Name: ______________________
Class: _________________
Date: _________
ID: A
Semester Final Exam Review for Algebra 2
Graph the exponential function.
1. y = 4 x
3. You open a savings account and deposit $1,000.
After 1 year of earning continuously compounded
interest, your balance is $1,018.16. After 2
years, the balance is $1,036.66. Assuming you
make no deposits or withdrawals, find the
equation for the best-fitting exponential function
to represent the balance of the account after x
years. How much money will be in the account
after 10 years?
2. Suppose you invest $1600 at an annual interest
rate of 4.6% compounded continuously. How
much will you have in the account after 4 years?
Write the equation in logarithmic form.
4. 2 5 = 32
Write the equation in exponential form.
5. log 4
1
= −2
16
Graph the logarithmic equation.
6. y = log 5 x
Write the expression as a single logarithm.
7. log 7 50 − log 7 5
a.
log 45
b.
log 7 45
c.
log 7 10
d.
log 10
Expand the logarithmic expression.
8. log b
a.
b.
c.
d.
9. Solve ln 2 + ln x = 5. Round to the nearest
tenth, if necessary.
a. 50,000
b. 74.2
c. 10
d. 3
57
74
1
1
log b 57 +
log b 74
2
2
1
1
log b 57 −
log b 74
2
2
log b 57 − log b 74
log b
1
(57 − 74)
2
1
Name: ______________________
ID: A
Find any points of discontinuity for the rational function.
10. y =
a.
b.
(x + 3)(x − 5)(x + 7)
(x + 1)(x + 4)
x = 1, x = 4
x = –1, x = –4
c.
d.
x = 3, x = –5, x = 7
x = –3, x = 5, x = –7
11. Describe the vertical asymptote(s) and hole(s)
(x − 3)(x − 1)
for the graph of y =
.
(x − 1)(x − 5)
a. asymptote: x = 5 and hole: x = 1
b. asymptote: x = –5 and hole: x = –1
c. asymptote: x = –3 and hole: x = 5
d. asymptote: x = 5 and hole: x = –1
What is the graph of the rational function?
2
12. y =
2
x − 4x + 3
2
x −9
2
13. Find the least common multiple of x − 7x + 6 and x + 3x − 4.
Simplify the sum.
14.
4
5
+ 2
m +9
m − 81
Generate the first five terms in the sequence using the explicit formula.
15. y n = −5n − 5
17. What is the 15
given formula?
16. c n = 12n − 11
th
term in the sequence using the
c n = 3n − 1
18. The table shows the predicted growth of a particular bacteria after various numbers of hours. Write an explicit
formula for the sequence of the number of bacteria.
Hours (n)
Number of
Bacteria
1
2
3
4
5
21
42
63
84
105
Write the explicit formula for the geometric sequence. Then find the fifth term in the sequence.
19. a 1 = −4, a 2 = 8, a 3 = −16
20. Use summation notation to write the series 2 + 4
+ 6 + 8 + ... for 10 terms.
2
Name: ______________________
ID: A
Does the infinite geometric series diverge or converge? Explain.
21.
1
1
1
1
+
+
+
+…
5
10
20
40
22. Graph x 2 + y 2 = 36. What are its lines of
symmetry?
Identify the center and intercepts of the conic section. Then find the domain and range.
23.
28. This is a spinner used in a board game. What is
the probability that the spinner will land on a
multiple of 3 and 4?
Is the pair of events dependent or
independent? Explain
24. Write an equation of a parabola with a vertex at
the origin and a directrix at y = 5.
25. An elliptical track has a major axis that is 80
yards long and a minor axis that is 72 yards long.
Find an equation for the track if its center is (0,
0) and the major axis is the x-axis.
29. For events A and B, P(A) =
Also, P(A and B) =
26. A yogurt shop offers 6 different flavors of frozen
yogurt and 12 different toppings. How many
choices are possible for a single serving of frozen
yogurt with one topping?
27. In how many different orders can you line up 8
cards on a table?
3
1
15
.
4
13
and P(B) =
1
4
.
Name: ______________________
ID: A
30. A class of 24 students wants to choose 3 students
at random to bring food for a class party. Any set
of 3 students should have an equal chance of
being chosen. Which of the following strategies
will result in a fair decision?
a.
b.
c.
