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Copyright © 2017

Education Time Courseware, Inc.

Justin S. Grover

John R. Mazzarella

Richard G. Schiller

Homework Book

Second Edition

Education Time Courseware Inc. Copyright 2017

AUTHORS: v2.1

Justin Grover Mathematics Chairperson/Teacher, St. John the Baptist DHS

John Mazzarella Adjunct Professor Mathematics, Molloy College

Mathematics Teacher (Retired)

Richard Schiller Adjunct Professor Mathematics, Molloy College

Mathematics Teacher, St. John the Baptist DHS

COPYRIGHT 2017

Education Time Courseware, Inc.

83 Twin Lane North

Wantagh, NY 11793

PHONE: (516) 784-7925

ISBN: 0-943749-86-8

COMMON CORE HOMEWORK

BOOK


Table of contents:

Study Guide ................................................................................................................................................. 6 Unit 1 – Introduction to Algebra ................................................................................................................. 9

Homework 1: Signed Numbers (A.SSE.1,2, A-APR.1) ...................................................................... 9

Homework 2: Rational and Irrational Numbers (N-RN.3) ............................................................. 11

Homework 3: Order of Operations (A-SSE.1,2) .............................................................................. 13

Homework 4: Distributive, Commutative & Associative Properties (A-SSE.1,2) ........................ 15

Homework 5: Terminology and Writing Expressions (A-SSE.1,2) ................................................ 17

Homework 6: Unit 1 Review .............................................................................................................. 21

Unit 2 – Exponents, Radicals, and Polynomials ....................................................................................... 23

Homework 1: Exponent Basic Properties (A-SSE.1,2) .................................................................... 23

Homework 2: Zero and Negative Exponents (A-SSE.1,2) ............................................................... 25

Homework 3: Fractional Exponents (A-SSE.1, 2) ........................................................................... 27

Homework 4: Simplifying, Adding and Subtracting Radical Expressions (N.RN.3, A-SSE.2) ... 29

Homework 5: Multiplying and Dividing Radical Expressions (N.RN.3, A-SSE.2) ....................... 31

Homework 6: Addition and Subtraction of Polynomials (A-SSE.2, A-APR.1) ............................ 33

Homework 7: Multiplication of Polynomials (A-SSE.2, A-APR.1) ............................................... 35


Homework 9: Post Test Cumulative Review ................................................................................... 39

Unit 3 – Factoring ..................................................................................................................................... 41

Homework 1: GCF (A-SSE.1,2) ......................................................................................................... 41

Homework 2: DIFFERENCE OF TWO SQUARES (A-SSE.2) ..................................................... 43

Homework 3: Basic Trinomial Factoring (A-SSE.2) ....................................................................... 45

Homework 4: Factoring by Grouping (A-SSE.2) ............................................................................. 47

Homework 5: Advanced Trinomial Factoring (A-SSE.2) ............................................................... 49

Homework 6: Factoring Completely (A-SSE.2) ............................................................................... 51


Homework 8: Post Test Cumulative Review .................................................................................... 55

Unit 4 – Solving ........................................................................................................................................ 57

Homework 1: Solving Linear Equations (A-CED.1, A-REI.1,3) .................................................... 57

Homework 2: Fractional and Decimal Equations (A-CED.1, A-REI.1,3) ..................................... 59

Homework 3: Literal Equations (A-SSE.1, A-CED.1,4, A-REI.1,3) .............................................. 61

Homework 4: Writing Equation Applications (A-SSE.1, A-CED.1, A-REI.1,3, A-APR.1) ......... 63

Homework 5: Consecutive Integers (A-SSE.1, A-APR.1, A-CED.1, A-REI.3) ............................. 65

Homework 6: Factoring to Solve Equations (A-SSE.2,3, A-CED.1, A-REI.1,3,4) ........................ 69

Homework 7: Sets, Inequalities, and Notation (A-CED.1, A-REI.1,3) .......................................... 71

Homework 8: Solving Inequalities (A-CED.1, A-REI.1,3) .............................................................. 73


Homework 9: Compound Inequalities (A-CED.1, A-REI.1,3) ....................................................... 75

Homework 10: Writing Inequalities (A-CED.1, A-REI.1,3) ........................................................... 77

Homework 11: Unit 4 Review ............................................................................................................ 79

Homework 12: Post Test Cumulative Review .................................................................................. 81

Unit 5 – Linear Equations ..........................................................................................................................83

Homework 1: Functions (A-REI.10, F-IF.1) .................................................................................... 83

Homework 2: Evaluating Functions (A-CED.1, A-REI.3,10, F-IF.1,2) ......................................... 87

Homework 3: What is a Line? (A-SSE.1, A-REI.10) ....................................................................... 90

Homework 4: The Slope Formula (F-IF.4,6) .................................................................................... 94

Homework 5: Intercepts (A-REI.10, F-IF.4) .................................................................................... 96

Homework 6: The Point-Slope Formula (A-CED.1, A-REI.3) ....................................................... 98

Homework 7: Graphing Lines from Slope and Intercepts (A-REI.10, F-IF.4,7) ........................ 100

Homework 8: Modeling Linear Functions (A-SSE.1, A-CED.1, A-REI.3, F-IF.2, F-LE.5) ...... 104

Homework 9: Average Rate of Change (F-IF.6) ............................................................................ 106

Homework 10: Unit 5 Review .......................................................................................................... 110

Homework 11: Post Test Cumulative Review ................................................................................ 112

Unit 6 – Solving Systems .........................................................................................................................114

Homework 1: Linear Systems Graphically (A-REI.6,10,11) ........................................................ 114

Homework 2: Linear Systems Algebraically: Substitution (A-SSE.1, A-CED.2, A-REI.5,6).... 118

Homework 3: Linear Systems Algebraically: Elimination (A-SSE.1, A-CED.2, A-REI.6) ....... 122

Homework 4: Modeling Linear Systems (A-CED.2,3, F-LE.5) .................................................... 126

Homework 5: Graphing Inequalities (A-REI.12) .......................................................................... 128

Homework 6: Inequality Systems Graphically (A-REI.12) .......................................................... 132

Homework 7: Inequality System Modeling (A-SSE.1, A-CED.2,3, A-REI.12) ........................... 136



Unit 7 – Quadratics ..................................................................................................................................144

Homework 1: Quadratic Graphs (A-REI.10, F-IF.2,4,7, A-APR.3) ............................................ 144

Homework 2: The Axis of Symmetry (F-IF.2,4) ............................................................................ 148

Homework 3: Transformations (F-IF.2, F-BF.3) ........................................................................... 152

Homework 4: Comparing Functions (A-REI.10, F-IF 2,4,6,9) ..................................................... 156

Homework 5: Quadratic Function Modeling (A-SSE.1,3, A-REI.4)............................................ 158

Homework 6: Piecewise Functions (A-REI.10, F-IF.2) ................................................................. 160




Unit 8 – Solving Quadratics .................................................................................................................... 168

Homework 1: Solving Factorable Quadratics (A-CED.1, A-REI.3,4, F-IF.8) ............................. 168

Homework 2: Solving Quadratic Linear Systems (A-CED.2, A-REI.4,6,11) .............................. 170

