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End of Course Algebra I Study Guide:
November, 2011
End-of-Course Exam Items
Estimated Number of Questions (37 questions)
Numbers & Operations
Function Characteristics
Linear Functions & Inequalities
Data Analysis
6-8
11-13
11-13
6-8
Format of exam questions: 29 multiple choice, 5 completion items and 3 short answer questions. A
graphing calculator can be used for all questions and will be cleared at the start of the exam. A formula
sheet and graph paper will be provided in the test booklet.
Study Guide Organization:
On the next page is the Review Topic Index. These items are key “green” standards that will be assessed
for graduation. The number of test questions for each topic heading is shown to help guide your study
priorities. Following the Index are Sample Questions for each topic. After each set of questions, there
are STUDY NOTES that explain the answers.
Review Topic Index:
(A2, A7) Numbers & Operations
6-8 questions
Compare & Order Real Numbers
Evaluate expressions with variables
Exponents, roots, use properties to evaluate
Evaluate Exponential Functions
Arithmetic & Geometric Sequences
Solving equations with several variables: ex. (A = prt)
(A3) Function Characteristics
11-13 questions
Relations and Functions
Domain, range, finding roots
Functions defined piecewise
Independent, dependent variables
Multiple representations: symbolic, graph, table, words
Connections between representations
Evaluate f(x) at a, solving f(x)
1
(A4) Linear Functions & Inequalities
11-13 questions
Write & Solve Linear Equations
Graph Linear Equations
Point slope forms, translate between
Interpret Slope and intercepts
Parallel & perpendicular lines
Write & Solve systems of two linear equations
Absolute value in equations
Graphing Absolute value
(A6) Data Analysis
Summary Statistics
Valid Inferences
Univariate Data
Linear Transformations
Effect on Center & Spread
Fit an equation to a line
Best fit lines
Predicting from data
Correlation of data in Scatterplots
6-8 questions
2
(A2A) Numbers & Operations
Compare & Order Real Numbers
Sample Question A2A1:
1. Order the following from greatest to least.
, 3 , 8.9, 8,
o
A. 8, 8.9,
o
B.
o
C. 9.3
o
D. 3 , 9.3
, 9.3
, 9.3
,3
, 3 , 8, 8.9,
,3 ,
, 9.3
,
,
, 8.9, 8
,
, 8.9, 8
Sample Question A2A2:
2. A star’s color gives an indication of temperature and age. The chart shows four types of stars and the
lowest temperature of each type. List the temperatures in order from lowest to highest.
Type
A
Lowest Temperature
1.35
Color
Blue-White
B
2.08
Blue
G
9.0
Yellow
P
4.5
Blue
o
A. 1.35
, 2.08
, 4.5
, 9.0
o
B. 1.35
, 4.5
2.08
, 9.0
o
C. 9.0
, 1.35
, 2.08
, 4.5
o
D. 9.0
, 4.5
1.35
, 2.08
,
STUDY NOTES:
When ordering numbers, always be sure which order, greatest to least or least to greatest.
Real numbers include scientific notation, fractions, decimals, exponents and radicals. (Subsets of real
numbers are natural, whole, counting, integers, rational and irrational numbers... ) Not included:
imaginary numbers 4i.
When comparing or ordering numbers, the fastest way is to use your calculator and convert all numbers
to decimal approximations, then order. Scientific notation is a way to write very large or very small
numbers. The base number is always written as one place followed by a decimal: 1.2343
, not as
123.43
.
3
To convert from
to standard, move the decimal 3 places to the right, From 5.2 to 5200
To convert from
Radicals:
to standard, move the decimal 3 places to the left, From 5.2 to 0.0052
: to approximate, use your calculator. It is the opposite of exponents, it undoes an
exponent.
To convert from a fraction to a decimal, divide the top by the bottom
Sample Question A1.2.A/M1.6.A
3. Which numbers are both less than 
5
6?
6
5
O A.
-2.1 and 
O B.

O C.
-0.65 and -1.2
O D.

2
3

and
3
4
2
3 and -0.8
(A2B) Numbers & Operations
Evaluate expressions with variables
Sample Question A2B1:
4. For what values of a is
o
o
o
o
A.
B.
C.
D.
an integer?
a = 1, 0
a≤1
a = 1, a ≠ 0
a>1
Sample Question A2B2:
5. Evaluate 2w + 6y2 for w = 4 and y = 3.
o
o
o
o
A.
B.
C.
D.
330
62
60
44
4
STUDY NOTES:
Evaluate means to find the value of an algebraic expression by substituting a number for each variable
and simplifying by using order of operations. PEMDAS. Do (P)arenthesis first, then any (E)xponents,
(M)ultiply and (D)ivide from left to right, then (A)ddition and (S)ubtraction left to right.
is undefined.
=5
distributive property: ex. 2(c + 4) = 2c + 8 ex. 3f(f – g3) = 3f2 – 3fg3
Test hint: rewrite the expression with the substitution, then use your calculator for each step. Verify
order of operations one step at a time. Do not just use your calculator left to right.
-a2 does not equal (-a)2
= 3,
= 3, Absolute value is the distance from 0 to the expression inside the brackets.
(A2C) Numbers & Operations
Exponents, roots, use properties to evaluate
Sample Question A2C1:
6. Simplify: 2-2325
223-352
o
o
o
o
A. 35
52
B. 24325
C. 35
245
D. 32
245
Sample Question A2C2:
7. Simplify:
o
A.
o
B. 4
o
C.
o
D.
5
Sample Question A2C3:
8. Simplify the expression using positive exponents.
o
o
o
o
A.
B
C.
D.
x3
x5
x20
x4
STUDY NOTES:
24 = 2×2×2×2 = 16
When simplifying exponents, change negative exponents to fraction form as shown below.
1
23
To simplify square or cube roots, look for factors that are perfect squares:
=
=2
Or perfect cube roots:
(A7AB) Numbers & Operations
Evaluate Exponential Functions, approximate solutions using graphs or tables.