Arrange the students in a line. Start at one
end and have each student flip a coin. The
first three students to flip heads can bring the
food.
Assign a number to each student. Write the
numbers on slips of paper and put them all in
a hat. Randomly choose three slips of paper.
The students with those three number can
bring the food.
Ask the students to volunteer. The first three
students to raise their hands can bring the
food.
Find the mean, median, and mode of the data set. Round to the nearest tenth.
31. 2, 10, 6, 9, 1, 15, 11, 10, 15, 13, 15
Use a calculator to find the mean and standard deviation of the data. Round to the nearest tenth.
32. 20, 16, 18, 14, 9, 20, 16
Find the measure of the angle.
34.
33. The scores on a final exam were approximately
normally distributed with a mean of 82 and a
standard deviation of 11. If 85 students took the
exam, and above a 60 is a passing grade, how
many students failed the exam?
4
Name: ______________________
ID: A
Sketch the angle in standard position.
35. –95°
37. Find the exact value of sin 120°.
36. Find the cosine and sine of 180°. Round your
answers to the nearest hundredth if necessary.
38. Find the degree measure of an angle of –
radians.
39. Find the period of the graph shown below.
5
π
6
Name: ______________________
ID: A
40. Find the amplitude of the sine curve shown below.
6
ID: A
Semester Final Exam Review for Algebra 2
Answer Section
1. ANS:
PTS: 1
DIF: L2
REF: 7-1 Exploring Exponential Models
OBJ: 7-1.1 To model exponential growth and decay
NAT: CC A.SSE.1.b| CC A.CED.2| CC F.IF.7.e| A.1.b| A.2.f| A.2.g
TOP: 7-1 Problem 1 Graphing an Exponential Function
KEY: exponential function
2. ANS:
$1,923.23
PTS: 1
DIF: L2
REF: 7-2 Properties of Exponential Functions
OBJ: 7-2.2 To graph exponential functions that have base e
NAT: CC A.SSE.1.b| CC A.CED.2| CC F.IF.7| CC F.IF.7.e| CC F.IF.8| CC F.BF.1| CC F.BF.1.b| N.3.f| G.2.c|
A.1.b| A.2.d| A.2.h TOP: 7-2 Problem 5 Continuously Compounded Interest
KEY: continuously compounded interest
3. ANS:
0.018t
A = 1000 ⋅ e
, $1,197.22
PTS:
OBJ:
NAT:
A.1.b|
KEY:
1
DIF: L4
REF: 7-2 Properties of Exponential Functions
7-2.2 To graph exponential functions that have base e
CC A.SSE.1.b| CC A.CED.2| CC F.IF.7| CC F.IF.7.e| CC F.IF.8| CC F.BF.1| CC F.BF.1.b| N.3.f| G.2.c|
A.2.d| A.2.h TOP: 7-2 Problem 5 Continuously Compounded Interest
compare properties of two functions | continuously compounded interest
1
ID: A
4. ANS:
log 2 32 = 5
PTS:
OBJ:
NAT:
TOP:
KEY:
5. ANS:
4 −2 =
1
DIF: L2
REF: 7-3 Logarithmic Functions as Inverses
7-3.1 To write and evaluate logarithmic expressions
CC A.SSE.1.b| CC F.IF.7.e| CC F.IF.8| CC F.IF.9| CC F.BF.4.a| G.2.c| A.2.h| A.3.h
7-3 Problem 1 Writing Exponential Equations in Logarithmic Form
write a function in different but equivalent forms
1
16
PTS:
OBJ:
NAT:
TOP:
KEY:
6. ANS:
1
DIF: L4
REF: 7-3 Logarithmic Functions as Inverses
7-3.1 To write and evaluate logarithmic expressions
CC A.SSE.1.b| CC F.IF.7.e| CC F.IF.8| CC F.IF.9| CC F.BF.4.a| G.2.c| A.2.h| A.3.h
7-3 Problem 1 Writing Exponential Equations in Logarithmic Form
write a function in different but equivalent forms
PTS:
OBJ:
NAT:
TOP:
7. ANS:
OBJ:
TOP:
8. ANS:
OBJ:
TOP:
9. ANS:
OBJ:
TOP:
KEY:
1
DIF: L2
REF: 7-3 Logarithmic Functions as Inverses
7-3.2 To graph logarithmic functions
CC A.SSE.1.b| CC F.IF.7.e| CC F.IF.8| CC F.IF.9| CC F.BF.4.a| G.2.c| A.2.h| A.3.h
7-3 Problem 4 Graphing a Logarithmic Function
KEY: logarithmic function