Homework 3: The Quadratic Formula (A-CED.1, A-REI.3,4) .................................................... 174

Homework 4: Completing the Square (A-CED.1 A-REI.1,3,4, A-SSE.3, F-IF.8) ....................... 176

Homework 5: Vertex Form (A-SSE.2,3, A-CED.1, A-REI.3, F-IF.8) .......................................... 180



Unit 9 – Exponential, Absolute Value, and Other Functions .................................................................. 188

Homework 1: Exponential Functions (A-REI.3, F-IF.2,4,6,7, F-LE.1) ........................................ 188

Homework 2: Exponential vs Linear (A-CED.1, A-REI.3, F-BF.1, F-LE.1,2) ............................ 190

Homework 3: Percent Rate of Change (A-SSE.1,2,3, A-CED.1, A-REI.3, F-IF.2, F-LE.1,5) .... 192

Homework 4: Absolute Value Functions (A-REI.6,10,11, F-IF.2,4,6,7, F-BF.3) ......................... 194

Homework 5: Square Root, and Step Functions (A-REI.10, F-IF.2,4,7, F-BF.3, F-LE.1) ......... 198

Homework 6: More Piecewise Practice (A-REI.6,10,11, F-IF.2,6, F-BF.3) ................................. 200

Homework 7: Arithmetic Sequences (A-CED.1, A-REI.3, F-IF.3, F-BF.1) ................................. 202

Homework 8: Geometric Sequences (A-CED.1, A-REI.3, F-IF.3, F-BF.1) ................................. 205

Homework 9: Recursive Sequences (A-CED.1, F-IF.3, F-BF.1) ................................................... 208


Homework 11: Post Test Cumulative Review ................................................................................ 214

Unit 10 – Statistics .................................................................................................................................. 216

Homework 1: Center and Spread (S-ID.2) ..................................................................................... 216

Homework 2: Quartiles (S-ID.1)...................................................................................................... 220

Homework 3: Tables and Plots (S-ID.1,3) ...................................................................................... 224

Homework 4: Two Way Tables (S-ID.5) ......................................................................................... 228

Homework 5: Linear Correlation (S-ID.6,7,8,9, A-CED.1, A-REI.3) .......................................... 230

Homework 6: Residuals (S-ID.6) ..................................................................................................... 234

Homework 7: Regression Models (S-ID.6, A-CED.1, A-REI.3) ................................................... 238



Regents Reference Sheet ......................................................................................................................... 246


Study Guide

Properties of Real Numbers

Addition Multiplication

Commutative: a + b = b + a a•b = b•a

Associative: a + (b + c) = (a + b) + c a(bc)=(ab)c

Distributive: a(b + c) = ab + ac

Identity: a + 0 = a a•1 = a

Inverse: a + (-a) = 0 a•1

a= 1

Zero Property: a•0 = 0

Scientific Notation:

9.4 ×103 = 9400 9.4× 10

-3 = .0094

Absolute Value

| 3 | = 3 | - 3 | = 3

TO SOLVE EQUATIONS :

Remove parentheses( DISTRIBUTE)

Remove decimals or fractions

Combine LIKETERMS on the same sideof the= sign

Movethe required variable tothe same sideof the=

ISOLATE thevariableby additionor subtraction

Then DIVIDE by thecoefficient of thevariabletoend

LITERAL EQUATIONS:

ISOLATE the REQUIRED VARIABLE:

Example: Solve for a:

ab + c = d

- c = - c

ab = d - c

ab d c

b b

d ca

b

INEQUALITIES

ISOLATE the REQUIRED VARIABLE

Same process as equations

NOTE: ONE MAJOR INEQUALITY FACT

WHEN MULTIPLY or DIVIDE both sides by

A NEGATIVE NUMBER

MUST CHANGE the DIRECTION of the

INEQUALITY

Example. -3x ≤ 15

Divide both sides by -3

x ≥ - 5

3 15

3 3

x

INTERVAL NOTATION

( 2 , 5 ) represents 2 < x < 5

[ 2, 5 ] represents 2 ≤ x ≤ 5

[ 2, 5 ) represents 2 ≤ x < 5

MULTIPLICATION ( FOIL or DISTRIBUTE)

( x + 4) ( x – 2) = x2 – 2x + 4x – 8

= x2 + 2x + 8

( a + b)2 = (a + b)( a + b)= a

2 + 2ab + b

2

( a – b)2 = ( a – b)( a – b) = a

2 – 2ab + b

2

( a – b)( a + b) = a2 – ab + ab + b

2 = a

2 – b

2

EXPONENTS

xa •x

b= x

a+b x

0 = 1

aa-b

b

x= x

x x

-2 =

2

1

x

(xa)b = x

ab

(xy)a = x

a•x

b (-5)

2 ≠ -5

2


EQUATIONS OF LINES m = slope

y = mx + b slope – intercept form

y – y1 = m(x – x1) point – slope form

SLOPE:

2 1

2 1

y yvertical change risem

hoeizontal change run x x

PARALLEL LINES have EQUAL SLOPES

PERPENDICULAR LINES slopes are NEGATIVE

RECIPROCALS

SYSTEMS of EQUATIONS:

y – 3x = 3 SUBSTITUTE one variable into the second equation.

y + 3x = 9 ADD or SUBTRACT to eliminate a variable.

INEQUALITY SYSTEMS – graph EQUALITY , DOTTED(< or >),

or SOLID (≤ or ≥)lines and SHADE the SOLUTION side.

LINEAR QUADRATIC SYSTEM:

SUBSTITUTE linear into the quadratic and solve.

FACTORING

1) Look for GCF First (greatest

common factor number or variable)

2) Difference of TWO perfect squares

A2 – B

2

3) Trinomial x2 + Ax + M

( 2 #’s add to A and multiply to M)

( x # ) ( x # )

QUADRATIC EQUATIONS

Set = 0 x2 – 2x – 8 = 0

Factor ( x – 4) ( x + 2) = 0

T chart x – 4 = 0 | x + 2 = 0

Solve for x x = 4 x = - 2

These are the ROOTS of the equation.

PARABOLAS

y = ax2 + bx + c

Axis of Symmetry x = b

a

Roots are the x intercepts, where the

parabola crosses the X AXIS

FRACTIONS

UNDEFINED: N

Dwhen D = 0

2

5x is undefined when x = 5 (D = 0)

ADDITION/ SUBTRACTION

Need COMMON DENOMINATOR

3 4

2 3

3 3 2 4

3 2 2 3

9 8 17

6 6 6

x x

x x

x x x

MULTIPLICATION

“MULTIPLY ACROSS” 2 5 10

3 7 21

DIVISION

“INVERT and MULTIPLY”

2 5 2 7 14

3 7 3 5 15

ALWAYS FACTOR FIRST!

FUNCTIONS

Every x value is assigned ONE and ONLY ONE y value.

{ (2,3), (4,5), (6,8) } YES { (2,3) , (4,5), ( 2, 8)} NO

f(x) = 2x+1 find f(3) = 2(3) + 1 = 7

A GRAPH that passes the VERTICAL LINE TEST is a function.

