x

48a. growth

48b. decay

49a. 8

3n

49b. 2

5n

50a. 3 25

50b. 11 54

50c. 2 5 57 49

51.

|

| 0

D x x

R y y

52. 9

1log 5

59,049

53. 4 1

96561

54a. 2

54b. 3

54c. 5

54d. 7

55a. 729x

55b. 3x

55c. 125n

55d. 3y

55e.

11

11 is extraneous

base cannot be negative

b

b

56.

8 15 8sin ; cos ; tan

17 17 15

17 17 15csc ; sec ; cot

8 15 8

57.

345 8 69 5sin ; cos ; tan

69 69 8

345 69 8 5csc ; sec ; cot

5 16 5

58a. 50 , 15.6, 11.9C b c

58b. 47 , 43 , 19.1C A c

59a. 10 , 15.2, 2.6P q p

59b. 38 , 52 , 24.5P R r

60a. 568.1 feet

60b. 3.4 meters

60c. 4.3 meters

60d. 1.3 inches

61. 5 41

cos41

62a. TS 140

IS

62a. TS

240 IS

62a. 640

IS

TS

62a. TS

IS

63a. 20

63b. 30

63c. 285

64a. 2

64b. 20

9

64c. 5

4

65a. 1

2

65b. 2

65c. 3

3

65d. undefined

65e. 2

2

65f. 2

66a.

5 2 5 1sin ; cos ; tan

5 5 2

5csc 5 ; sec ; cot given

2

66b.

60 11sin ; cos ; tan given

61 61

61 61 11csc ; sec ; cot

60 11 60

66c.

3 1sin ; cos ; tan given

2 2

2 3 3csc ; sec 2 ; cot

3 3

66d.

3 10 10sin ; cos ; tan 3

10 10

1csc given ; sec 10 ; cot

3

67a.

3 4 3sin ; cos ; tan

5 5 4

5 5 4csc ; sec ; cot

3 4 3

67b.

2 5 5sin ; cos ; tan 2

5 5

5 1csc ; sec 5 ; cot

2 2