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End of Course Algebra I Study Guide:

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 Compare & Order Real NumbersEvaluate expressions with variablesExponents, roots, use properties to evaluateEvaluate Exponential FunctionsArithmetic & Geometric SequencesSolving equations with several variables: ex. (A = prt)

(A3) Function Characteristics Relations and FunctionsDomain, range, finding rootsFunctions defined piecewiseIndependent, dependent variablesMultiple representations: symbolic, graph, table, wordsConnections between representationsEvaluate f(x) at a, solving f(x)

(A4) Linear Functions & Inequalities Write & Solve Linear EquationsGraph Linear Equations Point slope forms, translate betweenInterpret Slope and interceptsParallel & perpendicular linesWrite & Solve systems of two linear equationsAbsolute value in equations

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Graphing Absolute value

(A6) Data Analysis

Fit an equation to a lineBest fit linesPredicting from dataCorrelation of data in Scatterplots

(A2A) Numbers & OperationsCompare & 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

2

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

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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 & OperationsEvaluate expressions with variables

Sample Question A2B1: For what values of a, if an integer?

o A. a = 1, 0o B. a ≤ 1o 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. 330o B. 62 o C. 60o D. 44

Answer: B

STUDY NOTES:

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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 & OperationsExponents, roots, use properties to evaluate

Sample Question A2C1: Simplify: 2 -2 3 2 5 223-352

o A. 3 5 52

o B. 24325 o C. 3 5

245 o D. 3 2

245

Answer: C

Sample Question A2C2: Simplify:

o A.

o B. 4

o C. (correct)

o D.

5

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

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To simplify square or cube roots, look for factors that are perfect squares: = = 2

Or perfect cube roots:

(A7AB) Numbers & OperationsEvaluate 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; $242o B. B; $222 o C. A; $4,338o 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)

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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 thesePoints (1,1) (1,2) (2,4)

Items may be presented as graphs, in tablesor coordinates. Functions can be increasing, decreasing, be positive or negative,

(A7D) Numbers & OperationsSolving equations in several variables: ex. (A = prt)

Sample Question A7D1: Solve A = p + prt for p

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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 + 1B. y = x2 + 1; x = 0C. x2 + y2 = 1; x = yD. 3x – 2y = 7; y = 2x

Answer: D

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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 -1O C. All real numbers greater than -1O 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 = 3B. 2x – y = 0; x = y2 C. 34 xy ; y = 3x

D. 3 xy ; 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

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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 y1 31 42 53 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 1)( 2 xxf 93 xy x2 + y2 = 1 3y


Domain and Range

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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 zeroB. Domain: distance car traveled Range: hours car traveled Restriction: distance must be greater than or equal to zeroC. Domain: hours car traveled Range: miles per hour car traveled Restriction: hours must be greater than or equal to zeroD. 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 5)( xxf ?A. Domain: x can be all real numbers Range: f(x) can be all real numbersB. Domain: x = 5 Range: f(x) = 0C. Domain: 5x Range: f(x) is less than or equal to zeroD. Domain: 5x Range: f(x) is greater than or equal to zero

Answer: D

Sample Question A3A6: Below is the graph of 512)( xxf . Determine the domain and range of this function.

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A. Domain: 0x Range: 0)( xfB. Domain: 5x Range: 5)( xfC. Domain: x can be all real numbers Range: 5)( xfD. 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 8Graph: 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

56 x (estimation) and the range would be from 5)(2 xf (the range could also be written as 52 y ).

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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,

24)( xxf has no division sign and no square root sign so the domain is all real numbers.

However, 9)( xxf 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, x

xf 9)( 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. 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

9)( xxf has a range that is greater than or equal to zero because putting in a 9, 10, 11, etc.

would give you zero or a positive number. Likewise, x

xf 9)( can be all numbers but zero since we

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.

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Functions Defined Piecewise

Sample Question A3A7: 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 $.44Up to 2 oz $.61Up to 3 oz $.78Up to 3.5 oz $.95Weights larger than 3.5 oz, large envelope prices apply.

How much would a letter weighing 2.5 ounces cost?A. $.61B. $.78C. $1.05D. $1.53Answer: B

Sample Question A3A8: 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 $59.99900 minutes for $79.991350 minutes for $99.992000 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. $79.99B. $99.99C. $139.98D. $239.97

Answer: B

Sample Question A3A9: 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. $78B. $130C. $140D. $210Answer: B

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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 EOC we do not deal with piecewise equations.