DIRECT VARIATION

Occurs if one variable increases then the other increases also or if

one variable decreases, the other variable decreases also.

Expressed as y = kx or k =y

x where k is called the CONSTANT of

VARIATION.

EXPONENTIAL GROWTH and DECAY

GROWTH: y = a(base)x where a is positive and the base is

greater than 1. ( Growth RATE is greater than 100%)

DECAY: y = a(base)x where a is positive and the base is also

positive but less than . (Decay RATE is less than 100%)

PYTHAGOREAN THEORM

SIDES OF A RIGHT TRIANGLE

a2+b

2 = c

2

c is the HYPOTENUSE, a and b legs

TRIPLETS: 3,4,5 AND 5,12,13


STATISTICS

MODE = most frequent score

MEDIAN = middle number of an

ORDERED list

MEAN = average = Sum divided by

number of items

RANGE = high score – low score

OUTLIERS = values far away from rest of

data

NUMBER SUMMARY:

Min, Quart 1, Median, Quart #3, Max

QUARTILES divide data into 4 (25%)

equal parts.

PERCENTILES: Percentage of scores at or

below this percent=

#100

#

of scores below

total of scores

BOX AND WHISKER

Min and Max at ends of “WHISKERS”

Quartile 1, Median, Quartile 3 form the box

1 2

1 2

( ) ( )AverageRateof change

Linear Functions: averagerateof changeisa constant.

QuadraticFunctions: averagerateof changeis not constant

GRAPHS

f x f x

x x


Unit 1 – Introduction to Algebra

Homework 1: Signed Numbers (A.SSE.1,2, A-APR.1)

Tip: Think how to combine like terms, and watch out for parentheses!

Examples: 7 3 7 3 7 3 7 3 7 3 7 3

= 10 = 4 10 4 10 4

1) Choose the word that makes the statements true:

The word product represents the _______________ operation. The product of two positive

(multiplication, division)

terms is _______________. The product of two negative terms is _______________. The product of a

(positive, negative) (positive, negative)

positive term and a negative term is _______________. The rules for products are the same as the rules

(positive, negative)

rules for the ______________ operation.

(multiplication, division)

2) Perform the following operations with signed numbers. (No calculator is suggested)

a) 10 5 b) 15 – (–12) c) –25 – 18 d) –30 – (–12)

e) 8 + 6 f) 10 + 5 g) 14 – 8 h) –15 + 8

3) Perform the following operations with signed numbers. (No calculator is suggested)

a) 5 12 b) 8 ( –11) c) –6 6 d) –12 (– 10)

e) 72 f) ( –9)

2 g) –7

2 h) (–3)

3

i) 15

3

j)

28

7

k)

49

7

l)

36

3


HW 1-1 continued

4) One morning, the temperature was 8° below zero. By noon, the temperature rose 18° Fahrenheit (F)

and then dropped 6° F by evening. What was the evening temperature?

5) Jason started his own lawn service. The table below shows the profit or loss each month for four

months. Jason represented profit as positive numbers and loss as negative numbers. How much profit

or loss did Jason have after being in business for four months?

Month June July August September

Profit/Loss - 380 600 - 800 330

a) loss of $250 b) profit of $250 c) loss of $2110 d) profit of $2110

6) Kim states that 6 + (- 4) is the same as 6 - 4.

Do you agree with Kim? Justify your answer.

7) James was observing the number of students entering and leaving the library at school. He observed

12 students leave the library and 9 students enter. Later he observed 4 more students enter the library

and 14 students left. What was the net increase or decrease in the number of students in the library?

8) Explain why 2

and a a a are equivalent, and determine if the solution to both is positive or

negative.



Homework 2: Rational and Irrational Numbers (N-RN.3)

REVIEW: 1) Explain why 225 and 5 are not equivalent.

Definition: A rational number is any number that can be expressed as the quotient or fraction p

q of two

integers, a numerator p and a non-zero denominator q.

Definition: An Irrational number is a real number than cannot be written as a ratio (fraction) between

two integers and is not an imaginary number.

2) Determine and state which of the following numbers are rational or irrational.

1)

3a )b ) 8c ) 9d

1)

3e

1)

4f )

2g

)h

7)

16i

5)

20j

2

) 17k ) 0l

3) Determine if the following statements are True, False, or Sometimes True. If Sometimes is the

answer, give an example of when it would be true.

a) The sum of a rational number and an irrational number is rational

b) The product of a rational number and an irrational number is rational

c) An irrational number divided by an irrational number is rational

d) The difference of two irrational numbers is rational

e) An irrational number squared is rational

f) The sum, difference, and product of any two rational numbers is rational

g) The square of an irrational number is rational

h) The product of two irrational numbers is rational

i) Integers and whole numbers are both rational


HW 1-2 continued

4) Which does not represent a rational number?

5a) 196 b) .16 c) d) 8

3

5) Given the following expressions:

I. 8 II. 2 III. 1 IV. 5

Which of the following would result in a rational number?

a) I + II b) I + III c) III + IV d) II + IV

Which of the following would r

2

esult in an irrational number?

a) I + II b) I II c) III IV d) I

6) Which number is irrational?

Why is the number you chose an irrational number?

7) Determine a value of that would make the expression 5 have a rational result.

Is there more than one value of that would create a rational result?

Explain your answer.

r r

r

5 116, , , 3

49



Homework 3: Order of Operations (A-SSE.1,2)

REVIEW 1) Given the expression 2, determine two values of that would result in

a rational expression. Explain your answer.

x x

Notes: The proper order of operations to simplify an expression is:

PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)

These operations should be done in this order to correctly simplify an expression. However,

multiplication and division have the same priority. They are completed in the order they occur from left

to right and the same for addition and subtraction.

Example: 5 5 5 5 , if all multiplication is done first then division your answer is 1, which is incorrect.

The correct execution is:

1 2 3

5 5 5 5 25 5 5 5 5 25

st nd rd

, the correct solution is 25

2) Simplify each of the following expressions:

) 7 7 7 7a ) 3 3 3 3b ) 5+5 5 5 5c

) 1 1 7 3 4d 2) 5 3e

2) 5 3f

2

) 4 1 3 2g ) 2 4 1 6h

22 3 3

) 1 4

i

22) 5

4j

1 3) 2

5k

23 16

) 4 4 1

l


HW 1-3 continued

3) Caleb simplified the expression 2

3 5 3 1 as follows:

2

Line 1: 3 5 3 1

2

Line 2: 3 2 1

2

Line 3: 6 1

Line 4: 36 1

Line 5: 37

State if Caleb is correct or not. If you believe he is incorrect, between which lines did he make an error?

4) Evaluate the following expressions if 3 and 2 :x y

) a x y 2 2) b x y ) 5c x y

) 4 3d x y 2

) e x y y 2) 6 4f x y

4)

xg

y

) h xy yx 2 2) i x xy y

5) If and are both negative integers, and , which expression has the smallest value?

) ) ) )

f g g f

a f g b f g c g f d g f

2 26) If 3, are the expressions and x x x equivalent or not? Explain why.

2 27) If 4, are the expressions 2 and 2yy y equivalent or not? Explain why.