C
PTS: 1
DIF: L2
REF: 7-4 Properties of Logarithms
7-4.1 To use the properties of logarithms
NAT: CC F.LE.4| N.1.d| A.3.h
7-4 Problem 1 Simplifying Logarithms
B
PTS: 1
DIF: L4
REF: 7-4 Properties of Logarithms
7-4.1 To use the properties of logarithms
NAT: CC F.LE.4| N.1.d| A.3.h
7-4 Problem 2 Expanding Logarithms
B
PTS: 1
DIF: L4
REF: 7-6 Natural Logarithms
7-6.2 To solve equations using natural logarithms
NAT: CC F.LE.4| A.3.h
7-6 Problem 2 Solving a Natural Logarithmic Equation
natural logarithmic function
2
ID: A
10. ANS: B
PTS: 1
DIF: L2
REF: 8-3 Rational Functions and Their Graphs
OBJ: 8-3.1 To identify properties of rational functions
NAT: CC A.CED.2| CC F.IF.7| CC F.BF.1| CC F.BF.1.b| A.2.h
TOP: 8-3 Problem 1 Finding Points of Discontinuity
KEY: rational function | point of discontinuity | removable discontinuity | non-removable points of
discontinuity
11. ANS: A
PTS: 1
DIF: L3
REF: 8-3 Rational Functions and Their Graphs
OBJ: 8-3.1 To identify properties of rational functions
NAT: CC A.CED.2| CC F.IF.7| CC F.BF.1| CC F.BF.1.b| A.2.h
TOP: 8-3 Problem 2 Finding Vertical Asymptotes
KEY: rational function
12. ANS:
PTS: 1
DIF: L3
REF: 8-3 Rational Functions and Their Graphs
OBJ: 8-3.2 To graph rational functions
NAT: CC A.CED.2| CC F.IF.7| CC F.BF.1| CC F.BF.1.b| A.2.h
TOP: 8-3 Problem 4 Graphing Rational Functions
KEY: rational function
13. ANS:
(x − 6)(x − 1)(x + 4)
PTS: 1
DIF: L2
REF: 8-5 Adding and Subtracting Rational Expressions
OBJ: 8-5.1 To add and subtract rational expressions
NAT: CC A.APR.7| N.5.e| A.3.c| A.3.e
TOP: 8-5 Problem 1 Finding the Least Common Multiple
14. ANS:
4m − 31
(m − 9)(m + 9)
PTS: 1
DIF: L2
REF: 8-5 Adding and Subtracting Rational Expressions
OBJ: 8-5.1 To add and subtract rational expressions
NAT: CC A.APR.7| N.5.e| A.3.c| A.3.e
TOP: 8-5 Problem 2 Adding Rational Expressions
3
ID: A
15. ANS:
–10, –15, –20, –25, –30
PTS: 1
DIF: L3
REF: 9-1 Mathematical Patterns
OBJ: 9-1.1 To identify mathematical patterns found in a sequence
NAT: CC A.SSE.4| A.1.a
TOP: 9-1 Problem 1 Generating a Sequence Using an Explicit Formula
KEY: sequence | term of a sequence | explicit formula
16. ANS:
1, 13, 25, 37, 49
PTS:
OBJ:
NAT:
TOP:
KEY:
17. ANS:
44
1
DIF: L3
REF: 9-1 Mathematical Patterns
9-1.1 To identify mathematical patterns found in a sequence
CC A.SSE.4| A.1.a
9-1 Problem 1 Generating a Sequence Using an Explicit Formula
sequence | term of a sequence | explicit formula
PTS: 1
DIF: L2
REF: 9-1 Mathematical Patterns
OBJ: 9-1.1 To identify mathematical patterns found in a sequence
NAT: CC A.SSE.4| A.1.a
TOP: 9-1 Problem 1 Generating a Sequence Using an Explicit Formula
KEY: sequence | term of a sequence | explicit formula
18. ANS:
a n = 21n
PTS: 1
DIF: L2
REF: 9-1 Mathematical Patterns
OBJ: 9-1.2 To use a formula to find the nth term of a sequence
NAT: CC A.SSE.4| A.1.a
TOP: 9-1 Problem 4 Using Formulas to Find Terms of a Sequence
KEY: sequence
19. ANS:
a n = −4 ⋅ (−2) n − 1 ; –64
PTS:
OBJ:
NAT:
KEY:
20. ANS:
1
DIF: L4
REF: 9-3 Geometric Sequences
9-3.1 To define, identify, and apply geometric sequences
CC A.SSE.4| A.1.a
TOP: 9-3 Problem 2 Analyzing Geometric Sequences
geometric sequence
10
∑ 2n
n = 1
PTS:
OBJ:
TOP:
KEY:
1
DIF: L2
REF: 9-4 Arithmetic Series
9-4.1 To define arithmetic series and find their sums
NAT: CC F.IF.3| A.1.a| A.3.g
9-4 Problem 3 Writing a Series in Summation Notation
series | finite series | limits
4
ID: A
21. ANS:
It converges; it has a sum.
