End of Course Algebra I Study Guide: November, 2011
End-of-Course Exam Items Estimated Number of Questions (37 questions)
Numbers & Operations 6-8
Function Characteristics 11-13
Linear Functions & Inequalities 11-13
Data Analysis 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)
(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 6-8 questions
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
(A2A) Numbers & Operations
Compare & Order Real Numbers
Sample Question A2A1: Order the following from greatest to least.
, 3, 8.9, 8, , 9.3
o A. 8, 8.9, , 9.3, 3
o B. , 3, 8, 8.9, , 9.3
o C. 9.3, 3,, , 8.9, 8
o D. 3,9.3, , , 8.9, 8
Answer: D
Sample Question A2A2: 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 / Lowest Temperature / ColorA / 1.35 / 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,
Answer: C
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
to convert from to standard, move the decimal 3 places to the right, From 5.2 to 5200
to convert from to standard, move the decimal 3 places to the left, From 5.2 to 0.0052
Radicals: : 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 item for Performance Expectation A1.2.A/M1.6.A
Which numbers are both less than ?
O A. -2.1 and
O B. and
O C. -0.65 and -1.2
O D. and -0.8
Answer: A
(A2B) Numbers & Operations
Evaluate expressions with variables
Sample Question A2B1: For what values of a is an integer?
o A. a = 1, 0
o B. a ≤ 1
o C. a = 1, a ≠ 0
o D. a > 1
Answer: C
Sample Question A2B2: Evaluate 2w + 6y2 for w = 4 and y = 3.
o A. 330
o B. 62
o C. 60
o D. 44
Answer: B
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: Simplify: 2-2325
223-352
o A. 35
52
o B. 24325
o C. 35
245
o D. 32
245
Answer: C
Sample Question A2C2: Simplify:
o A.
o B. 4
o C. (correct)
o D.
Sample Question A2C3: Simplify the expression using positive exponents.
o A. x3
o B x5
o C. x20
o D. x4
Answer: C
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: 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 A. A; $242
o B. B; $222
o C. A; $4,338
o D. B; $268
Answer: D
Sample Question A7B1: Select the set of ordered pairs that represents an exponential function.
o A. (0,0) (-2,2) (1,1) (2,2)
o B. (4,6) (2,3) (6,8) (10,12)
o C. (1,1) (3,9) (2,4) (0,0)
o D. (0,0) (-1, 2) (-2, 3) (2,3)
Answer: C
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: Solve A = p + prt for p
o A.
o B.
o C.
o D.
Answer: A
Sample Question A7D2: Solve for r
o A.
o B.
o C.
o D.
Answer: A
(A3) Function Characteristics
Relations and Functions
Sample Question A3A1: Which of the following equations listed below determine y as a function of x?
A. x = 3; y = 2x + 1
B. y = x2 + 1; x = 0
C. x2 + y2 = 1; x = y
D. 3x – 2y = 7; y = 2x
Answer: D
Sample item for Performance Expectation A1.3.A/M1.2.A
The equation of a function is shown.
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
Answer: A
Sample Question A3A2: 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 = 3x
D. ; x2 + y2 = 9
Answer: C
Sample Question A3A3: 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 / 1Score on test / 85% / 92% / 90% / 70%
D.
Answer: B
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 / 4y / 3 / 3 / 4 / 5
x / y
1 / 3
1 / 4
2 / 5
3 / 6
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 x2 + y2 = 1
(A3) Function Characteristics
Domain and Range
Sample Question A3A4: 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
Answer: A
Sample Question A3A5: What is the domain and range of ?
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:
Range: f(x) is less than or equal to zero
D. Domain:
Range: f(x) is greater than or equal to zero
Answer: D
Sample Question A3A6: Below is the graph of . Determine the domain and range of this function.
A. Domain:
Range:
B. Domain:
Range:
C. Domain: x can be all real numbers
Range:
D. Domain: x can be all real numbers
Range: f(x) can be all real numbers
Answer: C
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 / 4y / 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 (estimation) and the range would be from (the range could also be written as ).
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, has no division sign and no square root sign so the domain is all real numbers. However, 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, has a division sign where x is in the bottom so the domain can be everything except what makes the bottom zero. In this case the domain is everything but 0.