Homework 4: Distributive, Commutative & Associative Properties (A-SSE.1,2)

2REVIEW 1) If 4 and 3, what is the value of 2x y y x y

Notes: Basic Number Properties of real numbers

The Distributive Property – Multiply through the parenthesis. Example : a b c ab ac

The Commutative Property – Swap the order of terms and keep equivalence, It holds true for addition

and subtraction but not division and subtraction! Example : , a b b a a b b a

The Associative Property – Change the grouping (association) of terms and keep equivalence, also holds

true for addition and multiplication only! Example : a b c a b c

Operation Inverse – Additive inverse causes two terms to add to zero (additive identity); Multiplicative

inverse (reciprocals) causes two terms to multiply to one (multiplicative identity).

1

Example : + 0, 1a a bb

Properties of Zero – Addition property of zero causes terms to sum to themselves, Multiplication

property of zero causes terms to multiply to zero. Example : 0 , 0 0a a b

3) Logan rewrote an expression as shown step by step below.

State the property he used to change from the previous step.

original expression: 3 (1 5) 4

Step 1: 3 1 5 4 property used:________________

______________

Step 2: 3 15 12 property used:______________________________

Step 3: 3 12 15 property used:______________________________

4) Explain why there is no communitive property of subtraction.


HW 1-4 continued

5) Fill in the box with a value that makes the statement true and name the property being used.

a) 4 3 3 property used:______________________________

b) 8 3 3 property used:______________________________

c) 2 3 4 2 3 property used:______________________________

d) 4 3 4 4 property used:______________________________a a

e) 6 0 property used:______________________________

f) 4 1 property used:______________________________

6) Henry solved the following equation. At each step explain how Henry got from the previous ste

7) When solving the equation 2 23(2 5) 7 5 4x x , James wrote 2 26 15 7 5 4x x as his first

step. Which property justifies James's first step?

a) addition property of equality b) commutative property of addition

c) multiplication property of equality d) distributive property of multiplication over addition

3 5 5 4 5 30

3 15 5 4 5 30 ________________________________________________

3 15 5 4 5 30 ________________________________________________

3 15 5 9 30 ___________________________________________

x x

x x

x x

x x

_____

3 5 15 9 30 ________________________________________________

2 24 30 ________________________________________________

2 6 ________________________________________________

3 ______________________

x x

x

x

x

__________________________



Homework 5: Terminology and Writing Expressions (A-SSE.1,2)

REVIEW 1) Katy solved an equation as shown. Which properties did she use?

2 4 3 18

2 8 3 18

5 8 18

5 10

2

x x

x x

x

x

x

2) Identify whether or not each of the following are mathematical expressions. (YES or NO)

a) 4xy + 7 b) 3x2y

5 – 3x c) -6abc

2 + 7a - 3 d) 4x

2 + 5 = 9

e) 3x + 1 > 4 f) 3

5 2

x

y g) 6

12

x h) x

3) Write what the numerical coefficient is and what the variable(s) is(are) in each expression.

a) 5x b) -3y2 c) st

3 d) -xy

4) Determine and state what the base is and what the exponent is in each expression.

a) 73 b) x

4 c) 3y

2 d) (st)

3

e) -42 f) (2y)

4 g) ( -8)

2 h) -5t

5

5) Determine and state how many terms are found in each of the following expressions?

a) 5abc b) 5 + x c) 3a – 4b + 5c d) 7x3y

2 + 4


. 3 4 3

. 5

. 7 10

I x y z

II x

III x

HW 1-5 continued

6) Determine and state the degree of each.

a) 3x2 – 4 b) 2x + 1 c) 4x

3 – 5x

2 +2x + 9

d) 9 – 4y2 e) 4 f) 7x

5 + 3y

2 – 2xy + 12

7) How many terms are in the expression 6 4 25 3 4 7a a a

a) 1 term c) 2 terms

b) 3 terms d) 4 terms

8) Which of the following is an example of a mathematical expression?

1) I only 2) II only 3) III only 4) I and III

9) For the following, write each as an algebraic expression, represent the number with n:

a) eight less than a number b) a number increased by 6

c) 5 more than a number d) one-half a number increased by 7

e) 9 is subtracted from 4 times a number f) The sum of 4 times a number and 6

g) the sum of the number and twice the number h) three times the number decreased by one –third

that number


HW 1-5 continued

10) If x represents a positive integer, which expression represents an odd integer?

2a) b) 3 c) 2 1 d) 3 1x x x x

11) Which expression represents “3 less than twice a number”?

2a) 3 2 b) 3 c) 2 3 d) 3 2x x x x

12) Which verbal expression can be represented by 3 7 ?x

a) triple the difference of x and seven

b) seven less than triple x

c) three times x decreased by seven

d) the product of x and three, minus seven

13) Which expression represents the number of cents in d dimes, n nickels, and p pennies?

a) c) 10 5

b) d) 10 5

d n p d n p

d np dnp

.

14) Ben is three times as old as Logan. If x represents Logan’s age, which expression represents how old

Ben will be in 4 years?

a) 4 c) 4 3

b) 3 4 d) 12

x x

x x


15) Caleb buys two toys that cost d dollars each. He gives the cashier $50. Which expression represents

the change he should receive?

16) To watch a school play, spectators much buy a ticket at the door. The cost of an adult ticket is $4.00

and the cost of a student ticket is $2.50. If the number of adult tickets sold is represented by a and

the number of student tickets sold is represented by s, which expression represents the amount of

money collected at the door from the ticket sales?

17) A regular pentagon has five equal length sides. If each side is represented by 2 3,x which

expression represents the perimeter of the pentagon?

2

2

a) 50 2 c) 50

b) 50 2 d) 50

d d

d d

a) 10

b) 6.5

c) 4 2.5

d) 4 2.5

as

a s

a s

a s

5

a) 5 2 3

b) 5 2 3

c) 2 3

d) 2 3 5

x

x

x

x



Homework 6: Unit 1 Review

1) Write each as an algebraic expression, represent the number with n:

a) Three times a number increased by two b) Two less than five times a number

c) Six times the sum of three and a number. d) The product of 5 and a number squared.

2) Find the value of each of the following.

2 2a) If a 3, b 4 and c 5, find the value of 3a b 2c

3(x y)

b) Find the value of , if x 2, y 5 and z 9.z

3) Bob solved the following equation. At each step explain how Bob got from the previous step:

2 5 4 24

10 2 4 24 _______________________________________________________

2 10 4 24 _______________________________________________________

2 14 24 _____________________________________________________

x

x

x

x

__

2 10 _______________________________________________________

5 _______________________________________________________

x

x

4) Using the digits 1,2,3 and 4, create an expression that evaluates to the following numbers. Only

addition and multiplication are used and each number appears only once. You may use grouping

symbols (parentheses).

a) 18 b) 20 c) 25


HW 1-6 continued

5) Place parentheses to make each statement true.

a) 4 3 2 4 5 b) 4 4 4 4 4 4

6) What is the additive inverse of the expression x – 3 ?

a) 3 b) 3 c) 3 d) 3x x x x

7) Which of the following is a mathematical expression?