2

)(x

xxf if

33

xx


Independent & Dependent Variables

Sample Question A3A10: 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. feet, fB. height, hC. feet per height, f/hD. height per feet, h/f

Answer: A

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Sample Question A3A11: 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. height, hB. years, yC. height per year, h/yD. year per height, y/h

Answer: B

Sample Question A3A12: 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. salary, sB. years, yC. salary per year, s/yD. year per salary, y/s

Answer: A

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.

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Multiple Representations

Sample Question A3B1: 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 8Total 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 + 200C. T(c) = 100(c + 200)D. T(c) = c + 200

Answer: A

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Sample Question A3B2: A population of bacteria doubled every hour, starting with 4 bacteria. The chart below represents the total population, p, at a given hour, t.

010203040506070

1 2 3 4

Number ofbacteria perhour

Which equation best represents the total population (P) of bacteria at a given hour, t.A. P(t) = 2t B. P(t) = 2t + 4C. P(t) = 4t

D. P(t) = 4(2t)

Answer: D

Sample Question A3B3: 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 8Banking Angle 560 280 160 140

Which equation best represents the banking angle B for the turning radius r?A. B(r) = 28rB. B(r) = 2r – 28C. 3134)( rrB

D. r

rB 112)(

Answer: D

STUDY NOTES:

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hour 0 1 2 3Number of bacteria 4 8 16 32

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 4f(x) 5 6 7


Connections Between Representations

Sample Question A3B4: Which equation listed below best represents the set of data in the table?x -2 -1 1 3 6y -4 -1 5 11 20

A. y = 3x + 2B. y = – 3x + 2C. y = x – 2D. y = – x – 6

Answer: A

Sample Question A3B5: Which function includes all of the ordered pairs in the table?x -2 0 1 2 3f(x) 5 -3 -1 5 15

A. f(x) = x – 3B. f(x) = x + 7C. f(x) = 2x2 – 3D. 27)( xxf

Answer: C

Sample Question A3B6: Which equation listed below best represents the set of data in the table?x -3 -1 2 3 5y -8 -4 2 4 8

A. x – y = 2B. 2x – y = 2C. x + y = – 2D. x + 2y = – 2

Answer: B

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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 5y -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).


Evaluate f(x) at a, solving f(x)

Sample Question A3C1: For the function 13)( xxf , find )5(f .A. – 12B. – 2.4C. 12D. 18

Answer: C

Sample Question A3C2: For the function 64)( xxf , find )2(f .A. – 14B. – 7C. – 2 D. 2

Answer: A

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Sample Question A3C3: For the function 3)( xxf , find )6(f

A. 21

B. 6C. 3D. 9

Answer: C

Sample Question A3C4: For the function 63)( xxf , find x when 0)( xf .A. – 2 B. 0C. 6D. 9

Answer: A

Sample Question A3C5: For the function 179)( xxf , find x when 1)( xf .A. – 17B. – 8C. 1D. 2

Answer: D

Sample Question A3C6: For the function 92)( xxf , find x when 3)( xf . A. – 9B. – 3C. 3D. 6

Answer: D

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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.

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 9)( xxf .Answer: Replace the x with a 4. f (4) = 94

5594 . So f (4) = 5.

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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 + 14 = 4x4/4 = 4x/41 = xSo 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 + 2028 = 2x28/2 = 2x/214 = xSo x = 14 when f(x) = 8.

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(A4) Core Content: Linear functions, equations, and inequalities

(A4A) Write and solve linear equations and inequalities in one variable

Sample Question A4A1 :

Solve the equation and chose the correct answer:

y 12 4

A. y = -16B. y = -8C. y = 8D. y = 16

Answer: C

STUDY NOTES: Add 12 to both sides of the equation.


Solve the inequality and chose the correct answer:

x 58

A. x > -3B. x > 3C. x > 11D. x < 3

Answer: B

STUDY NOTES: Add -5 to both sides of the inequality.

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George has eight more than twice the number of marbles you have. Write an inequality for the number of marbles George has.

Chose the correct answer:

A. 8 – 2nB. 2n - 8C. 8 + (-2n)D. 2n + 8

Answer: D

STUDY NOTES: Let n be the unknown so twice the unknown would be 2n . Eight more than twice the unknown would be 2n + 8.


Write an absolute value equation for all the numbers 2 units from 7.

Chose the correct answer:

A.

x 7 2B.

x 2 7C.

x 7 2D.

x 2 7

Answer: C

STUDY NOTES:

x b a is an absolute value representing all the numbers that are a units from b.

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The equation

2 x 1 10 4 has two real solutions.

Determine the negative solution of the equation and chose the correct answer.

A. -1B. -2C. -3D. -4

Answer: B

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 or x-1=-3. The negative solution is -2.