Sample Question A7A1:
9. You won a door prize and are given a choice between two options. A: $150 invested for 10 years at
4% compounded annually, or B: $200 invested for 10 years at 3% compounded annually. Which plan is
best and what is the final amount of the investment? Investment = P(1+r)t
o
o
o
o
A.
B.
C.
D.
A; $242
B; $222
A; $4,338
B; $268
Sample Question A7B1:
10. Select the set of ordered pairs that represents an exponential function.
o
o
o
o
A.
B.
C.
D.
(0,0) (-2,2) (1,1) (2,2)
(4,6) (2,3) (6,8) (10,12)
(1,1) (3,9) (2,4) (0,0)
(0,0) (-1, 2) (-2, 3) (2,3)
6
STUDY NOTES:
An exponential function has the form f(x) =abx where initial amount a
, and the base ratio b≠ 0, b > 0
Typical examples are compound interest and growth rates. The graph of an exponential functions is not
linear, it increases at a progressively faster rate. (Exponentially)
For example: population growth of crickets: f(x) =2(4)x models the growth in weeks (x) of 4 initial
crickets, with a base growth rate of 2 In 8 weeks, we will have f(8) =4(2)8 = 1,024 crickets.
An example of an exponential graph of these
Points (1,1) (1,2) (2,4)
Items may be presented as graphs, in tables
or coordinates. Functions can be increasing,
decreasing, be positive or negative,
(A7D) Numbers & Operations
Solving equations in several variables: ex. (A = prt)
Sample Question A7D1:
11. Solve A = p + prt for p
o
A.
o
B.
o
C.
o
D.
Sample Question A7D2:
12. Solve
for r
o
A.
o
B.
o
C.
o
D.
7
(A3) Function Characteristics
Relations and Functions
Sample Question A3A1:
13. Which of the following equations listed below determine y as a function of x?
o A. x = 3; y = 2x + 1
o B. y = x2 + 1; x = 0
o C. x2 + y2 = 1; x = y
o D. 3x – 2y = 7; y = 2x
Sample Question A1.3.A/M1.2.A
14. The equation of a function is shown.
f ( x)  x  1
What is the domain of f(x)?
O A.
All real numbers
O B.
All real numbers except -1
O C.
All real numbers greater than -1
O D.
All real numbers between -1 and 1
Sample Question A3A2:
15. Which of the following equations listed below determine y as a function of x?
A. y = 3/x; x = 3
B. 2x – y = 0; x = y2
C. y  4 x  3 ; y = 3x
D. y 
x  3 ; x2 + y2 = 9
8
Sample Question A3A3:
16. Which of the following tables, situations, or graphs represent a function?
A. The age in years of each student in your math class and each student’s shoe size.
B. The number of degrees a person rotates a spigot and the volume of water that comes out of the
spigot.
C.
Hours studied 2
2
3
1
Score on test 85% 92% 90% 70%
D.
STUDY NOTES:
A function is a relation where every input has exactly one output. Functions can be expressed in several
ways:
A story problem. Every student in the class and their eye color would be a function since everyone only
has one eye color. It is OK that two students have the same color. However, eye color and every
student would not be a function since the color blue might have more than one student associated with
it.
A graph. To figure out if a graph is a function, we do what is called the vertical line test. If you take a
vertical line and sweep the entire graph from right to left and it only hits the graph once at all times,
then it is a function. So the first graph is a function, but the second graph isn’t:
A table. The left hand side of the table if in columns (or the top of the table if in rows) is usually the
input, or x values of the function. The right hand side (or bottom) is usually the output. If there are any
repeats of the input or x values where the output is different, then it is not a function. The first table is a
function, the second table is not.
x 1 2 3 4
y 3 3 4 5
x
1
1
2
3
y
3
4
5
6
9
An equation: f(x) This notation, read “f of x”, says that f is a function where x is the input. Any time
you see an equation written as f(x) = it is saying it is a function.
y If the equation has only y’s and x’s in it, then this is a bit harder. As long as you can
manipulate the equation so it is a pure y = then it is a function. y2 = doesn’t count.
The first 3 are functions, the last 2 are not:
4y – 3x = 2y + 1
f ( x)  x 2  1
y  3 x  9
x2 + y2 = 1
y 3
(A3) Function Characteristics
Domain and Range
Sample Question A3A4:
17. A function f(n) = 60n is used to model the distance in miles traveled by a car traveling 60 miles per
hour in n hours. Identify the domain and range of this function. What restrictions on the domain of this
function should be considered for the model to correctly reflect the situation?
A. Domain: hours car traveled
Range: distance car traveled
Restriction: hours must be greater than or equal to zero
B. Domain: distance car traveled
Range: hours car traveled
Restriction: distance must be greater than or equal to zero
C. Domain: hours car traveled
Range: miles per hour car traveled
Restriction: hours must be greater than or equal to zero
D. Domain: distance car traveled
Range: miles per hour car traveled
Restriction: distance must be greater than or equal to zero
Sample Question A3A5:
18. What is the domain and range of f ( x) 
x5 ?
A. Domain: x can be all real numbers
Range: f(x) can be all real numbers
B. Domain: x = 5
Range: f(x) = 0
C. Domain: x  5
Range: f(x) is less than or equal to zero
D. Domain: x  5
Range: f(x) is greater than or equal to zero
10
Sample Question A3A6:
19. Below is the graph of f ( x)  2 x  1  5 . Determine the domain and range of this function.
A. Domain: x  0
Range: f ( x)  0
B. Domain: x  5
Range: f ( x)  5
C. Domain: x can be all real numbers
Range: f ( x)  5
D. Domain: x can be all real numbers
Range: f(x) can be all real numbers
STUDY NOTES:
Domain is the input, normally x, of the function. Sometimes it would be easier to think of what can’t be
in the domain. Range is the output, normally y or f(x). There are four ways to think of domain and
range:
Table: The x column (or row depending on how the table is oriented) shows the values of the domain
and the y column (or row) shows the values of the range. Below the domain would be 1, 2, 3, 4 and the
range would be 5, 6, 7, 8.
x 1 2 3 4
y 5 6 7 8
Graph: Look along the x-axis. If there is a graph drawn above, through, or below the x-axis at that spot,
then that x-value is in the domain. Likewise, look along the y-axis. If there is a graph drawn to the left,
through, or to the right of that spot, then that y-value is in the range. Below the domain would be from
 6  x  5 (estimation) and the range would be from  2  f ( x)  5 (the range could also be
written as  2  y  5 ).