2a) 3 2 9 b) 2 4 7 c) 3 d) 2 2x x x y x y

2


5 3I. II. 2 III. IV. 4

9 2


a) II III b) I + III c) I + IV d) III

not

9) An appliance repairman charges $65 per hour for the labor and a $45 service charge just to come to

the site. If M represents the total charges in dollars and h represents the number of hours worked,

which formula can be used to calculate the total charges for the repairman?

a) 65 25 b) 65 20 c) 45 65 d) 45 65M h M h M h M h

10) Which statement is not always true?

a) The product of two irrational numbers is irrational.

b) The product of two rational numbers is rational.

c) The sum of two rational numbers is rational.

d) The sum of a rational number and an irrational number is irrational.


Unit 2 – Exponents, Radicals, and Polynomials

Homework 1: Exponent Basic Properties (A-SSE.1,2)

Notes Basic Exponent Properties

Multiplication of a common base – Rule: addition of exponents. 2 5 2 5 7Ex: x x x x

A base and exponent raised by another exponent – Rule: multiply exponents. 5

2 2 5 10Ex: x x x

Division of a common base – Rule: subtraction of exponents. 5

5 2 3

2Ex:

xx x

x

2) Simplify each:

a) 4 2x x b)

4 3y y y c) 5 73 3

d) 3 2

2 2x x e) 6

2

x

x f)

16

4

y

y

g) 9

3

4

4 h)

( )

( )

8

2

3x

3x i)

23x

j) 3

2 32x y k)

23 42

x y5

l) 3

52x

REVIEW 1) Write an expression to represent "five less than a number squared" and determine

an irrational number that when substituded into the expression results in a rationl number.


HW 2-1 continued

m) 4 3 5 33 4x y x y n) ( )

( )

10 5 7

5 5 9

24a b c

6a b c

o) 2 2a b

p)x

y

3

3 q)

23x

2

r)

324x y

3a

8

4 4

8 7

3) What is half of 2 ?

a) 1 c) 2

b) 1 d) 2

3

3

2

7 7

4 4

24) The expression is equivalent to

4

a) 4 c) 2

1b) 2 d)

2

x

x

x x

x x

2 2 45) Stephen thinks , explain why Stephen is incorrect.x x x

5 10

4 12

5

2

6) Determine a value of , that would make the following equations true:

a)

b) 2 8

c)

x

x

x

x

a a a

a a

a aa

a



Homework 2: Zero and Negative Exponents (A-SSE.1,2)

REVIEW 1) Simplify: 2 4 4

3 2

2x y 6 x z

3z 4 y 2) Simplify

33

2

2x

3y

Notes:

Zero Exponents – Any non zero base raised to the zero power equals one. 0

0 2 3Ex: 1, 2 1x x y

Negative Exponents – Reciprocals its base. 1

Ex: ,

n nn

n

x yx

y xx

3) Simplify each: Express all answers with positive exponents where applicable:

a) 03 b) 0

2x c) 04x d) 24 e) 32

f) 22x g)

12

3

h) 2

2 3x y

i)

1

2

2

x

j)

02

3

5

4

x

y

k) 2 3 5 53x y 5x y l)

23 2 34a b c m)

5 -2 3

3 4 -2

12r s t

18r s t n)

( ) ( )

( )

2 1 2 3

3 2

5x y 3x

15x y

4) Write the following in scientific notation.

a) 0.0000567 b) 38,200,000 c) 2.57632

5) Solve for , in the equation 5.2 10 .00052nn


HW 2-2 continued

3 2

3 2 3

2 3 2 3 2

6) Which expression is equivalent to ?

1a) b) c) d)

x y

x y x

y x y x y

2

3

6 5 6 6

7) Which expression is equivalent to 4 ?

1 1 1 4a) b) c) d)

16 16 4

x

x x x x

0 2 18) Find the value of 3 , when .

4x x x

0 1

9) Find the value of 3 1 , when 4.x x x

10) Bob thinks that dividing the term 5x by itself is the same as raising

5x to the zero power.

Explain why you agree or disagree with Bob.

11) Michelle thinks a positive base if raised by any exponent cannot become negative. Explain why you

agree or disagree with Michelle.



Homework 3: Fractional Exponents (A-SSE.1, 2)

REVIEW 1) 0 1Find the value of 4 2 , when 1.xx x

Notes: Fractional Exponents – Power over root. Example: or

aab a bbx x x

2) Simplify each of the following (Without a calculator is recommended)

a) 2

327 b)

3

281 c)

1

327

d)

2

38

e)

1

24

9

3) Rewrite each as expressions without fractional exponents.

a)

1

2x b) 2

3y c)

1

5a

d) 1

225x e)

1

3

3

x

y

f)

3

2xy

4) Rewrite each as expressions without radicals.

4 53a) b) c)x y a

63

5

1 1d) e) f )y

x a


HW 2-3 continued

1

2

2


7a) 7 b) 7 c) 7 d)

x

x x xx

3

4 2

6 6

6 6


1 1a) b) c) d)

x

x xx x

12 2

4

3

3


1 1a) b) c) d)

x

x

x xxx

5

3 22

3 2 55

5 5 32


a) b) c) d)

x

x x x x

11

312

19) Find the value of 8x 10 x , when x 2.

4x

10) Explain why a positive base raised by a fraction exponent can get closer to zero, but never equal to it.



Homework 4: Simplifying, Adding and Subtracting Radical Expressions (N.RN.3, A-SSE.2)

Notes: Simplifying Irrational Radicals – Radicals can be simplified if the number inside has a square

factor. A term is in simplest radical form when the radical term has no square factors.

Example: 20

4 5 Split into two factors, one factor must be square

4 5 Split each factor into its own radical

2 5 Simplify the square radical, 5 has no square factors therefore 2 5 is in simplest radic

al form

Watch out for rational radicals such as 16 which just simplifies to 4 with no radical.

An original radical and its simplest radical form have the same decimal value, checking with a

calculator can be helpful. 20 4.472... , 2 5 4.472...

Radicals with Exponents:

12 6Even Exponent Example: Simply divide exponent by 2 and remove the radicalx x

7 3 1Odd Exponent Example: Divide exponent by 2, remainder remains inside the radicalx x x

2) Simplify each expression into simplest radical form:

a) 25 b) 18 c) 32

d) 252 e) 5 28 f) 3 144

4 5 12g) h) i)

9 16 25

2 16 9j) k) l) x x x

4 6 5 3 7 8m) n) o) 4 25x y x y x y

1 3

0 2 2 2Review 1) Given 4, list the following expressions least to greatest: , , , x x x x x


HW 2-4 continued

Notes: Adding/Subtracting Radicals: must have the same radicand before you can combine them.

Example: 20 45 32

2 5 3 5 4 2 Simplify each term first into simplest radical form

5 5 4 2 Combine line radical terms by adding/subtracting coefficients

3) Simplify each expression as much as possible:

a) b)

c) d)

e) f)

g) h)

4) Determine the value of if 27 3 7 3x x

5) Determine the perimeter of a rectangle if the length is 5 2 and width is 18.

