Algebra 2 Semester Exam Study Guide
Algebra 2 Semester Exam 2015-2016
30 questions
Topics:
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Equations and inequalities
Systems of Equations
Systems of Inequalities
Quadratic Functions (including complex numbers)
Polynomial Functions (including operations on polynomials)
Rational and Radical Functions (include turning radical expressions
into expressions with rational exponents)
Reporting Category 1: (60% of the Semester Exam)
Algebra and Modeling
Reporting Category 2: (20% of the Semester Exam)
Functions and Modeling
Reporting Category 3: (20% of the Semester Exam)
Statistics, Probability, and the Number System
 This Study Guide includes sample problems from the textbook that
students can use to help prepare for the exam.
 They are NOT the test items.
 This is geared towards the DISTRICT semester exam.
 There will not be a study guide that is geared towards any state
assessment (End of Year EOC).
 Answers will not be provided by District Office for the Study Guide.
Algebra 2 Semester Exam Study Guide
BENCHMARK
MAFS.912.A-REI.1.1
DOK
Skill/Concepts
M
Determine an
effective step in
solving rational
equations
Example problem or problems to support studying
Given the rational equations below, explain your reasoning on how to
solve.
a.
b.
2(𝑥+8)
𝑥
−4𝑥
=
𝑥−1
= 10
(𝑥+2)
3𝑥
a. The period T of the object’s oscillation is given by the
𝑚
formula 𝑇 = 2𝜋√ 𝑘 where k is the spring constant, which
MAFS.912.A-CED.1.4
MAFS.912.A-CED.1.3
MAFS.912.F-IF.2.6
M
Solve for a specified
variable
M
Determine the set of
inequalities that
represents the
problem’s context
L
Determine the
average rate of
change on a
quadratic function
describes the stiffness of the spring. Solve for k.
b. An object with mass m1 and an object with mass m2 whose
centers of mass are a distance r apart experience an
attractive gravitational force F given by the formula 𝐹 =
𝐺𝑚1 𝑚2
where G is the universal gravitational constant. Solve
𝑟2
this formula for G, r, m1, and m2.
a. You need to buy some filing cabinets. You know that Cabinet
X costs $10 per unit, requires 6 square feet of floor space,
and holds 8 cubic feet of files. Cabinet Y costs $20 per unit,
requires 8 square feet of floor space, and holds 12 cubic feet
of files. You have been given $140 for this purchase, though
you don't have to spend that much. The office has room for
no more than 72 square feet of cabinets. How many of
which model should you buy, in order to maximize storage
volume? Write the system of inequalities to represent the
situation.
b. In order to ensure optimal health, a lab technician needs to
feed the rabbits a daily diet containing a minimum of 24
grams (g) of fat, 36 g of carbohydrates, and 4 g of protein.
But the rabbits should be fed no more than five ounces of
food a day. Rather than order rabbit food that is customblended, it is cheaper to order Food X and Food Y, and blend
them for an optimal mix. Food X contains 8 g of fat, 12 g of
carbohydrates, and 2 g of protein per ounce, and costs $0.20
per ounce. Food Y contains 12 g of fat, 12 g of
carbohydrates, and 1 g of protein per ounce, at a cost of
$0.30 per ounce. Write the system of inequalities to
represent the situation.
a. What is the average rate of change of 𝑓(𝑥)over the interval
[−3, −1]? (Graph on left)
b. What is the average rate of change of 𝑓(𝑥)over the interval
[−3, 1]? (Graph on right)
Algebra 2 Semester Exam Study Guide
MAFS.912.N-RN.1.2
M
MAFS.912.F-LE.2.5
M
MAFS.912.A-REI.3.6
M
MAFS.912.A-REI.3.7
M
MAFS.912.N-CN.1.1
M
MAFS.912.N-CN.1.2
M
MAFS.912.N-CN.3.7
M
MAFS.912.A-REI.2.4
M
MAFS.912.F-IF.3.8
M
Use rational
exponents to
represent radical
functions
Simplify the following expressions using rational exponents.