Equation: Looking at domain first, there are only two things you can’t do: divide by zero or take the
square root of a negative number. Otherwise the domain can be all real numbers. So for example,
f ( x)  4 x  2 has no division sign and no square root sign so the domain is all real numbers.
However, f ( x)  x  9 has a square root sign over the x so the domain can only be numbers greater
than or equal to 9, since if I were to put an 8 into the equation that would be the square root of negative
1. Likewise, f ( x ) 
9
has a division sign where x is in the bottom so the domain can be everything
x
except what makes the bottom zero. In this case the domain is everything but 0.
11
Now consider range. This is much harder in an equation. First determine the domain and then think
of all the values that function can be for those values in the domain. Taking our previous examples
f ( x) 
x  9 has a range that is greater than or equal to zero because putting in a 9, 10, 11, etc.
9
would give you zero or a positive number. Likewise, f ( x )  can be all numbers but zero since we
x
can have both positive and negative x-values.
Story Problem: Whatever is the input, or the independent value, of the story problem is the domain. If
there is a story problem about people, then your domain can’t be negative or fractions since part of a
person doesn’t make since. Likewise, if the story problem is about miles, then your domain must be
greater than or equal to zero since it doesn’t make sense to have negative miles. With range, it is the
same consideration as domain except range is the output, or the dependent value. So if we have a
function where the input is hours and the output is miles, both domain and range would be greater than
or equal to zero.
(A3) Function Characteristics - Functions Defined Piecewise
Sample Question A3A7:
20. According to the USPS government website, dated January 2011, the going rate of a letter
depended on its weight in ounces and could be priced as follows:
Up to 1 oz $.44
Up to 2 oz $.61
Up to 3 oz $.78
Up to 3.5 oz $.95
Weights larger than 3.5 oz, large envelope prices apply.
How much would a letter weighing 2.5 ounces cost?
A. $.61
B. $.78
C. $1.05
D. $1.53
Sample Question A3A8:
21. AT&T’s website lists prices for cell phone plans. As of January 2011, the cell phone prices for an
individual plan depending on minutes used can be priced as follows:
450 minutes for $59.99
900 minutes for $79.99
1350 minutes for $99.99
2000 minutes for $119.99
How much would an individual cell phone plan cost if a person wants to make sure they are covered for
1000 minutes?
A.
B.
C.
D.
$79.99
$99.99
$139.98
$239.97
12
Sample Question A3A9:
22. Ms. Namtvedt wants to register her math club for the Washington State Mathematics Competition
held on March 9, 2011. The fees for competing are as follows:
$15 per participant for the first 6 participants
$100 for 7 – 10 participants
$130 for 11 – 15 participants
$115 for 16 – 20 participants
$175 for more than 20 participants
How much would math club be charged if registering 14 participants?
A. $78
B. $130
C. $140
D. $210
STUDY NOTES:
Functions Defined Piecewise
Piecewise functions are used when a function changes from one equation to another. This may be
represented graphically, such as below. Notice the function changed from a
line to a curve when x = 3.
Piecewise functions can also be represented in a story problem. Part of the story problem has a
rule or equation for certain values in the independent, or x, values which then change to a different rule
or equation for other values in the independent, or x, values. A common example is a cell phone plan
which charges one fee for 0 – 400 minutes of use but then changes fees if using more than 400 minutes.
Piecewise functions can be represented as an equation too, as shown below. However, for the
x
EOC we do not deal with piecewise equations. f ( x)  
x
2
if
x3
x3
(A3) Function Characteristics
Independent & Dependent Variables
Sample Question A3A10:
23. Brad measured the size of each of his students’ feet, f, and their corresponding height, h. He then
formed a function relating the two variables. What is the dependent variable?
A.
B.
C.
D.
feet, f
height, h
feet per height, f/h
height per feet, h/f
13
Sample Question A3A11:
24. Marilyn kept track of how tall her son was every year since he was born. If Marilyn were to write a
function relating height, h, and years, y, what would be the independent variable?
A.
B.
C.
D.
height, h
years, y
height per year, h/y
year per height, y/h
Sample Question A3A12:
25. Tammy wanted to be a professional basketball player. Ever since she was a child she would keep
track of what was the average salary for a female basketball player since the year 1995. If Tammy were
to write a function relating salary, s, and years, y, what would be the dependent variable?
A.
B.
C.
D.
salary, s
years, y
salary per year, s/y
year per salary, y/s
STUDY NOTES:
Independent & Dependent Variables
To analyze independent and dependent variables, it is often easier to figure out the dependent
variable first. The dependent variable depends on the independent variable. For example, if a function
is relating miles and hours driven one could see that the number of miles would depend on how long
one is on the road, hours. Therefore miles would be the dependent variable and hours would be the
independent variable. Likewise, a height of a tree depends on the number of years it has grown, not the
other way around, so height would be the dependent variable and years would be the independent
variable.
If a relationship is written in function notation for our tree example so that h(t)=3t where t is
time in years and h(t) is height of the tree in inches, then the dependent variable would be the variable
inside the h(t), namely t, and the independent variable would be outside of the notation, namely h.
14
(A3) Function Characteristics
Multiple Representations
Sample Question A3B1:
26. The chart shows the amount of total salary (commission plus base salary) paid to employees of a car
dealership in one week.