6) Express in simplest radical form: 3 2 500 2 72

6 7 10 7 12 5 3

4 5 3 5 8 72

24 3 3 4 12x x x 2 316x x x x

2 16

33 3 24 64 6x x x x



Homework 5: Multiplying and Dividing Radical Expressions (N.RN.3, A-SSE.2)

Review 1) Determine the value of if 3 5 9 5x x

Notes: Multiplying and Dividing Radicals:

Multiply/Divide the outside (coefficient) and Multiply /Divide the inside (radicand).

Example: 2 5 3 8

2 3 5 8 Multiply inside parts and outside parts

6 40

12 10 Simplify into simplest radical form


a) 2 10 b) 5 3 6 c) 3 3 4 12

2 2 2

d) 10 e) 5 3 f) 2 5x

20 24 5 18 180g) h) i)

2 2 5 2 3 5

2

2

4 8 2 3 3 4 25j) k) l)

56 2 3

x

x


HW 2-5 continued

3) Express in simplest radical form: 3 2 10 8

4) Determine the area of a square with a side length of 2 5.

5) Determine the area of a rectangle if the length is 5 2 and width is 18.

6) Determine the area of a triangle with a base of 4 2 and a height of 6.

8) Determine the value of if 2 20 4 15x x

2

9) Tom thinks 2 8 and 2 8 have the same value.

Explain why you agree or disagree with Tom.

7) Express in simplest radical form: 10 5 2 15 2 3



Homework 6: Addition and Subtraction of Polynomials (A-SSE.2, A-APR.1)

Review 1) Express in simplest radical form: 2 6 3 5 10

Definition:

Like Terms – Terms with the same variables with the same exponents or the same radicals. Coefficients

can be different.

Addition of Polynomials – combine like terms by adding the coefficients and keeping the variables.

2 2 2 2

2

Example: 3 5 7 2 1 3 5 2 7 1

4 3 8 Final answer should have no remaining like terms

x x x x x x x x

x x

Subtraction of Polynomials – subtract each term of the polynomial, then combine like terms.

2 2 2 2

2 2

2

Example: 3x 5x 7 x 2x 1 3x 5x 7 x 2x 1

Remember that subtracting negatives becomes adding positives 2x 2x

3x 5x 7 x 2x 1

2x 7x 6 Final answer should have no remaining like terms

2) Combine like terms in the following expressions:

2 2

2 2 2 2

a) 5 6 4 b) 3 2 7 2

c) 2 3 d) 3 4

x x x x x x

a b b c ac b c a b y x y x

3) Perform the indicated operation and write each expression in simplest form:

2 2

2 2 2

a) 3 7 2 8 b) 4 3 2 6 5

c) 3 4 2 3 d) 5 4 2 2

x x x x x

x y x y xy x y xy a b c c b a


HW 2-6 continued

2 24) Subtract 3 2 5 from 5 7, express the result as a trinomial.x x x x

2 25) If 5 7 5 and 4 8 5, then find the value of each of the following:

a) The sum of and . b) Subtract from . c) Subtract from .

J x x K x x

J K J K K J

6) State the property that proves and J K K J has the same value.

2

2 2 2 2

7) The expression 5 2 1 2 1 is equivalent to

a) 7 6 b) 8 5 c) 8 5 d) 10 5

x x x

x x x x x x x

8) The length of a rectangle is two less than seven times a number, and the width is five more than

four times that same number. Create an expression in simplest form to represent the perimeter

of the rectangle.

2 29) If 5 +3 and 2 1, Maria thinks that the value of is the same as .

Explain why you agree or disagree with Maria.

A x B x A B B A

2 2 410) A common mistake is for students to think . Explain why this is incorrect. x x x



Homework 7: Multiplication of Polynomials (A-SSE.2, A-APR.1)

Review 1) Express the perimeter of a regular pentagon (5 sides of equal length) if each side

is represented by the expression 3 1.x

Notes: Multiplication of Polynomials – each term from one polynomial gets distributed to each term of

the other polynomial. (Each With Each), then combine like terms and write in exponent descending

order. 2 2Example: 5 2 3 3 5 3 2 3x x x x x x x x

3 2 2 3 23 5 15 2 6 8 13 6x x x x x x x x

A polynomial Squared – Anything squared is being multiplied by itself.

2

2 2

Example: 5 5 5 5 5 5

5 5 25 10 25

x x x x x x

x x x x x

For a binomial squared a helpful shortcut is to remember a pattern of

“Square the front, times each other doubled, and square the back” 2 2 2Example: 2a b a ab b

2) Perform the indicated operation and write each expression in simplest form:

2 2

2 2 2

222 2

a) 5 2 8 b) 2 6 5 c) 3 5 4

d) 2 5 e) 4 7 f) 2 1 6

g) 3 h) 4 i) 3 2

j) 5 4 k) 2 3 1 l) 2 3

x x x x x x

x x x x x x

x x x

x x x x x x x


HW 2-7 continued

3) The length of a rectangle is five more than a number and the width is one less than triple

that same number. Write an expression in simplest form to represent the area of the rectangle.

4) The side length of a square is represented by 5 1,

Write an expression in simplest form to represent the area and perimeter of the square.

x

5) Write an expression in simplest form to represent the area of a triangle with a base of 4

and a height of 2 3.

x

x

226) If 5 3 2 is subtracted from 2 3 , what is the result, written standard form?x x x

2

7) The expression 4 2 is equivalent to:

a) 16 4 c) 16 16 4

b) 16 4 d) 8 8 2

x

x x x

x x x

2

2

2

2

8) Caleb's work for simplifying an expression is shown below.

2 3 2

1 2 3 9 2

2 2 6 18 2

3 2 6 20

In which step did he make an error in his work?

a) 1 b) 2

x

Step x x

Step x x

Step x x

Step Step

c) 3 d) He made no errorsStep




1) Simplify each:

a) 4 0x x b)

2 2

3

4

2

y y

y c) 25x

d) 3

2 24x e) 1

3 2 2x x

f)

17 3

1

x

x

3 0 22) Find the value of , when 2.x x x x

3) Simplify each expression into simplest radical form:

16 3 9a) 2 150 b) 4 c) x x y


3 2a) 2 18 3 72 b) 3 5 5 20x x x x

5) Determine the perimeter in simplest radical form of a regular hexagon if the length of a side is 32.


HW 2-8 continued

6) Express in simplest radical form: 3 2 75 12

2

7) Express in simplest radical form: 5 3 2

8) Explain why 1n

n x

9) If 2 4 and 5 3, determine the value of:

a) Subtract from . b) The product of and . c) Square the sum of and .

A x B x

A B A B A B

10) A square has a side length that is 5 less than three times a number.

a) Create an expression to represent the perimeter of the square.

b) Create an expression to represent the area of the square.

11) Express the following as a polynomial in simplest form:

2

2 3 2 3 (2 3)x x x



Homework 9: Post Test Cumulative Review

0 1

23

1) Given the four expressions below:

52 7 , , , 4

2

The sum of which who expressions would result in a rational expression? Explain your answer

2

2) Logan needs to simplify the following expression:

2 3 4 5 1

What should Logans do first to simplify the expression?

