Excellence is not an act, but a habit. Aristotle · Excellence is not an act, but a habit. – Aristotle Dear Algebra II Student, First of all, Congrats! for making it this far in

Excellence is not an act, but a habit. – Aristotle

Dear Algebra II Student,

First of all, Congrats! for making it this far in your “math career.” Passing Algebra II is a huge mile-stone Give yourself a pat on the back, for giving it your best thus far. If you’re hesitating on giving yourself that pat on the back, well, buckle up! We’ve got a lot of math practice to catch up on.

How to use this Algebra II - Semester 2 Study Packet

1. Complete all review homework assigned by teacher

2. Re-learn and/or study topic, as needed

3. After review is complete, take the corresponding practice test

4. For multiple choice exams in math, for most question types, cover answer choices and complete

the problem. Double check. Then look for the correct answer choice.

5. Come to tutoring for answer key and for multiple choice test taking strategies

Algebra II Topics Included

Multiple Choice Practice Tests

These are basic concept questions, level easy to medium. These practice tests are not designed to teach

concepts. They are designed to review concepts after studying and practice multiple choice style exams.

Note: Graphing is minimal on this review, but may be tested on your exam! (Discuss with your teacher!)

Powers, roots, complex numbers (important semester 1 review)

Quadratics (important semester 1 review)

Functions & Transformations

Conics

Polynomials

Exponential and logarithmic expressions

Sequences, series, probability

Statistics

Trigonometry

Happy Studying,

Kristy

Kristy Arthur Tutoring Services [email protected] 714.401.5088

mailto:[email protected]

Excellence is not an act, but a habit. – Aristotle Page 2 of 16

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Powers, Roots, and Complex Numbers

Subtract .

(11 – y2)

(11 – y)(11+y)

(11 – y)

11 – a

Multiply .

10x2 – 11x –6

10x – 11 – 6

10x – – 6

10x – 209 + 6

Simplify • by multiplying and

factoring.

62.61

Divide .

Simplify (f4/3g–5/6)–12.

f16g10

Evaluate the third root of –343.

–49

49

7

–7

Rationalize the denominator of .

Solve the equation 22 – x = .

13

no real solution

13 and 36

–4

Rewrite (4x5)3/7 without rational exponents

and simplify if necessary.

2x

16x11

x2

x2

Find .



Quadratic Equations

Find the quadratic equation whose

solutions are + 5i and – 5i.

x2 – x +

x2 + 25

x2 – 25

x2 – x –

Substitute values from the following

equation into the quadratic formula: x2 – 2x – 6 = 0.

Solve 3x2 + 3x = 8x +12.

–3 and

0, –1 and –

– and 3

–1 and –

Solve z4 – 6z2 + 5 = 0.

–1 and –5

1 and 5

1 and

1, –1, and –

Solve x2 + 8x + 15 = 0.

–1 and –15

–3 and –5

3 and 5

1 and 15

What should you do as a first step in solving this equation x2 – 4x = –7 by

completing the square?

add 4 to both sides

square –7

add 4x to both sides

add 2 to both sides

Suppose a coin that is tossed upward can

be modeled by the quadratic function h(t) = –16t2 + 24t, where h(t) is the height

in feet and t is the time in seconds. At

what time will the coin be at a height of 5

ft? If necessary, round your answer(s) to

the nearest hundredth of a second.

–0.25 s and –1.25 s

–280 s

0 s and 1.5 s

0.25 s and 1.25 s

Determine the nature of the solution(s) of –5x2 + 2x – 1 = 0.

two real

two complex

two fake

one real



Solve x2 + 26 = 10x.

–5 + i and –5 – i

– + and – –

.5 + i and 5 – i – 2

–13 + 2 and –13

What must be true of a and c so

that is a real number?

either a = 0 or c = 0

both a < 0 and c < 0

either a > 0 or c > 0

ac < 0

Quadratic Functions and Transformations

Which of the following is symmetric with

respect to the origin?

y2 = 9 – (x + 2)2

x2 = y + 3

x = x – y

y = x3 – 3x

Choose the graph that represents y ≤

x2 + .



Graph the set of functions on the same set

of axes. f(x) = x2 and h(x) = (x – 2)2

Here is a graph of y = f(x).

How would the graph of y = –4f(x) be

different?

The peaks would be at (–6, 16),

(0,16), and (6, 16); the valleys would be

at (–4, –16) , (2, –16) , and (8, –16).

The peaks would be at (–16, 4), (8,4),

and (32, 4); the valleys would be at (–24,

–4) , (0, –4) , and (24, –4).

The peaks would be at (–24, 4), (0,4),

and (24, 4); the valleys would be at (–16,

–4) , (8, –4) , and (32, –4).

The peaks would be at (–4, 16),

(2,16), and (8, 16); the valleys would be

at (–6, –16) , (0, –16) , and (6, –16).



Graph the set of functions on the same set

of axes. f(x) = x2 and h(x) = x2 – 4

Which of the following is not symmetric with respect to the y–axis?

3x2 – 6y2 = 42

5x + 2y = 8

y = x4 – 9

y – 9 = x4 – 10x2

Consider the graph of y = x2.