7
8
a. ( √𝑥 3 )
15
b. √(𝑗 −3 )−2
a. The function 𝐶(𝑡) = 1.25𝑡 + 8.5 models the total cost C, in
dollars, for a large cheese pizza with t toppings from a local
Determine the
restaurant. What does the t intercept mean?
meaning of the xb. A car traveling along a long straight highway at a constant
intercept of an
speed passes a rest area. The distance d, in miles, the car
equation
must travel to reach the next rest area as a function of the
time t, in hours, is modeled by 𝑑(𝑡) = 100 − 65𝑡. What
does the t-intercept mean?
a. A small business has three employees who decorate
pastries. Carlotta earns $11 per hour and decorates 12
pastries each hour on average. James earns $10 per hour
and decorates 11 pastries each hour on average. Melissa
was recently employed and earns $8 per hour. She
decorates 7 pastries each hour on average. In one week, the
Solve a system of
employees worked for 96 total hours, decorated 1016
equations with three
pastries, and earned a total of $960 in wages. How many
variables
hours did each employee work?
b. Danielle and Inder brought apples, bananas, and oranges to
a fruit sale. The bananas were sold for $0.50 each, while the
apples and oranges were sold for $0.75 each. They sold 50
pieces of fruit and earned $33.50 total. If Danielle and Inder
sold twice as many bananas as oranges, how many apples
did they sell?
Find the intersection a. What is the intersection of 2𝑥 + 𝑦 = −12 and 𝑦 = 𝑥 2 − 5?
of a linear and a
b. What is the intersection of 𝑦 = −2𝑥 − 5 and 𝑦 = 𝑥 2 + 4𝑥 −
quadratic function
21?
a. Write a number that is equivalent to the complex number
Determine an
(5 + 3𝑖)(9 + 8𝑖).
number equivalent to
b. Write a number that is equivalent to the complex number
a complex number
(4 − 8𝑖)(5 − 6𝑖)?
a. Write an expression that is equivalent to the following:
Combine like terms
(8 + 3𝑖) + (7 + 5𝑖).
with complex
b. Write an expression that is equivalent to the following:
numbers
(8 + 3𝑖) − (7 + 3𝑖).
Solve a quadratic for a. Solve the equation: 7𝑥 2 − 2𝑥 + 9 = 2𝑥 2 − 5𝑥 + 8.
the roots
b. Solve the equation: 𝑥 2 + 14𝑥 + 65 = −2.
a. Write the function 𝑓(𝑥) = 2(𝑥 − 4)2 + 3 in the form
Determine the an
equation equivalent
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐.
to a quadratic
b. Write the function 𝑓(𝑥) = 𝑥 2 + 6𝑥 + 4 in vertex form.
a. The quadratic function that approximates the height of a
javelin throw is ℎ(𝑡) = −0.08𝑡 2 + 4.48, where t is the time
in seconds after it is thrown. How can the function be
Rewrite a quadratic
rewritten to calculate the time at which the javelin hits the
function to find the
ground?
zeros
Algebra 2 Semester Exam Study Guide
MAFS.912.F-IF.3.7
M
Describe the features
of the function
MAFS.912.A-APR.1.1
M
Operations on
binomials/trinomials
MAFS.912.A-APR.2.2
M
Dividing polynomials
MAFS.912.A-APR.2.3
M
Determine the
number of x
intercepts of a
polynomial
MAFS.912.A-APR.3.4
M
Polynomial identities
b. In a football game, Tony attempts to kick a field goal at a
distance of 40 yards from the goal post. The path of the
kicked football is given by the function 𝑓(𝑥) = −0.02𝑥 2 +
0.9𝑥 where 𝑥 is the horizontal distance in yards and 𝑓(𝑥) is
the vertical distance in yards. How can the function be
rewritten to calculate the time at which the ball hits the
ground?
Describe the following characteristics (maxima, minimum,
intercepts, zeros, end behavior) of the functions below.
a. 𝑓(𝑥) = 𝑥 2 − 4𝑥 + 5
b. 𝑓(𝑥) = 2𝑥 2 + 4𝑥 + 1
c. 𝑓(𝑥) = 4𝑥 2 + 8𝑥 − 5
a. The school board allows for signs to be built such that, for
some integer 𝑥, they are feet (𝑥 + 1) high and 𝑥 2 + 7𝑥 +
10 feet wide. In order for painters to paint a sign, they must
first calculate the area. Write an expression that represents
the area, in square feet, of each sign.
b. Find the product. (3𝑥 − 2)(2𝑥 2 − 5𝑥 + 1)
a. Divide: 2𝑥 3 + 4𝑥 2 + 5 by 𝑥 − 3
b. Divide: 3𝑥 3 + 14𝑥 2 − 𝑥 + 20 by 𝑥 + 5
c. Divide: 5𝑥 3 − 8𝑥 2 − 𝑥 − 4 by 𝑥 − 2
Determine the number of x-intercepts for the functions below.
a. 𝑓(𝑥) = 𝑥 3 + 2𝑥 2 − 5𝑥 − 6
b. 𝑓(𝑥) = 𝑥 3 − 𝑥 2 − 8𝑥 − 12
c. 𝑓(𝑥) = 𝑥 3 − 𝑥 2 − 4𝑥 + 4
a. Use the polynomial identity (𝑥 2 − 𝑦 2 )2 + (2𝑥𝑦)2 =
(𝑥 2 + 𝑦 2 )2 with 𝑥 = 4 and 𝑦 = 1 to generate a Pythagorean
triple.
b. Use the polynomial identity (𝑎 − 𝑏)3 = 𝑎3 − 3𝑎2 𝑏 +
3𝑎𝑏 2 − 𝑏 3 to determine the coefficients of the terms in the
expanded form of a binomial raised to the third power
without cubing the binomial. What is the coefficient of the
x-term in the expanded form of (2𝑥 − 5)3 ?