4) An appliance repairman charges $65 per hour for the labor and a $45 service charge just to come to

the site. If c represents the total charges in dollars and h represents the number of hours worked, which

formula can be used to calculate the total charges for the repairman?

a) c = 65 + 25h b) c = 65h + 20 c) c = 45 + 65h d) c = (45 + 65)h

3) An equation is solved below, match each step with the property used to change from the previous step.

2Original Equation: x 3 6

3

2Step 1 x 2 6 a) Additive

3

Inverse

2Step 2 x 4 b) Mulitplicative Inverse

3

Step 3 x 6 c) Distribution Property


HW 2-9 continued

2 25) What must be added to 2x 6x 12 in order to get 27x 2x 15?

6) The simplest form of (3 7)(2 1) (5 1)( 3)x x x x is a trinomial with positive coefficients. Find

the trinomial and the sum of the coefficients of the trinomial.

7) From the sum of 7x2 – 4x + 5 and 2x

2 – 8x - 7 subtract 5x

2 + 2x + 5.

a) 4x2 - 10x – 7 b) 4x

2 - 14x + 3 c) 4x

2 - 14x – 7 d) 14x

2 - 10x + 3

8) The length of a rectangle is one more than twice the width. If w represents the width of the rectangle,

which expressions represent the perimeter and the area of the rectangle?

a) P = 6w +2, A = 2w2 + w

b) P = 4w + 1, A = 2w2

c) P = 6w + 2, A = 2w2

d) P = 4w + 1, A = 2w2 + w

9) The expression 3 5 3 5 is equivalent to:

a) 9 25

b) 9 25

c) 9 25

d) 9 25

x x

x

x

x

x

10) What is 32

4 expressed in simplest radical form?

a) 2 c) 8

b) 2 2 d) 8

2


Unit 3 – Factoring

Homework 1: GCF (A-SSE.1,2)

Notes: Factoring is a form of division and there are many different methods that will be discussed in this

chapter. When a polynomial is factored, it is being rewritten as a product of factors.

Greatest Common Factoring (GCF) is used when each term has a common factor, meaning each term

can be divided by something in common. The expression can be rewritten as the product of the GCF and

the quotient (expression after the GCF has been divided out for each term). The quotient should have no

common factors in the end.

Example 1: Each term in the expression 2 6 can be divided by 2.

Therefore 2 6 can be factored to 2 3

x

x x

3 2

3 2 2

Example 2: Each term in the expression 6 can be divided by .

Therefore 6 can be factored to 6 1

x x x x

x x x x x x

3 2 2 2

3 2 2 2

Example 3: Each term in the expression 8 6 can be divided by 2 .

Therefore 8 6 can be factored to 2 4 3

x y x y x y

x y x y x y x y

1) Factor each of the following expressions using the GCF method.

a) 3 12x b) 4 20x c) 2 28 2x y

d) 2 3x x e) 3 23x x x f) 22 5x y y

g) 22 6x x h) 2 23 9xy x i) 3 24 6 2x x x

j) 2 25 15xy x y k) 3 2 3 22 10 6x y xy xy l) 3 6 3xyz x xy


HW 3-1 continued

3 2 2 5 2 4 3

2 2

2 2

2) Which of the following is the greatest common factor of the expression: 8 16 12

a) 8 c) 4

b) 4 d) 4

x y z x y z x y

xy x y z

xyz x y

2 2

2

3) Which of the following is a common factor of: 20 10

a) 5 c)

b) 10 d)

not x y xy

x

y

2

4) Which of the following is a factor common factor of: 2 3 5 3

a) 2 c)

b) 3 d) 3

x x

x

x

5) When solved for , the equation 2 is written as:

a) b)

c) d)

x ax c bx c

c cx x

a b a b

cx x c b a

ab

2

2

2

6) Which of the following is :

a) The product of the expressions 2 and 5 is 2 10

b) The expression 2 10 factors to 2 5

c) The expression 2 is a factor of 2 10

d) The expression 5 is

false

x x x x

x x x x

x x x

x

2a factor of 2 10x x



Homework 2: DIFFERENCE OF TWO SQUARES (A-SSE.2)

REVIEW 1) Factor each of the following expressions:

2 2 2a) 5 15 b) 7 c) 4 6x x x x y xy

Notes: Factoring the Difference of Two Squares (DOTS), is a form of factoring that can only be done

with binomial expressions that have difference (subtraction) of two square terms (square coefficients

and/or variables with even exponents). The result of the factoring is the product of two binomials. The

terms in each of the factors are the square root of the terms in the original expression. One binomial is

the sum of them, and the other is the difference.

2 2Example 1: factors to a b a b a b 2Example 2: 9 factors to 3 3x x x

2Example 3: 4 25 factors to 2 5 2 5x x x

4 6 2 3 2 3Example 4: 9 4 factors to 3 2 3 2x y x y x y

2) Factor each of the following expressions:

2 2 2a) 16 b) 1 c) 64x x x

2 2 2 2d) 9 e) f) 4 81x x y x

2 4 8g) 1 9 h) 25 g) 100x x x

2 2 16 10 64h) 4 i) 121 j) 196 225x y x x y


HW 3-2 continued

3 2 2

3) Which of the following is a difference of two squares:

: 16 : 16 : 8

a) only b) only c) & only d) , ,&

not

I x II x III x

I II I II I II III

2 24) Stephen thinks that the expressions 9 and 9 are equivalent.

Explain why you agree or disagree with Stephen.

x x

25) Elizabeth thinks 25 factors to 5 5 . Prove Elizabeth is incorrect.x x x

2

2

4

2

6) Which of the following is ?

a) The expression 9 factors to 3 3

b) The expression 16 factors to 4 4

c) The expression 8 can not be factored

d) The expression 25 can not be factore

true

x x x

x x x

x

x d

2

2 2

2

2


a) The expression 4 16 can factor to 2 4 2 4

b) The expression 4 16 can factor to 4 4

c) The expression 4 16 can factor to 4 2 2

d) The expression 4 16 can

false

x x x

x x

x x x

x not be factored



Homework 3: Basic Trinomial Factoring (A-SSE.2)

REVIEW 1) Factor each of the following expressions:

2 2 2 2a) 81 b) 4 c) 1x x x x y

Notes: A Trinomial (polynomial with three terms) in the form 2x bx c with b and c being rational

numbers, is factorable to the product of two binomials if two numbers exist that add to b and

multiply to c.

2Example 1: 6 8 factors to 2 4 because 2 4 6 & 2 4 8x x x x




2 2 2a) 6 5 b) 8 15 c) 9 14x x x x x x

2 2 2d) 5 4 e) 18 9 f) 10 25x x x x x x

2 2 2g) 12 h) 12 i) 4 12x x x x x x

2 2 2j) 5 6 k) 6 l) 6x x x x x x


HW 3-3 continued

2 2 2

3) Which of the following are factorable:

: 2 : 2 1 : 6


not

I x x II x x III x x

I II I III I II III

2

2

2

2


a) The expression 4 5 factors to 1 5


c) The expression 4 5 can not be factored

d) The expression 5 4 can not

false

x x x x

x x x x

x x

x x be factored

2

2

2

2


a) The expression 4 12 factors to 6 2


c) The expression 20 factors to 5 4

d) The expression 8 16 fac

incorrect

x x x x

x x x x

x x x x

x x

2

tors to 4x

26) Factor the expression +x a b x a b

7) Mary thinks the expression 2 3 4x x is not factorable.

