ALGEBRA II
WITH
TRIGONOMETRY: MIDTERM EXAM REVIEW 2015
UNIT RF – RELATIONS
AND
FUNCTIONS
DUE DATE: ______________
Directions: Complete ALL questions for full credit. Do all work on notebook paper
and/or graph paper.
RF-1 Relations & Functions
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Express relations as tables, mappings, graphs, rules
Determine if a relation is a function and be able to explain why or why not.
Given two variables determine the independent and the dependent variable.
Determine if a relation is discrete or continuous.
Find the domain and range of relations (using inequalities or interval notation)
Identify x and y-intercepts
Determine symmetry and if a function is even, odd or neither.
Evaluate using function notation algebraically and graphically.
RF-2 Function Operations & Inverses
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Add, subtract, multiply, divide and compose functions
Find the domain of a function from its equation and its graph.
Find the inverse of a function.
Verify inverses algebraically using composition.
Determine if a function is 1-1.
RF-3 Transformations of Parent Functions
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Identify and graph the transformations of different parent functions without a calculator.
RF-4 Piece-wise Defined Functions
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Graph, evaluate, and apply piece-wise defined functions.
RF-5 Linear and Absolute Value Inequalities
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Graph and shade appropriately linear and absolute value inequalities.
RF-6 Systems of Linear and Non-linear Equations
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Solve systems of linear and non-linear equations by graphing, substitution and elimination.
RF-7 System of Equations in Three Variables
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Solve system of equations with three variables using the graphing calculator.
MD/PH/CR/RS 1-14
Page 1 of 10
ALGEBRA II
WITH
TRIGONOMETRY: MIDTERM EXAM REVIEW 2015
PROBLEM SET
Part I
1)
Relations & Vocabulary
Consider the relation {(-3,-5), (0, 1), (2,5), (4, 9)}.
a) State the domain.
b) State the range.
c) Is the relation a function? Why or why not?
d) What is the rule?
e) State the inverse.
2)
Determine which variable is the independent variable and which is the dependent
variable. For each situation, if it were graphed, would it be discrete or continuous.
a)
The faster you drive your car, the longer it will take to come to a complete stop.
b)
The prom committee is selling tickets to the Junior Prom. The more tickets that
they sell, the greater the amount of money they can spend for decorations.
Part II: Functions
Determine if a relation is a function and be able to explain why or why not.
1) (-2, 3), (0, 3), (-2, 4), (1, 6), (-5, -1)
3)
6)
2) (5, -6), (-2, -6), (1, -4), (2, -6), (0, 3)
4)
5)
State the domain of for each function:
a)
f(x) = 8 - x
MD/PH/CR/RS 1-14
b)
f(x) = x2 – 3x + 2
c)
f(x) =
x
2x 2 - 7x + 3
Page 2 of 10
ALGEBRA II
WITH
TRIGONOMETRY: MIDTERM EXAM REVIEW 2015
7)
Determine the symmetry of the following functions and whether they are odd,
even or neither. Show the algebraic work to defend your answer.
a)
f(x) = -6x4 + x2 + 2
Part III
Domain and Range
b)
f(x) = x3 – x + 1
For each graph below, determine the domain and range and the x and y intercepts.
1)
2)
3)
4)
MD/PH/CR/RS 1-14
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ALGEBRA II
Part IV
1)
WITH
TRIGONOMETRY: MIDTERM EXAM REVIEW 2015
Operations of Functions
If f(x) = x2 + 3 and g(x) = 4x – 1, find the following:
a)
f(x) – g(x)
e)
If g(x) = 9, solve for x.
Part V
b) f(x)  g(x)
c)
f(g(x))
d)
g(f(x))
f)
4f(x – 2)
g)
f(2x)  g(3x)
Inverses
1)
Find the inverse of the function f(x) = 4x - 2. Is the inverse a function?