Solve for x: 2(x – 3) + 4x = 14 + 2x

Choose the correct answer:

A. 2B. 5C. 4D. 6

Answer: B

STUDY NOTES: 2(x - 3) + 4x = 14 + 2xdistribute 2x – 6 +4x = 14 + 2xcombine like terms 6x - 6 = 14 + 2xadd -2x -2x -2x

4x – 6 = 14add 6 4x = 20divide by 4 x =5

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x 1 3


Solve

x 6 4Choose the correct answer: (and be able to graph the solution on a number line)

A. 2 < x < 10B. 2 > x > 10C. x < 2 and x < 10D. x > 2 and x > 10

Answer: B

STUDY NOTES:

means x – b < c and x – b > -c

STUDY NOTES:

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 sideor a-c=b-c , subtract the same number from each side.or a x c = b x c , multiply the same number by each side

and if c is not 0 then

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.

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x b c

ac

bc

(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 item for Performance Expectation A1.4.B/M1.3.D

Which equation represents the line that passes through the points (2, 2) and (4, 1)?

O A.

O B.

O C.

O D.

Answer: B

Sample Question A4B1:

Find an equation for a line with y-intercept equal to -2 and slope equal to 3.

A. y = -2x + 3B. y = -2x - 3C. y = 3x - 2D. y = 2x – 2

Answer: C

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 :

Find an equation for a line with a slope of -1 that goes through the point (-3, 2).

A. y = -x + 1B. y = -x – 1C. y = 2x - 1 D. y = 3x – 2

Answer: B

STUDY NOTES:

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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.


Find an equation for a line that goes through the points (1, 8) and (-2, -1)

A.

y 3x 5B.

y 3x 5

C.

y 13x 7 2

3

D.

y 13x 7 2

3

Answer: A

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.


Use the equation

y 72(x 2) (without sketching the graph) to describe the graph.

A. the graph has a negative slope and y-intercept of -2B. the graph has a negative slope and y-intercept of 3C. the graph has a positive slope and y-intercept of -2D. the graph has a positive slope and y-intercept of 3

Answer: D

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 y-intercept of 3.


Write the equation

3x 2y 5 in slope-intercept form.

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Slope Intercept form is: _______________________

Answer:

STUDY NOTES: Solve the equation for y. First subtract 3x from both sides then divide both sides by 2.


Write the equation

y 12(x 2) in standard form.

A. 2x +y =-3 B. -2x + y = 3C.

2x y 3D. y = 2x - 3

Answer: C

STUDY NOTES: Standard form is written Ax +By = C, where A,B, and C are integers and A is positive.


Write the equation

y 12x 5 in standard form

A.

x 2y 10B.

x 2y 10

C.

y 12x 5

D.

y 12x 10

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Answer: B

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

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(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 item for Performance Expectation 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 monthlyO B. The amount of money originally in the accountO C. The number of months it would take to earn $100O D. The number of months it would take to reach $500

Answer: A 32

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: CSTUDY 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. Blue PlanB. Green PlanC. Red PlanD. Cannot be determined

Answer: A

STUDY NOTES: At 300 minutes:Blue plan- $40 Red plan- $50 Green plan- $60


What is the least number of minutes you can use and have the Green and Red Plans cost the same amount?

A. 100 minutesB. 200 minutesC. 250 minutesD. 300 minutesE. 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.


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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 PlanB. Green PlanC. Red PlanD. Cannot be determined

Answer: C

STUDY NOTES: If you continued the lines at the same slope to 600 minutes you would see that the Red plan would be paying the least.

Refer to the graph below that shows a square in the coordinate grid to answer the following questions.


The slope of sides AB and DC are

A. unrelatedB. parallelC. perpendicularD. opposite

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Answer: B

STUDY NOTES: Each slope (rise/run) is ½. They are parallel


The slope of sides AD and DC are

A. unrelatedB. parallelC. perpendicularD. opposite

Answer: C

STUDY NOTES: The slope of AD is -2 and the slope of DC is ½. They are opposite reciprocals so they are perpendicular.


The actual slope of side AB is:

A. ¼B. – ¼ C. ½ D. – ½E. 2F. -2

Answer: C

STUDY NOTES:

35

To go from point A to point B you go up (positive) 2 units and over a positive 4. The slope is ½.

36

(A4D) Write and solve systems of two linear equations and inequalities in two variables.

Sample item for Performance Expectation A1.4.D/M1.3.E

Two lines divide the coordinate plane into the four lettered regions shown.

One region represents the solution set to the system of inequalities given.

Which region represents the solution set for the system of inequalities given?