$1000
$900
$800
$700
$600
$500
$400
$300
$200
1 2 3 4 5 6 7 8 9
Number of cars sold
1
2
3
5
7
8
Total Salary
$200 $300 $400 $600 $800 $900
Which equation best represents total salary (T) that an employee makes for selling any number of cars,
c, in a week?
A. T(c) = 100c + 100
B. T(c) = 100c + 200
C. T(c) = 100(c + 200)
D. T(c) = c + 200
Sample Question A3B2:
27. A population of bacteria doubled every hour, starting with 4 bacteria. The chart below represents
the total population, p, at a given hour, t.
70
60
50
40
Num ber of
bacteria per
hour
30
20
hour
0 1 2 3
Number of bacteria 4 8 16 32
10
0
1
2
3
4
Which equation best represents the total population (P) of bacteria at a given hour, t.
A. P(t) = 2t
B. P(t) = 2t + 4
C. P(t) = 4t
D. P(t) = 4(2t)
15
Sample Question A3B3:
28. The graph below shows a model for the relationship between the banking angle and the turning
radius for a bicycle traveling at a particular speed. For the values shown, the banking angel B and the
turning radius r can be approximated by an inverse variation.
Radius
2
4
7
8
Banking Angle 560 280 160 140
Which equation best represents the banking angle B for the turning radius r?
A. B(r) = 28r
B. B(r) = 2r – 28
C. B(r ) 
r  3134
112
D. B ( r ) 
r
STUDY NOTES:
Multiple Representations
Every chart, table, equation, and story problem corresponds to a set of ordered pairs. To match
a graph to a table, or a table to an equation, or an equation to a graph, the ordered pairs in one form
must match the other form. For example, if a graph goes through the points (2, 5), (3, 6) and (4, 7) it
would correspond to the equation f(x) = x + 3 because f(2) = 5, f(3) = 6 and f(4) = 7. Likewise if a graph
goes through the same points, it would correspond to the table:
x
2 3 4
f(x) 5 6 7
(A3) Function Characteristics
Connections Between Representations
Sample Question A3B4:
29. Which equation listed below best represents the set of data in the table?
x -2 -1 1 3 6
y -4 -1 5 11 20
A.
B.
C.
D.
y = 3x + 2
y = – 3x + 2
y=x–2
y= –x–6
16
Sample Question A3B5:
30. Which function includes all of the ordered pairs in the table?
x
-2 0 1 2 3
f(x) 5 -3 -1 5 15
A. f(x) = x – 3
B. f(x) = x + 7
C. f(x) = 2x2 – 3
D. f ( x) 
x  27
Sample Question A3B6:
31. Which equation listed below best represents the set of data in the table?
x -3 -1 2 3 5
y -8 -4 2 4 8
A.
B.
C.
D.
x–y=2
2x – y = 2
x+y=–2
x + 2y = – 2
STUDY NOTES:
Connections Between Representations
For equations to match data, the equation must be true for all data points. In a table, the top
row (or left column if vertical) is what you input into the equation. The bottom row (or right column if
vertical) is what you should get out of the equation. If every data point matches for an equation, that
equation best represents the data. For example, consider the table:
x -2 0 3 4 5
y -2 -6 3 10 19
The equation for this table is y = x2 – 6 because it works for all values. y = x works for (-2, -2) but not for
(0, -6). y = x – 6 works for (0, -6) but not for (3, 3).
For equations to match graphs, the same thing applies. Each point along the graph can be
represented as (x, y). So an equation must generate all the points on the graph. Note that if an
equation is represented in function notation, f(x), then the equation works for data if when substituting
x into the equation it gives you f(x).
(A3) Function Characteristics - Evaluate f(x) at a, solving f(x)
Sample Question A3C1:
32. For the function f ( x)  3 x  1 , find f (5) .
A.
B.
C.
D.
– 12
– 2.4
12
18
17
Sample Question A3C2:
33. For the function f ( x)  4 x  6 , find f (2) .
A. – 14
B. – 7
C. – 2
D. 2
Sample Question A3C3:
34. For the function f ( x) 
A.
x  3 , find f (6)
1
2
B. 6
C. 3
D. 9
Sample Question A3C4:
35. For the function f ( x)  3 x  6 , find x when f ( x)  0 .
A. – 2
B. 0
C. 6
D. 9
Sample Question A3C5:
36. For the function f ( x)  9 x  17 , find x when f ( x)  1 .
A. – 17
B. – 8
C. 1
D. 2
Sample Question A3C6:
37. For the function f ( x)  2 x  9 , find x when f ( x)  3 .
A. – 9
B. – 3
C. 3
D. 6
STUDY NOTES:
Evaluate f(x) at a, solving f(x)
f(x) means the “function with respect to x”. It does NOT mean f times x. When you see f(x) you
automatically know that what follows the equal sign is a function. The independent variable, or the
input, is x and the dependent variable, or the output is f(x). Take the function f(x) = 3x + 7. Sometimes it
is easier just to think of it as y = 3x + 7 since that is how we write equations more often.
18
Evaluating f(x) = 3x + 7 at 2. In other words, find f (2).
Using the familiar equation y = 3x + 7, I could ask “what is the value of y if x is 2?” To answer this I would
substitute a 2 into the x and get y = 13. Another way to ask the question would be to use the first
notation and say “find f (2)”. I would substitute a 2 into the x and get f (2) = 13. See the following
examples for evaluating a function at a particular point:
Find f (3) if f(x) = 4x – 7 .
Answer:
Replace the x with a 3.
f (3) = 4(3) – 7
4(3) – 7 = – 1.
So f (3) = – 1.
Find f (4) if f ( x)  x  9 .
Answer:
Replace the x with a 4.
f (4) = 4  9
49  5  5.
So f (4) = 5.
Solving f(x) = 3x + 7 for x when f (x) = 1.