PTS: 1
DIF: L3
REF: 9-5 Geometric Series
OBJ: 9-5.1 To define geometric series and find their sums
NAT: CC A.SSE.4| A.1.a| A.3.g
TOP: 9-5 Problem 3 Analyzing Infinite Geometric Series
KEY: geometric series | converge | diverge
22. ANS:
Every line through the center is a line of symmetry.
PTS: 1
DIF: L2
REF: 10-1 Exploring Conic Sections
OBJ: 10-1.1 To graph and identify conic sections
NAT: CC G.GPE.1| CC G.GPE.2| CC G.GPE.3| G.4.c
TOP: 10-1 Problem 1 Graphing a Circle
KEY: conic sections
23. ANS:
The center of the ellipse is (0, 0).
The x-intercepts are (–3, 0) and (3, 0).
The y-intercepts are (0, 5) and (0, –5).
The domain is {x | –3 ≤ x ≤ 3}.
The range is {y | –5 ≤ y ≤ 5}.
PTS:
OBJ:
NAT:
TOP:
24. ANS:
1
DIF: L2
REF: 10-1 Exploring Conic Sections
10-1.1 To graph and identify conic sections
CC G.GPE.1| CC G.GPE.2| CC G.GPE.3| G.4.c
10-1 Problem 4 Identifying Graphs of Conic Sections KEY: conic sections
y = −
1 2
x
20
PTS:
OBJ:
NAT:
KEY:
1
DIF: L3
REF: 10-2 Parabolas
10-2.1 To write the equation of a parabola and to graph parabolas
CC G.GPE.2 TOP: 10-2 Problem 1 Parabolas with Equation y = ax^2
directrix
5
ID: A
25. ANS:
y2
x2
+
= 1
1600
1296
PTS:
OBJ:
TOP:
KEY:
26. ANS:
72
1
DIF: L4
REF: 10-4 Ellipses
10-4.1 To write the equation of an ellipse
NAT: CC G.GPE.3| G.4.g
10-4 Problem 3 Using the Foci of an Ellipse
ellipse | equation of an ellipse | major axis | minor axis | center of an ellipse
PTS: 1
DIF: L2
REF: 11-1 Permutations and Combinations
OBJ: 11-1.1 To count permutations
NAT: CC S.CP.9| D.4.e| D.4.j
TOP: 11-1 Problem 1 Using the Fundamental Counting Principle
KEY: Fundamental Counting Principle
27. ANS:
40,320
PTS:
OBJ:
TOP:
KEY:
28. ANS:
3
1
DIF: L2
11-1.1 To count permutations
11-1 Problem 2 Find the Number
Fundamental Counting Principle |
REF: 11-1 Permutations and Combinations
NAT: CC S.CP.9| D.4.e| D.4.j
of Permutations of n Items
permutation | n factorial
8
PTS:
OBJ:
NAT:
KEY:
29. ANS:
1
DIF: L3
REF: 11-2 Probability
11-2.1 To find the probability of an event using theoretical, experimental, and simulation methods
CC S.IC.2| D.4.c| D.4.d| D.4.j
TOP: 11-2 Problem 3 Finding Theoretical Probability
theoretical probability
No, A and B are not independent. If they were independent, then P(A and B) would be
P(A and B) =
PTS:
OBJ:
NAT:
TOP:
30. ANS:
OBJ:
NAT:
1
15
4
13
⋅
1
4
, or
1
13
. But
.