O A. Region MO B. Region NO C. Region PO D. Region Q

Answer: C

Sample Question A4D1 :

37

MN

PQ

What is the solution to the simultaneous linear system pictured above?

A. (0, 4)B. (-2, 0)C. (-3, -1)D. (-2, 0)

Answer: C

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 :

Solve the following simultaneous linear equations algebraically:

2x y 2 38

x y 1

A. (-1,-2)B. (-1/3,-4/3)C. (0,-1)D (-1,0).

Answer: D

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:

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 vansB. 3 vans and 2 carsC. 3 cars and 2 vansD. 3 vans and 1 car

Answer: B

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.

39

(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 :

Find a function that represents this situation.

A.

y 400x 10,000B.

y 400x 1,000C.

y 400x 10,000D.

y 400x 9,000

Answer: B

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.


If the flow rate increases from 400 to 500 gallons per minute, how will the graph of the function change?

A. The y-intercept will increaseB. The y-intercept will decreaseC. The slope will decreaseD. The slope will increase

Answer: D

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


40

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. The y-intercept will increaseF. The y-intercept will decreaseG. The slope will decreaseH. The slope will increase

Answer: A

STUDY NOTES: At 0 minutes the number of gallons is 2,000. Therefore, the y-intercept would now be (0,2000)

Refer to the following two functions to answer the next 3 questions.

Function A:

y 3 x Function B:

y 13x

Sample item for Performance Expectation A1.4.E/M1.3.B

Which statement best describes the transformation of the graph of to the graph of ?

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.

Answer: C


What can be said about the following two functions?

A. Functions A and B reflect exactly across the x-axisB. Functions A and B reflect exactly across the y-axisC. The functions do not reflectD. The functions cannot be graphed

Answer: C

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.

41


Which function’s end behavior is negative?

A. Function AB. Function BC. NeitherD. Both

Answer: B

STUDY NOTES: Because Function B opens downward, y decreases as x increases and decreases.


Which function has a greater rate of change?

A. Function AB. Function BC. NeitherD. Both

Answer: A

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 item for Performance Expectation A1.5.B/M2.2.B

42

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

Answer: B

Sample item for Performance Expectation A1.5.C/M2.2.D

What are the solutions to the equation ?

O A. or

O B. or

O C. or

O D. or

Answer: D

43

(A6A) Describing and Assessing Data SetsUse and evaluate the accuracy of summary statistics to describe and compare data sets.

Sample Question A6A1:

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.

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.

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Company A

Company B

Company C


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.

Explain your answer below:

Note: The scale goes from 0 to 10.

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.”

45


Calculate the mean and standard deviation of the follow data set:

15, 18, 21, 14, 23, 12, 15, 18, 20, 20, 19, 18

o A. mean = 17.75 standard deviation = 3.19o B. mean = 17.75 standard deviation = 3.06o C. mean = 18 standard deviation = 10.18o D. mean = 18 standard deviation = 3.19

Answer: A


Calculate the median of the follow data set:

15, 18, 21, 14, 23, 12, 15, 18, 20, 20, 19, 18

o A. median = 17.75o B. median = 13.5o C. median = 15o D. median = 18

Answer: D

(A6B) Inferences and ConclusionsMake valid inferences and draw conclusions based on data.

46


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:

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.


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:

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 broccoli that is causing the lower cholesterol, but another factor which is making the variable move (like exercise).

47


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.

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:

Answer: Fuel Cost = 0.84(50) + 2.13Fuel Cost = 44.13 (or $44,130)

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).

48

5 10 15 20 25 30 350

5

10

15

20

25

30

35

Fuel Cost vs Train Length

Fuel Cost (in thousands)

Number of Train Cars

(A6C) Linear TransformationsDescribe how linear transformations affect the center and spread of univariate data.


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 A. The mean and range both decrease.o B. The mean and range do not change.o C. The mean does not change but the range decreases.o D. The mean decreases but the range does not change.

Answer: A


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 A. The mean increases but the range does not change.o B. The mean does not change but the range increases.o C. The mean and range both increase.o D. The mean and range do not change.

Answer: C

49


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?

mean standard deviationo A. 20 7o B. 20 5o C. 22 5o D. 22 7

Answer: C

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.


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?

mean standard deviationo A. 20 10o B. 20 5o C. 40 5o D. 40 10

Answer: D

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.

50

Sample item for Performance Expectation A1.6.E/M1.3.G

A scatterplot is shown.

Which statement describes the correlation of the data graphed in the scatterplot?

O A. Strong negativeO B. Strong positiveO C. Weak negativeO D. Weak positive

Answer: B

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x

y

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