Using the familiar equation y = 3x + 7, I could ask “what is the value of x when y is 1?” To answer this I
would substitute a 1 for the y and solve for x. 1 = 3x + 7 would give me an answer of x = – 2. Another
way to ask the question would be to use the first notation and say “find x when f(x) = 1.” I would
substitute a 1 for the f(x) and get 1 = 3x + 7. See the following examples for solving a function for x
when f(x) is given:
For the function f(x) = 4x – 1, find x if f (x) = 3.
Answer:
Replace f(x) with a 3.
3 = 4x – 1.
3 + 1 = 4x – 1 + 1
4 = 4x
4/4 = 4x/4
1=x
So x = 1 when f(x) = 3.
For the function f(x) = 2x – 20, find x if f(x) = 8.
Answer:
Replace f(x) with an 8.
8 = 2x – 20
8 + 20 = 2x – 20 + 20
28 = 2x
28/2 = 2x/2
14 = x
So x = 14 when f(x) = 8.
19
(A4) Core Content: Linear functions, equations, and inequalities
(A4A) Write and solve linear equations and inequalities in one variable
Sample Question A4A1:
38. Solve the equation and chose the correct answer: y -12 = -4
A. y = -16
B. y = -8
C. y = 8
D. y = 16
STUDY NOTES:
Add 12 to both sides of the equation.
Sample Question A4A2:
39. Solve the inequality and chose the correct answer:
A.
B.
C.
D.
x+5³8
x > -3
x>3
x > 11
x<3
STUDY NOTES:
Add -5 to both sides of the inequality.
Sample Question A4A3:
40. George has eight more than twice the number of marbles you have. Write an inequality for the
number of marbles George has.
A. 8 – 2n
B. 2n - 8
C. 8 + (-2n)
D. 2n + 8
STUDY NOTES:
Let n be the unknown so twice the unknown would be 2n . Eight more than twice the unknown would
be 2n + 8.
Sample Question A4A4:
41. Write an absolute value equation for all the numbers 2 units from 7.
A. x + 7 = 2
B. x + 2 = 7
C. x - 7 = 2
D. x - 2 = 7
20
STUDY NOTES:
x - b = a is an absolute value representing all the numbers that are a units from b.
Sample Question A4A5:
42. The equation 2 x -1 -10 = -4 has two real solutions.
Determine the negative solution of the equation and chose the correct answer.
A. -1
B. -2
C. -3
D. -4
STUDY NOTES:
Isolate the absolute value first using the properties of equality. Add 10 to each side and divide by 2. You
will then have
x  1  3 x-1 = 3 or x-1=-3. The negative solution is -2.
Sample Question A4A6:
43. Solve for x: 2(x – 3) + 4x = 14 + 2x
A. 2
B. 5
C. 4
D. 6
STUDY NOTES:
2(x - 3) + 4x = 14 + 2x
distribute
2x – 6 +4x = 14 + 2x
combine like terms 6x - 6 = 14 + 2x
add -2x
-2x
-2x
4x – 6 = 14
add 6
4x = 20
divide by 4
x =5
Sample Question A4A7:
44. Solve x - 6 £ 4
Choose the correct answer: (and be able to graph the solution on a number line)
A. 2 < x < 10
B. 2 > x > 10
C. x < 2 and x < 10
D. x > 2 and x > 10
21
STUDY NOTES:
x b  c
means x – b < c and x – b > - c
When solving linear equations you must use the properties of equality.
If a=b then a + c = b + c, add the same number to each side
or a – c = b - c , subtract the same number from each side.
or a ´ c = b ´ c , multiply the same number by each side
a b and if c is not 0 then

c c
What you do to one side of the equation you do to the other.
Also remember to simplify each side of the equation by using the distributive property a(b + c) = ab + ac
and combining like terms for example 2x + 5x = 14 should be simplified to 7x.
Solving linear inequalities work the same way with one exception: when you multiply or divide each
side by a negative.
(A4B) Write and graph an equation for a line given the slope and the y-intercept, the slope and a
point on the line, or two points on the line, and translate between forms of linear equations.
Sample Question A1.4.B/M1.3.D
45. Which equation represents the line that passes through the points (2, 2) and (4, 1)?
O A.
O B.
O C.
O D.
Sample Question A4B1:
46. Find an equation for a line with y-intercept equal to -2 and slope equal to 3.
A.
B.
C.
D.
y = -2x + 3
y = -2x - 3
y = 3x - 2
y = 2x – 2
STUDY NOTES:
A line is written in slope intercept form if y=mx + b. m is the slope of the line and b is the y-intercept.
Sample Question A4B2:
47. Find an equation for a line with a slope of -1 that goes through the point (-3, 2).
A. y = -x + 1
B. y = -x – 1
C. y = 2x - 1
D. y = 3x – 2
22
STUDY NOTES:
Point-slope form of a line is (y – y1)= m(x – x1) where the point is (x1,y1) and the slope is m. So to write
the equation that has a slope of -1 and goes through the point (-3,2) you would start with y-2=-1(x+3)
and solve for y.
Sample Question A4B3:
48. Find an equation for a line that goes through the points (1, 8) and (-2, -1)
A. y = 3x + 5
B. y = -3x - 5
1
2
x+7
3
3
1
2
D. y = - x - 7
3
3
C. y =
STUDY NOTES:
Given two points, first find the slope using the formula m= (y2-y1)/(x2-x1). Once you have found the
slope, pick one of the points and use the point-slope equation (y – y1)= m(x – x1). The slope of this line is
(8+1)/(1+2) = 3. Then picking the point (-2,-1) the equation would be (y+1)=3(x+2). Solve for y to get
y=3x +5.