Which graph represents y = 3(x – 1)2?



Find the vertex and the line of symmetry of f(x) = –2(x + 4)2.

vertex: (–4, 0); line of symmetry: x =

–4

vertex: (–2, 4); line of symmetry: x =

–2

vertex: (4, 0); line of symmetry: x =

–4

vertex: (–2, –4); line of symmetry: x = –2

Which of the following functions is neither

even nor odd?

f(x) = x3 + x2 – 3x

f(x) = –x4 + 5x2

f(x) = x

f(x) = 3x3 – x

Consider the graph of y = |x|.

Which graph represents y = – ?



Conics

What is the equation for this hyperbola in

standard form?

Find the distance between (3, –7) and (–

4, 5).

Graph the ellipse and give the

coordinates of its foci.

Give the standard form for the equation of

the graph of the ellipse shown below.



Find an equation of a parabola that has a focus at (3, 5) and a directrix at x = 11.

(y – 5)2 = –(x – 3)

(y – 3)2 = –16(x – 5)

(y – 5)2 = –16(x – 7)

(y – 5)2 = –4(x – 11)

Write an equation for the conic y2 – 2y – 9x2 = 35 in standard form. Then

give the center of the conic.

; (0,1)

; (1,0)

; (3,1)

; (1,3)

Complete the square to find the center

and radius of the circle. x2 + y2 – 6x – 2y – 26 = 0

Center: (6, 2); radius:

Center: (3, 1); radius: 6

Center: (–3, –1); radius: 6

Center: (–6, –2); radius:

Write an equation of the circle with center

at (–8, –3) and radius 4 .

(x – 8)2 + (y – 3)2 = 576

(x – 8)2 + (y – 3)2 = 96

(x + 8)2 + (y + 3)2 = 96

(x + 8)2 + (y + 3)2 = 4

Find the coordinates of the midpoint of the

segment having the endpoints (–3, –2)

and (–1, –5).

(–2.5, –3)

(–2, –3.5)

(–2, –1.5)

(1, 3.5)

Find the vertex, focus, and directrix of y = –(x + 1)2 + 3.

vertex: (–1, 3); focus: (–1, 3.25); directrix: y = 2.75

vertex: (3, 1); focus: (2.75, –1); directrix: x = 3.25

vertex: (1, 3); focus: (–1, 2.75); directrix: x = 3.25

vertex: (–1, 3); focus: (–1, 2.75); directrix: y = 3.25



Exponential and Logarithmic Functions Suppose f(x) = –2x + 5 and g(x) = –6x2.

Find f(g(–7)).

–2166

–583

–486

593

If log3 2 ≈ 0.6309 and log3 3 = 1.0000, then approximate x = log3 48.

x ≈ 0.1584

x ≈ 2.5236

x ≈ 3.5236

x ≈ 17.2404

Which of the following are properties of

logarithms?

log (xy) = log x + log y

= log (x – y)

log (xn) = nlog x

log x log y = log (x + y)

IV only

I, II, and III

I and III

II and III

Convert logx 32,768 = 5 to an exponential

equation.

x5 = 32,768

5x = 32,768

x32,768 = 5

logx =

Convert 82/3 = 4 to a logarithmic equation.

= log84

log 8 = log 42/3

log 4 = log 8

log 8 = log4

Find an equation for f–1(x) if f(x) =

f–1(x) = x2 – 18

f–1(x) = x2 – 18; x ≥ 0

f–1(x) = x2 + 18; x ≥ 0

f–1(x) = x2 + 18; x > 0

Express 6logb x – 5logb y – 7 logb z as a

single logarithm.

logb

logb

logb (6x – 5y – 7z)

logb (x6 – y5 – z7)

Sequences, Series

Which of the following sequences are

arithmetic or geometric?

343, –49, 7, –1, …

1, 4, 9, 16, …

2, 9, 16, 23, …

2, 3, 5, 8, 13, …

Arithmetic: c and d

Geometric: a only

Arithmetic: c only

Geometric: a and b

Arithmetic: c and d

Geometric: a and b

Arithmetic: c only

Geometric: a only



A geometric series converges if its has

an absolute value that is less than 1.

common ratio

sigma notation

common difference

first term

Find S7 for the sequence 12, 14, 16, 18, …

126

2

24

420

Give a formula for the nth term of the

geometric sequence 16, 24, 36, 54, ….

16(1.6)n

26(1.5)n – 1

16

16(1.4)n – 1

Find the common ratio for the geometric

sequence , 3, 5, ….

5

This is not a geometric sequence.

Give the sum of the infinite geometric

series: 36 + + + + ….

45

44

36

Find the first five terms of the sequence an = –n– n.

–2, –4, –6, –8, –10

–1, – , – , – , –

–1, , – , , –

1, 4, 27, 256, 3125

Give the sum.

982.8

980.1

108

1093.5



Counting and Probability

A pizza parlor offers a choice of

mozzarella or Colby cheese. Available

toppings are mushrooms, olives, and

sausage. How many different medium-size

cheese pizzas with one topping can be

ordered?