2)
Verify that f and g are inverses of each other. (Show f(g(x)) = g(f(x)) = x)
f(x) = 6x - 5
Part VI
and
g(x) =
1
6
x+
5
6
Evaluating Functions
g(x)
f(x)
1) Using the graph answer the
following.
a)
h(-4)
b)
g(x) = 3, x = ?
c)
f(-3)
d)
f(g(-2))
e)
h(a) = 1,
find f(a + 1).
h(x)
MD/PH/CR/RS 1-14
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ALGEBRA II
2)
WITH
TRIGONOMETRY: MIDTERM EXAM REVIEW 2015
ìï2x + 1
Evaluate the function for the given value of x. f (x) = í 2
ïîx - 2
x < -1
x ³ -1
a)
d)
f(0)
Part VII
b)
f(-2)
c)
f(-1)
Graphs
1)
Sketch each function. Use graph paper.
a)
ì3x -2,
x £ -2
ïï
2
f (x ) = íx + 1, - 2 < x < 1
ï6,
x ³1
ïî
b)
h(x) = 2 x - 2 + 1
c)
f(x) = -4 | x + 2 | - 3
d)
g(x) = - ½ (x – 1)2 + 5
e)
ìï
y £ 3x - 2
í
ïî x + 4y > 8
Part VIII
f(5)
ì
2
ï y = -(x + 3) + 5
5) í
y ³ x +2
ïî
Systems of Linear and Nonlinear Equations
1)
Solve the following systems by graphing. Use graph paper.
ì
1
ïf(x) = x + 1
ïìf(x) = (x + 2)2 - 6
a) í
b) í
3
ïg(x) = - x - 2 + 3
îïg(x) = 6x - 3
î
2)
a)
Solve the following systems by substitution or elimination.
ìï y2 - 3x2 = 6
ìïx2 + (y - 5)2 = 25
b) í
í
ïî y = 2x - 1
ïî y = x2
MD/PH/CR/RS 1-14
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ALGEBRA II
c)
WITH
TRIGONOMETRY: MIDTERM EXAM REVIEW 2015
ìx + 2y + z = 10
ïï
í2x - y +3z = - 5
ï2x -3 y - 5z = 27
ïî
3)
Declare variables, write a system of equations, and solve using your graphing
calculator.
a) In the school cafeteria, Mike purchased a cookie, two chicken fingers, and a can of
Pepsi for $4.30. Jennifer purchased three cookies, one chicken finger, and two
cans of Pepsi for $5.30. Scott purchased two cookies, one chicken finger, and two
cans of Pepsi for $4.55. What is the individual price of each item?
b) Sarah has three parrots: Frank, Bob, and Steve. Steve is three times older than
Bob. Frank is twice the combined ages of Bob and Steve. The sum of their ages is
72. How old are Sarah’s parrots?
4)
Given the system of constraints and objective quantity shown below, graph the
feasible region. Identify the vertices and determine the maximum and minimum
values of the objective quantity.
Objective Quantity
MD/PH/CR/RS 1-14
C = 3x + 6y
ì x³0
ï y³0
ï
Constraints í
ïx + 4 y £ 1
ïî x + y £ 2
Page 6 of 10
ALGEBRA II
WITH
TRIGONOMETRY: MIDTERM EXAM REVIEW 2015
UNIT QF – QUADRATIC FUNCTIONS
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DUE DATE: ______________
Perform Operations and simplify expressions with Complex Numbers
Graph Quadratic Functions in Standard Form
Graph Quadratic Functions in Vertex Form
Write Quadratic Functions in Standard Form and Vertex Form
Graph Quadratic Inequalities
Solve Quadratic Equations by Factoring, finding Square Roots, Completing the Square,
or using the Quadratic Formula
Determine the number and nature of solutions using the discriminant
Applications
PROBLEM SET
Simplify:
1)
- 144
2)
- 28 +
- 63
3)
i15
4)
(3i)(2i)
5)
(2 + 3i) + (5 - 4i)
6)
(2 + 3i) - (5 - 4i)
7)
(2 + 3i)(5 - 4i)
8)
(2 + 3i)
(5 - 4i)
9)
(3i)5
10) (4 -5i)2
Factor:
1)
x2 - 7x + 6
2)
x3 - 8
3)
4y3 + 108
4)
3x3 - 24x2 + 21x
5)
8x3 + 27
6)
4x2 + 12x + 9
7)
x2 - 81
8)
64x3 - 27
9)
15x3 + 10x2 + 6x + 4
10) k3 + 4k2 – 9k – 36
MD/PH/CR/RS 1-14
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ALGEBRA II
WITH
TRIGONOMETRY: MIDTERM EXAM REVIEW 2015
Solve:
11) 3x2 - 24 = 0
12) 5x2 + 19x = 125 + 19x
13) x2 - 10x - 4 = 0
14) -2x2 + 3x -7 = -9
15) x2 + 5x + 7 = 0
16) 2x2 - 2x = -3
For each of the following, find: the vertex, axis of symmetry, y-intercept, and xintercepts. Then graph the function.