Sample Question A4B4:
49. Use the equation y - 7 = 2(x - 2) (without sketching the graph) to describe the graph.
A. the graph has a negative slope and y-intercept of -2
B. the graph has a negative slope and y-intercept of 3
C. the graph has a positive slope and y-intercept of -2
D. the graph has a positive slope and y-intercept of 3
STUDY NOTES:
Using the point slope form of the line, the graph has a positive slope of 2 and goes through the point
(2,7). Solving the equation for y you would get y=2x + 3. So, the graph has a positive slope and a yintercept of 3.
Sample Question A4B5:
50. Write the equation 3x + 2y = 5 in slope-intercept form.
Slope Intercept form is: _______________________
STUDY NOTES:
Solve the equation for y. First subtract 3x from both sides then divide both sides by 2.
23
Sample Question A4B6:
51. Write the equation y -1= 2(x - 2) in standard form.
A. 2x +y =-3
B. -2x + y = 3
C. 2x - y = 3
D. y = 2x - 3
STUDY NOTES:
Standard form is written Ax +By = C, where A,B, and C are integers and A is positive.
Sample Question A4B7:
52. Write the equation y =
1
x + 5 in standard form
2
A. -x + 2y =10
B. x - 2y = -10
1
x =5
2
1
D. y - x = 10
2
C. y -
STUDY NOTES:
Standard form is written Ax +By = C, where A,B, and C are integers and A is positive. Multiply both sides
of the equation by 2, getting 2y = x + 10, then subtract both sides of the equation by 2y and 10 getting x
– 2y = -10
Refer to the graph below that shows a square in the coordinate grid to answer the following questions.
18
B
14
10
8
6
4
2
A
C
D
2 4 6 8 10 14
18
24
Sample Question A4C5:
53. The slope of sides AB and DC are
A.
B.
C.
D.
unrelated
parallel
perpendicular
opposite
STUDY NOTES:
Each slope (rise/run) is ½. They are parallel
Sample Question A4C6:
54. The slope of sides AD and DC are
A.
B.
C.
D.
unrelated
parallel
perpendicular
opposite
STUDY NOTES:
The slope of AD is -2 and the slope of DC is ½. They are opposite reciprocals so they are perpendicular.
Sample Question A4C7:
55. The actual slope of side AB is:
A.
B.
C.
D.
E.
F.
¼
–¼
½
–½
2
-2
STUDY NOTES:
To go from point A to point B you go up (positive) 2 units and over a positive 4. The slope is ½.
25
(A4D) Write and solve systems of two linear equations and inequalities in two variables.
Sample Question A1.4.D/M1.3.E
56. Two lines divide the coordinate plane into the four lettered regions shown.
M
P
N
Q
Which region represents the solution set for the system of inequalities given?
O A.
Region M
O B.
Region N
O C.
Region P
O D.
Region Q
Sample Question A4D1:
y = 2x + 4
1
1
x + 3y = - 6
57. What is the solution to the simultaneous linear system pictured above?
A.
B.
C.
D.
(0, 4)
(-2, 0)
(-3, -1)
(-2, 0)
26
STUDY NOTES:
To solve a system of equations graphically, you would look for the point of intersection and then plug
the point into both of the equations to see if the point is a solution to both equations.
Sample Question A4D2:
58. Solve the following simultaneous linear equations algebraically:
-2x + y = 2
x + y = -1
A. (-1,-2)
B. (-1/3,-4/3)
C. (0,-1)
D (-1,0).
STUDY NOTES:
When solving a system of equations you have two choices:
Solving by substitution- solving one of the equations for either x or y and then substitute into the other
equation.
Solving by linear combinations- multiplying one or both of the equations by a number so that when you
add the equations together one of the variables is eliminated.
Sample Question A4D3:
59. An academic team is going to a state mathematics competition. There are 30 people going on the
trip. There are 5 people who can drive and 2 types of vehicles; vans and cars. A van seats 8 people, and
a car seats 4 people, including drivers. Let v = number of vans and c = number of cars.
v+c≤5
8v + 4c > 30
How many vans and cars does the team need for the trip?
A. 3 cars and 3 vans
B. 3 vans and 2 cars
C. 3 cars and 2 vans
D. 3 vans and 1 car
STUDY NOTES:
The only combination that would satisfy both inequalities would be 3 vans (24people) and 2 cars (8
people). You would also have the five drivers, 3 in vans and 2 in cars.
27
(A4E) Describe how changes in the parameters of linear functions and functions containing an
absolute value of a linear expression affect their graphs and the relationships they represent.
Use the following situation for the next three questions:
A gas station’s 10,000-gallon underground storage tank contains 1,000 gallons of gasoline. Tanker trucks
pump gasoline into the tank at a rate of 400 gallons per minute. How long will it take to fill the tank?
Sample Question A4E1:
60. Find a function that represents this situation.
A.
B.
C.
D.
y = 400x +10,000
y = 400x +1,000
y = -400x +10,000
y = -400x + 9,000
STUDY NOTES:
At 0 minutes, there is 1,000 gallons of gasoline and the rate of change is 400gallons per minute. This
information fits into y=mx + b format.
Sample Question A4E2:
61. If the flow rate increases from 400 to 500 gallons per minute, how will the graph of the function
change?
A.
B.
C.
D.
The y-intercept will increase
The y-intercept will decrease
The slope will decrease
The slope will increase
STUDY NOTES:
The rate of change increases from 400 to 500 gallons per minute, this is the slope. The starting point of
1,000 gallons remains unchanged. The equation would be y=500x+1000
Sample Question A4E3:
62. If the initial amount of gasoline in the tank changes from 1,000 to 2,000 gallons, how will the graph
of the function change?
E.
F.
G.
H.
The y-intercept will increase
The y-intercept will decrease
The slope will decrease
The slope will increase
STUDY NOTES:
At 0 minutes the number of gallons is 2,000. Therefore, the y-intercept would now be (0,2000)
28
Refer to the following two functions to answer the next 3 questions.
Function A: y = 3 x
Function B: y = -
1
x
3
Sample Question A1.4.E/M1.3.B
63. Which statement best describes the transformation of the graph of y  x to the graph
of y  x  2 ?