12

8

6

5

Evaluate 25C18.

480,700

342,014,400

3.07762104 × 1021

1.22802249913 × 1028

Find the number of distinguishable

permutations of the letters in the word

INFINITY.

20160

40320

336

3360

The symbol 6! is read as

"permutation six."

"six factorial."

"six chosen randomly."

none of the above

Evaluate 4P2.

6

12

16

155.76

Expand: (x + 2)6.

x6 + 6x5 + 15x4 + 20x3 + 15x2 + 6x + 1

x6 + 2x5 + 4x4 + 8x3 + 16x2 + 32x + 64

x6 – 12x5 + 60x4 – 160x3 + 240x2 –

192x + 64

x6 + 12x5 + 60x4 + 160x3 + 240x2 +

192x + 64

Ten names are put into a hat to be drawn

for four different door prizes. In how

many ways can four names be drawn from

the hat without replacement?

210

720

5040

151200

Find the 5th term of the expansion: (2x – 3y)7.

35x3y4

–210x3y4

22,680x3y4

–22,680x3y4



Statistics and Data Analysis

Find the mean, median, and mode of the

following set of data.

6, 9, 7, 4, 7, 6, 3, 5, 3, 1, 1, 6, 7, 8, 6, 7,

4, 2, 1, 7

mean : 5, median: 5, mode: 6




What is the range of this data?

894, 718, 241, 823, 197, 379, 593, 427

467

510

534

697

Suppose that a data set is normally

distributed with a mean of 40 and a

standard deviation of 5. Approximately

68% of the data values lie between what

two numbers?

25, 55

30, 50

35, 45

40, 70

Construct a frequency distribution for the

data using the intervals i) 0–24, ii) 25–49,

iii) 50–74, and iv) 75–99. Which interval

has a relative frequency of 20%?

4, 98, 81, 6, 18, 20, 26, 14, 89, 21, 17,

65, 2, 23, 5, 1, 89, 9, 15, 19, 0, 91, 84,

11, 7, 29, 16, 72, 24, 13

i

ii

iii

iv

The mean of the following data is 12.

Compute the standard deviation.

17, 11, 12, 9, 10, 13

0

2.58

6.67

44.49

What is the mean deviation of this data?

894, 718, 241, 823, 197, 379, 593, 427

223

534

697

6178497



Construct a stem-and-leaf diagram for the

data set.

45, 98, 35, 21, 79, 45, 23, 89, 65, 32, 78,

58, 54, 28, 89, 31, 67, 9, 20, 91, 84, 26,

37, 68, 24, 13

Construct a box-and-whisker plot of the

data.

15, 49, 42, 19, 12, 6, 1, 42, 42, 4, 7, 2,

10, 36, 43



Trigonometric Functions

Give the exact value of the trigonometric

ratio for tan 90°.

1

0

–1

Undefined

Given sin θ = 0.9239, find θ in degrees

and minutes, then convert the measure to

decimal form.

22°30', 22.50°

67°30', 67.50°

67°83', 67.50°

68°30', 68.50°

Find the sine, cosine, and tangent for q.

sin q = – ; cos q = ; tan q = –

sin q = ; cos q = ; tan q =

sin q = – ; cos q = ; tan q = –

sin q = ; cos q = – ; tan q = –

Use ΔABC to find cos A and tan B.

cos A = ; tan B =

cos A = ; tan B =

cos A = ; tan B =

cos A = ; tan B =

Convert 245° to radian measure.

Find cos θ, cot θ, and the length of side a for the following triangle in which θ

= 60° and c = 24.

cos θ = , cot θ = , a = 12

cos θ = , cot θ = , a = 12

cos θ = , cot θ = , a = 12

cos θ = , cot θ = , a = 12

For an angle measure of 809°, give an

equivalent angle measure between 0° and

360°, and tell in which quadrant the

terminal side lies.

89°, Quadrant I

89°, Quadrant II

209°, Quadrant III

209°, Quadrant IV



Find the reference angle for angle shown.

193°

167°

13°

–167°

Trigonometric Identities and Equations

Find two angles, in radians, between 0

and 2π which satisfy sin x = –

and

and

and

and

To find the exact value of sin 165° from a

sum or difference identity, choose a

suitable replacement for 165°.

180° – 15°

90° + 75°

145° + 20°

120° + 45°

Find a value of cos (A + B) if tan A =

and csc B = , both angles being acute.

The expression sin (A + B) is identical to

which expression?

sin A cos B + cos A sin B

sin A sin B – cos A cos B

sin A cos B – cos A sin B

cos A sin B – sin A cos B

Use the identity cos (a – b) ≡ cos a cos b +

sin a sin b to evaluate cos in simplest

form.

If sin θ = – and θ is an angle in the third

quadrant, find sin 2θ.

–

If cot θ = – and sin θ is negative, find

the value of tan θ.

–

2

–3

Documents

Excellence is not an act, but a habit. Aristotle · Excellence is not an act, but a habit. – Aristotle Dear Algebra II Student, First of all, Congrats! for making it this far in