excellence is not an act, but a habit. aristotle · excellence is not an act, but a habit. –...
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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
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 .
Excellence is not an act, but a habit. – Aristotle Page 3 of 16
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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
Excellence is not an act, but a habit. – Aristotle Page 4 of 16
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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 + .
Excellence is not an act, but a habit. – Aristotle Page 5 of 16
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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).
Excellence is not an act, but a habit. – Aristotle Page 6 of 16
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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?
Excellence is not an act, but a habit. – Aristotle Page 7 of 16
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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 = – ?
Excellence is not an act, but a habit. – Aristotle Page 8 of 16
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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.
Excellence is not an act, but a habit. – Aristotle Page 9 of 16
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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
Excellence is not an act, but a habit. – Aristotle Page 10 of 16
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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
Excellence is not an act, but a habit. – Aristotle Page 11 of 16
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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
Excellence is not an act, but a habit. – Aristotle Page 12 of 16
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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
Excellence is not an act, but a habit. – Aristotle Page 13 of 16
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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
mean : 5, median: 1, mode: 7
mean : 5, median: 6, mode: 7
mean : 7, median: 5, mode: 1
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
Excellence is not an act, but a habit. – Aristotle Page 14 of 16
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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
Excellence is not an act, but a habit. – Aristotle Page 15 of 16
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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
Excellence is not an act, but a habit. – Aristotle Page 16 of 16
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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