O A.
The graph shifts up 2 units.
O B.
The graph shifts down 2 units.
O C.
The graph shifts left 2 units.
O D.
The graph shifts right 2 units.
Sample Question A4E4:
64. What can be said about the following two functions?
A.
B.
C.
D.
Functions A and B reflect exactly across the x-axis
Functions A and B reflect exactly across the y-axis
The functions do not reflect
The functions cannot be graphed
STUDY NOTES:
Function A opens upwards in a “v” shape and the slope of the lines is +/-3. Function B opens downward
and has a slope of +/- 1/3. They do not reflect because their slopes are different.
Sample Question A4E5:
65. Which function’s end behavior is negative?
A.
B.
C.
D.
Function A
Function B
Neither
Both
STUDY NOTES:
Because Function B opens downward, y decreases as x increases and decreases.
Sample Question A4E6:
66. Which function has a greater rate of change?
A.
B.
C.
D.
Function A
Function B
Neither
Both
29
STUDY NOTES:
The slope of Function A is +/- 3 and the slope of Function B is +/- 1/3. Function A has a greater rate of
change.
Sample Question A1.5.B/M2.2.B
67. A quadratic function is represented by the given graph.
Which values represent the zeros of the function?
O A.
or
O B.
or
O C.
or
O D.
or
Sample Question A1.5.C/M2.2.D
68. What are the solutions to the equation
O A.
or
O B.
or
O C.
or
O D.
or
?
(A6A) Describing and Assessing Data Sets
Use and evaluate the accuracy of summary statistics to describe and compare data sets.
30
Sample Question A6A1:
69. A local company has a salary dispute. The workers claim that they are underpaid as compared to the
national average. Management claims that they are actually overpaid as compared to the national
average.
Things to consider when answering the following questions:
 In this company there are only a few of the workers earning salaries which are quite high; in fact
these salaries are way above the national average.
 Both of the claims made can be accurately supported by appropriate summary statistics.
Would it be in the managements’ best interest to use the mean or the median when quoting the
‘average’ salary of the workers at this local company? Explain.
What would be in the workers’ best interest when quoting the ‘average’ of local salaries? Explain.
Sample Question A6A2:
70. Each box-and-whisker plot shows the prices of used cars (in thousands of dollars) advertised for sale
at three different car dealers. If you want to go to the dealer whose prices seem least expensive, which
dealer would you go to? Use statistics from the displays to justify your answer.
Company A
Company B
Company C
31
Explain your answer below:
Note: The scale goes from 0 to 10.
Sample Question A6A3:
71. Calculate the mean and standard deviation of the follow data set:
15, 18, 21, 14, 23, 12, 15, 18, 20, 20, 19, 18
o
o
o
o
A. mean = 17.75
B. mean = 17.75
C. mean = 18
D. mean = 18
standard deviation = 3.19
standard deviation = 3.06
standard deviation = 10.18
standard deviation = 3.19
Sample Question A6A4:
72. Calculate the median of the follow data set:
15, 18, 21, 14, 23, 12, 15, 18, 20, 20, 19, 18
o
o
o
o
A. median = 17.75
B. median = 13.5
C. median = 15
D. median = 18
(A6B) Inferences and Conclusions
Make valid inferences and draw conclusions based on data.
Sample Question A6B1:
73. Mr. Shapiro found that the amount of time his students spent doing mathematics homework is
positively correlated with test grades in his class. He is concluding that doing homework makes his
students’ test scores higher.
Explain any flaws in Mr. Shapiro’s reasoning:
32
Sample Question A6B2:
74. Dr. Wilson noticed that as her patients ate more broccoli, their cholesterol levels dropped. She
concluded that eating broccoli causes lower cholesterol levels in the blood.
Explain any flaws in Dr. Wilson’s conclusion:
Sample Question A6B3:
75. The following scatter plot shows the total fuel cost for a train run from San Francisco to Seattle for
trains with different numbers of cars attached to the lead engine.
Fuel Cost vs Train Length
35
30
25
20
Fuel Cost (in
thousands)
15
10
5
0
5
10
15
20
25
30
35
Number of Train Cars
The associated least-squares regression equation is: Fuel Cost = 0.84(Number of Cars) + 2.13
Use the above equation to predict the total fuel cost for a train with 50 cars.
Comment on any concerns you may have with using this equation to make such a prediction:
33
STUDY NOTES:
Since this equation was created from data that had a range of 10 to 32 train cars, it is not a good idea to
use this equation to make estimates beyond this range. The trend may change, the data may not
continue to have the same pattern. (Danger of Extrapolation).
(A6C) Linear Transformations
Describe how linear transformations affect the center and spread of univariate data.
Sample Question A6C1:
76. Due to budget constraints at a particular company, every employee receives a 2% decrease to their
salary. What impact does this salary decrease have on the mean and on the range of employee salaries
at the company?
o
o
o
o
A. The mean and range both decrease.
B. The mean and range do not change.
C. The mean does not change but the range decreases.
D. The mean decreases but the range does not change.
Sample Question A6C2:
77. At a particular company, every employee receives a 4% cost-of-living increase to their salary. What
impact does this cost-of-living increase have on the mean and on the range of employee salaries at this
company?
o
o
o
o
A. The mean increases but the range does not change.
B. The mean does not change but the range increases.
C. The mean and range both increase.
D. The mean and range do not change.
Sample Question A6C3:
78. Suppose that a distribution of a set of numbers has a mean 20 and a standard deviation of 5. If 2 is
added to each score, what will the new mean and standard deviation of this set of numbers?
o
o
o
o
A.
B.
C.
D.
mean
20
20
22
22
standard deviation
7
5
5
7
34
STUDY NOTES:
If you use the data set {15, 20, 25} and add 2 to each number you get a new data set of {17, 22, 27}. It is
easy to see the new mean is 22 with a standard deviation of 5.