1
DIF: L3
REF: 11-3 Probability of Multiple Events
11-3.1 To find the probability of the event A and B
CC S.CP.2| CC S.CP.5| CC S.CP.7| D.4.a| D.4.b| D.4.c| D.4.h| D.4.j
11-3 Problem 1 Classifying Events
KEY: dependent events | independent events
B
PTS: 1
DIF: L3
REF: 11-5 Probability Models
11-5.1 To use probabilities to make fair decisions and analyze decisions
CC S.MD.6| CC S.MD.7
TOP: 11-5 Problem 1 Making a Fair Decision
6
ID: A
31. ANS:
mean = 9.7, median = 10, mode = 15
PTS: 1
DIF: L2
REF: 11-6 Analyzing Data
OBJ: 11-6.1 To calculate measures of central tendency
NAT: CC S.IC.6| D.1.a| D.1.b| D.2.c| D.1.e| D.2.a
TOP: 11-6 Problem 1 Finding Measures of Central Tendency
KEY: measure of central tendency | mean | median | mode
32. ANS:
mean = 16.1;
standard deviation = 3.6
PTS:
OBJ:
NAT:
TOP:
KEY:
33. ANS:
2
1
DIF: L2
REF: 11-7 Standard Deviation
11-7.1 To find the standard deviation and variance of a set of values
CC S.ID.4| CC S.IC.6| D.1.c
11-7 Problem 2 Using a Calculator to Find Standard Deviation
mean | standard deviation
PTS:
OBJ:
NAT:
TOP:
34. ANS:
235°
1
DIF: L4
REF: 11-10 Normal Distributions
11-10.1 To use a normal distribution
CC S.ID.2| CC S.ID.4| D.1.b| D.1.c| D.2.g
11-10 Problem 3 Analyzing a Normal Distribution
KEY: normal distribution
PTS:
OBJ:
TOP:
KEY:
1
DIF: L3
REF: 13-2 Angles and the Unit Circle
13-2.1 To work with angles in standard position
NAT: CC F.TF.2
13-2 Problem 1 Measuring Angles in Standard Position
standard position | initial side | terminal side
7
ID: A
35. ANS:
PTS:
OBJ:
TOP:
KEY:
36. ANS:
–1, 0
1
DIF: L3
REF: 13-2 Angles and the Unit Circle
13-2.1 To work with angles in standard position
NAT: CC F.TF.2
13-2 Problem 2 Sketching Angles in Standard Position
standard position | initial side | terminal side
PTS:
OBJ:
TOP:
KEY:
37. ANS:
1
DIF: L3
REF: 13-2 Angles and the Unit Circle
13-2.2 To find coordinates of points on the unit circle NAT: CC F.TF.2
13-2 Problem 4 Finding the Cosines and Sines of Angles
cosine of theta | sine of theta
sin = −
PTS:
OBJ:
TOP:
KEY:
38. ANS:
–30°
3
2
1
DIF: L3
REF: 13-2 Angles and the Unit Circle
13-2.2 To find coordinates of points on the unit circle NAT: CC F.TF.2
13-2 Problem 5 Finding Exact Values of Cosine and Sine
sine of theta
PTS: 1
DIF: L3
REF: 13-3 Radian Measure
OBJ: 13-3.1 To use radian measure for angles
NAT: CC F.TF.1| M.3.e
TOP: 13-3 Problem 1 Using Dimensional Analysis
KEY: central angle | intercepted arc | radian
39. ANS:
2π
PTS:
OBJ:
NAT:
TOP:
1
DIF: L3
REF: 13-4 The Sine Function
13-4.1 To identify properties of the sine function
CC F.IF.4| CC F.IF.7.e| CC F.TF.2| CC F.TF.5| M.3.c
13-4 Problem 2 Finding the Period of a Sine Curve
KEY: sine function | sine curve
8
ID: A
40. ANS:
4
PTS:
OBJ:
NAT:
TOP:
1
DIF: L3
REF: 13-4 The Sine Function
13-4.1 To identify properties of the sine function
CC F.IF.4| CC F.IF.7.e| CC F.TF.2| CC F.TF.5| M.3.c
13-4 Problem 3 Finding the Amplitude of a Sine Curve KEY: sine function | sine curve
9