6𝑥 3 +5𝑥 2 +2𝑥+7
?
2𝑥+3
𝑥 4 +2𝑥 3 +𝑥 2 +8𝑥−9
as the sum of a
𝑥 2 +4
a. What is equivalent to
MAFS.912.A-APR.4.6
M
MAFS>912.A-REI.1.1
M
MAFS.912.A-CED.1.3
M
Divide polynomials
b. Rewrite
polynomial and a
rational expression whose numerator is a constant.
Determine the error Explain the steps in solving the following equations.
in students work with a. 2𝑥 2 − 6 = 42
solving a quadratic
b. 𝑥 2 − 14𝑥 + 49 = 18
c. −3𝑥 2 + 18𝑥 = −30
equation
a. You can work at most 20 hours next week. You need to earn
at least $92 to cover you weekly expenses. Your dogDetermine which
walking job pays $7.50 per hour and your job as a car wash
system of
attendant pays $6 per hour. Write a system of linear
inequalities
inequalities to model the situation.
represents the
b. Jonah is going to the store to buy candles. Small candles cost
context
$3.50 and large candles cost $5.00. He needs to buy at least
20 candles, and he cannot spend more than $80. Write a
system of linear inequalities that represent the situation.
Algebra 2 Semester Exam Study Guide
3
MAFS.912.N-RN.1.1
MAFS.912.N-RN.1.2
L
M
Determine
equivalent forms of
radicals
Determine the
missing value
a. Write the radical expression in rational exponent form √𝑘 7.
b. Given that the fourth root of x is defined as a quantity that,
when raised to the fourth power, equals x, explain why it
4
1
makes sense that √𝑏 = 𝑏 4 .
1
𝑎
a. If √(8−4 )2 = 82 , what is the value of a?
1
𝑎
b. If √(53 )−4 = 54 , what is the value of a?
Determine the equation that represents the graphs below.
M
Determine the
quadratic equation of
the graph
MAFS.912.F-IF.3.8
M
Determine the axis of
symmetry give the
factors of a quadratic
MAFS.912.A-REI.3.6
M
Determine the ycoordinate of a
system of equations
MAFS.912.F-IF.3.7
a. The factors of a quadratic function are 4𝑥 + 3 and 𝑥 − 5.
What is the axis of symmetry of this function?
b. The factors of a quadratic function are 3𝑥 + 8 and 𝑥 − 2.
What is the axis of symmetry of this function?
What is the y-coordinate of the solution of system of equations
below?
𝑦 = 2𝑥 − 30
a. {1
1
𝑥 − 𝑦 = −1
5
b. {
MAFS.912.A-REI.3.7
M
MAFS.912.N-CN.1.2
M
MAFS.912.N-CN.3.7
M
MAFS.912.A-APR.1.1
\
M
Determine the
solution of a
quadratic and a
linear function
Multiply complex
expressions
Determine which
equation has a
complex solution
Combine like terms
2
2
𝑦 = −5𝑥 − 2
3
𝑦 = 2 𝑥 + 17
What are the solutions of the system of equations?
−4𝑥 + 3𝑦 = 1
a. {
𝑦 = 𝑥2 − 𝑥 + 1
𝑥 − 3𝑦 = 2
b. {
𝑦 = 𝑥 2 + 2𝑥 − 34
a. Simplify the expression: (𝑥 + 4𝑖)(𝑥 − 3𝑖)
b. Simplify the expression: (𝑥 − 2𝑖)(𝑥 + 5𝑖)
a. What equation has a solution of 4 ± 2𝑖√3 if a=1?
b. What equation has a solution of 5 + 𝑖√3 if a=1?
a. (3𝑥 5 + 4𝑥 4 + 𝑥 3 − 3𝑥 2 − 3𝑥 − 1) − (2𝑥 5 + 3𝑥 4 − 3𝑥 3 +
4𝑥 2 − 5𝑥 + 6)
b. (5𝑥 4 − 𝑥 3 + 2𝑥 + 1) + (2𝑥 3 + 3𝑥 2 − 4𝑥 − 7)