Sample Question A6C4:
79. Suppose that a distribution of a set of numbers has a mean 20 and a standard deviation of 5. If 2 is
multiplied to each score, what will the new mean and standard deviation of this set of numbers?
o
o
o
o
mean
20
20
40
40
A.
B.
C.
D.
standard deviation
10
5
5
10
STUDY NOTES:
If you use the data set {15, 20, 25} and multiply each number by 2 you get a new data set of {30, 40, 50}.
It is clear that the new mean is 40 with a standard deviation of 10.
Sample Question A1.6.E/M1.3.G
80. A scatterplot is shown.
y
Which statement describes the correlation of the data graphed in the scatterplot?
O A.
Strong negative
O B.
Strong positive
O C.
Weak negative
O D.
Weak positive
35
x
1. Answer: D
2. Answer: C
3. Answer: A
4. Answer: C
5. Answer: B
6. Answer: C
7. Answer: C
8. Answer: C
9. Answer: D
10. Answer: C
11. Answer: A
12. Answer: A
13. Answer: D
14. Answer: A
15. Answer: C
16. Answer: B
17. Answer: A
18. Answer: D
19. Answer: C
20. Answer: B
21. Answer: B
22. Answer: B
23. Answer: A
24. Answer: B
25. Answer: A
26. Answer: A
27. Answer: D
28. Answer: D
29. Answer: A
30. Answer: C
31. Answer: B
32. Answer: C
33. Answer: A
34. Answer: C
35. Answer: A
36. Answer: D
37. Answer: D
38. Answer: C
39. Answer: B
40. Answer: D
41. Answer: C
42. Answer: B
43. Answer: B
44. Answer: B
36
45. Answer: B
46.
47.
48.
49.
Answer:
Answer:
Answer:
Answer:
C
B
A
D
50. Answer: y  
3
5
x
2
2
51. Answer: C
52. Answer: B
53. Answer: B
54. Answer: C
55. Answer: C
56. Answer: C
57. Answer: C
58. Answer: D
59. Answer: B
60. Answer: B
61. Answer: D
62. Answer: A
63. Answer: C
64. Answer: C
65. Answer: B
66. Answer: A
67. Answer: B
68. Answer: D
69. Answer:
The management would want to use the mean. The few high salaries would pull the mean up and not
affect the median. This would cause the mean to be greater than the median and support their case that
the workers were being overpaid as compared to the national average.
The workers would want to use the median for the same explanation given above. The median would be
lower than the mean and used as support for the claim that the workers are being underpaid.
70. Answer:
Company C should be chosen. The graph needs to be used when providing support of the choice.
Specifically, the quartiles should be compared. For example, “Company C should be picked. This is
because since it has the lowest third quartile, 75% of its cars are cheaper than 75% of the cars from the
other two companies.”
71. Answer: A
72. Answer: D
73. Answer:
Correlation does not imply causation. For example, students who are doing well in mathematics and/or
enjoy it may therefore do the homework more frequently than those who are struggling and/or do not
enjoy the subject. Thus, it might not be the act of doing the homework that is causing the high scores,
but another factor which is making both of the variables (test scores and homework completion) go up.
74. Answer:
Correlation does not imply causation. For example, patients who are eating broccoli may be doing other
things that contribute to lowering cholesterol levels, like exercise. Thus, it might not be the act of eating
37
broccoli that is causing the lower cholesterol, but another factor which is making the variable move (like
exercise).
75. Answer:
Fuel Cost = 0.84(50) + 2.13
Fuel Cost = 44.13 (or $44,130)
76. Answer: A
77. Answer: C
78. Answer: C
79. Answer: D
80. Answer: B
38
NOTE: THE FOLLOWING SAMPLES WERE MOVED TO THEN END OF THIS DOCUMENT BECAUSE THEY
REQUIRE A COLOR COPIER TO INTERPRET THE GRAPHS.
(A4C) Identify and interpret the slope and intercepts of a linear function, including equations for
parallel and perpendicular lines.
Refer to the graph below to answer the following questions about three different cell phone plans: Blue,
Green, and Red. The graphs relate total minutes per month used for each cell phone plan and monthly
cost in dollars.
Sample Question A1.4.C/M1.3.C
Mary is going to deposit an equal amount of money into a checking account each month
until she has saved $500. The amount of money, y, in the account after x months can be
modeled by the equation
.
What does the slope of the graph of the equation represent?
O A.
The amount of money deposited monthly
O B.
The amount of money originally in the account
O C.
The number of months it would take to earn $100
O D.
The number of months it would take to reach $500
Answer: A
39
Sample Question A4C1:
What can be said about the initial cost of each cell phone plan?
A. All the plans begin with the same cost.
B. The Red Plan costs the most.
C. The Blue Plan costs the most.
D. The Green Plan costs the most.
Answer: C
STUDY NOTES:
The initial cost for each plan is indicated by the cost at 0 minutes per month.
Red plan-$20
Green plan- $30Blue plan $40
Sample Question A4C2:
Which plan costs the least if you have 300 minutes of cell phone use in the month?
A.
B.
C.
D.
Blue Plan
Green Plan
Red Plan
Cannot be determined
Answer: A
STUDY NOTES:
At 300 minutes: Blue plan- $40
Red plan- $50
Green plan- $60
Sample Question A4C3:
What is the least number of minutes you can use and have the Green and Red Plans cost the same
amount?
A.
B.
C.
D.
E.
100 minutes
200 minutes
250 minutes
300 minutes
400 minutes
Answer: A
STUDY NOTES:
The Red and Green lines intersect twice. The least number of minute would be 100 minutes and you
would pay $30 for each plan.
Sample Question A4C4:
If someone needed to use their cell for more than 600 minutes every month, which plan would you
recommend so they paid the smallest monthly fee?
A. Blue Plan
40
B. Green Plan
C. Red Plan
D. Cannot be determined
Answer: C
41