17) y = x2 - 4x + 3
18) y = 2x2 - x – 6
19) Graph the quadratic inequality:
y > x 2 + 6x + 6
20) The length of a rectangle is 16 cm longer than its width. The area of the rectangle
is 65 m2. Find the dimensions of the rectangle, rounding to the nearest hundredth of a
meter.
21) Marcus is shooting of a rocket from a 160 foot cliff at a velocity of 48 ft/sec.
a) Find the time it takes the rocket to hit the ground?
b) What is the maximum height the rocket reaches?
c) How long does it take for it to reach the height in problem b?
MD/PH/CR/RS 1-14
Page 8 of 10
ALGEBRA II
WITH
TRIGONOMETRY: MIDTERM EXAM REVIEW 2015
UNIT PF – POLYNOMIAL FUNCTIONS
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DUE DATE: ______________
Polynomial division – long and synthetic
Remainder & Factor Theorems
Rational Zero Theorem
Connection between zeros/factors/solutions
Fundamental Theorem of Algebra
Writing polynomial equations/functions from zeros
Graphing polynomial functions. Identify key points.
Applications
PROBLEM SET
1)
Find each quotient.
a) (12x4 - 4x2 - 3)  (2x2 + 1)
c)
(6x3 + 11x2 - 4x - 9)  (3x - 2)
b)
(8x3 + 2x2 - 5)  (x + 1)
d)
(3x4- 5x3 + 15x2 – 4x + 3)  (x2 - x + 4)
2)
What is the remainder when (x3 - 2x2 - 9)  (x + 5)?
3)
Is (x + 4) a factor of (x3 - 12x + 16)?
4)
Find the missing factors for each of the following.
a)
x3 - 5x2 - 12x + 36 = (x + 3)(
b)
2x3 - 13x2 - 13x + 42 = (x - 7)(
)(
)
)(
)
5)
List the possible rational zeros given by the Rational Zero Test for the function:
f(x) = 4x3 + 15x2 – 8x – 3
6)
How many solutions does the equation 2x3 – 4x2 + 6x = 7 have? Explain your answer.
7)
Two of the zeros of f(x) = x3 + 3x2 – 10x – 24 are 3 and – 4. The third zero must be
what kind of a number? Why?
8)
Write a polynomial function that has the given zeros and a leading coefficient of 1.
a) -6, 4, 2
b) 4, 3i
MD/PH/CR/RS 1-14
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ALGEBRA II
9)
WITH
TRIGONOMETRY: MIDTERM EXAM REVIEW 2015
One zero of f(x) = x3 – 2x2 – 9x + 18 is x = 2. Find the other zeros.
10) Factor f(x) = 2x3 + 11x2 + 18x + 9 given that f(-3) = 0.
11) Find all the real zeros of f(x) = 10x4 – 3x3 – 29x2 + 5x + 12.
12) Find all the complex zeros of the polynomial function f(x) = x4 + x3 + 2x2 + 4x – 8.
Graphing Polynomial Functions - Find the following characteristics and graph the
polynomials.
A)
x-intercept(s)
B)
y-intercept
C)
Right End Behavior x ® ¥
D)
Left End Behavior x ® -¥
E)
Table of Values
F)
Turning Points
13) f(x) = x3- 6x2 +8x
14) g(x) = 2x4- 18x3 - 5x2 + 42x + 27
15) The volume of a rectangular prism is 300 cubic inches. If the height of the prism is
x, the length is (x + 5), and the width is (x + 1), what is the height of the prism?
16) In a study, scientists found that a person’s score, S, on the step-climbing exercise
test was related to a persons amount of hemoglobin, x (in grams per 100 ml. of blood),
according to this function. S = -0.015x3 + 0.6x2 – 2.4x + 19 Approximately how much
hemoglobin was found in person who scores 75?
MD/PH/CR/RS 1-14
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