Explain why you agree or disagree with Mary.



Homework 4: Factoring by Grouping (A-SSE.2)

REVIEW 1) Factor each of the following expressions: 2 2 2a) 2 35 b) 4 9 c) 3x x x x x

Notes: Factoring by Grouping is a factoring method that groups terms in a polynomial into smaller

groups that have a GCF, each group must have a common quotient after the GCFs have been factored

out. This method required the polynomial to have an even amount of terms.

3 2

3 2

2

Example: Factor the expression 5 4 20

First group the expression into two groups 5 4 20

Factor each group using the GCF method 5 4 5

Both groups have a common quotient of 5, so we kno

x x x

x x x

x x x

x

2

w we are correct so far

Next factor the common quotient out to get 5 4

Lastly check to see if either binomial can be factored, in this case we have a DOTS

The final factorization of the expression i

x x

n simplest form is 5 2 2 . x x x

2) Factor each of the following into simplest form:

3 2 3 2a) 3 4 12 b) 3 3 x x x x x x

3 2 3 2c) 2 6 3 d) 3 2 3 2 x x x x x x

3 2 3 2e) 5 5 1 f) 3 80 5 48x x x x x x


HW 3-4 continued

3) Factor each of the following into simplest form:

2 3a) 4 4 b) 4 4 x xy y x y x y x

3 2 3 2


: 4 4 : 2 6 3

a) only b) only c) & d) Neither

not

I x x x II x x x

I II I II

3 25) Which of the following is not a factor of 4 9 36

a) 4 c) 3

b) 4 d) 3

x x x

x x

x x

3 2 26) Factor the expression in simplest form: 3 5 15 36 108x x x x x



Homework 5: Advanced Trinomial Factoring (A-SSE.2)

REVIEW 1) Factor each of the following expressions into simplest form:

3 2 2 2 2a) 2 2 b) 4 c) 2x x x x y x x

Notes: A Trinomial in the form 2ax bx c with a, b and c being rational numbers, and 1a is

factorable to the product of two binomials, like basic trinomials, but requires factoring by grouping.

Since grouping can only be used on polynomials with an even amount of terms we need to expand the

trinomial into four terms then use grouping. The way to expand it into four terms is by splitting the

middle term into two terms that add to the b term and multiply to the product of a and c. This is often

referred to as AC factoring.

2

2 2 2

2

Example: Factor the expression 3 5 2

First we determine the product of and ,3 and 2, which is 3 2 6

Then we determine terms that add to the term, 5 , and multiply to the term, 6

Th

x x

a c x x x

b x ac x

2

2

ese terms are 6 and 1 because the sum of 6 and 1 is 5 , and the product is 6

The term is replaced by these two terms to give the expression 3 6 1 2

Which then gets factored using groupin

x x x x x x

b x x x

g 3 2 1 2

To get the final factored expression 3 1 2

x x x

x x


2 2a) 3 7 2 b) 2 3 x x x x

2 2c) 2 3 5 d) 6 4 5 x x x x


HW 3-5 continued

2 2


: 3 4 : 3 4

a) only b) only c) & d) Neither

not

I x x II x x

I II I II

4) Justin claims the expression 22 6 20x x can be factored using the AC factoring method. Stephen

claims the same expression can be factored using GCF and then the basic trinomial factoring method.

Explain who you agree with and why.

2 25) Factor the expressin 2 11 15 using the factoring method and grouping.

Check your answer by multiplying the factors together.

x y xy AC

4 3 26) Factor the expression: 2 7 8 12x x x x



Homework 6: Factoring Completely (A-SSE.2)

REVIEW 1) Factor each of the following expressions into simplest form: 3 2 2 2a) 4 4 16 b) 4 9 c) 4 4 1x x x x x x

Notes: Factoring completely simply means factor into simplest form. Sometimes expression can be

factored more than once as a combination of multiple types of factoring. GCF should always be the first

type of factoring used when possible.

2

2

Example 1: Factor 3 75

First a GCF can be factored out to become 3 25 , then factoring DOTS yields 3 5 5

x

x x x

4

2 2 2

Example 2: Factor 16

Factoring DOTS yields 4 4 , then factoring DOTS again yields 4 2 2

x

x x x x x

An expression is factored completely when none of its factors can be factored further

2) Factor completely each of the following expressions:

2 3 2a) 5 20 b) c) 2 16 30 x x x x x

4 4 4d) 81 e) 16 1 f) 2 32 x x x

3 2 3 2 2g) 3 15 3 15 h) 2 8 i) 6 22 8 x x x x x x x x


HW 3-6 continued

5

2

3) Which of the following are factors of 81

:

: 3

: 3

: 9

a) only b) & only

c) , & only d) , , , &

x x

I x

II x

III x

IV x

I II III

I II III I II III IV

4 6

2 3 2 3

2 3 2 3

2 3 2 3

2 3 2 3

4) Which expression represents 16 64 factored completely?

a) 4 8 4 8

b) 16 2 2

c) 4 2 4 2 4

d) 8 32 8 32

x y

x y x y

x y x y

x y x y

x y x y

4 3 2

2

2 2

2

2

5) Which expression represents 4 10 6 factored completely?

a) 4 2 3

b) 2 2 3

c) 2 2 1 3

d) 2 2 1 3

x x x

x x x

x x x x

x x x

x x x

4 3 2 2 3 46) Factor 4 6 4 6 completely.x y x y x y xy




4 3 2 2 4 3

2 2 2 4

1) Which of the following is the greatest common factor of the expression: 6 12 9

a) 3 z b) 6 c) 3 d) 3

x y z x yz x z

x x yz x yz x z

2

2

8

2


a) The expression factors to 1


c) The expression 1 can be factored

d) The expression 6 can not be factored

false

x x x x

x x x x

x

x x


a) 4 c) 5

b) 4 d) 5

x x x

x x

x x

2

2

2 2

4) Which of the following is :

a) The product of the expressions and 3 is 3

b) The expression 3 factors to 3

c) The expression is a factor of 3

d) The expression 3 is a factor

false

x x x x

x x x x

x x x

x

2 of 3x x

5) When solved for , the equation is written as:

a) b)

c) d)

x ax c bx

c cx x

a b a b

cx x c b a

ab


HW 3-7 continued

26) Which expression represents 2 10 factored completely?

a) 2 5 2 b) 2 2 5

c) 2 5 2 d) 2 5 2

x x

x x x x

x x x x

4 3 2

2 2

3


a) 2 2 b) 2 2

c) 4 2 d) 2 2

x x x x

x x x x x x

x x x x x x

8 4 6

8) Which of the following are a difference of two squares:

: 1 : 6 : 4


not

I x II x III x

I II I II I II III


a) 2 b) 3 1

c) 2 d) 2

x x x

x x

x x

4 3 2

2

2

10) Which of the following are factors of 5 25 125

:

: 5

: 10 25

: 25

a) only b) & only

c) , & only d) , , , &

x x x x

I x

II x

III x x

IV x

I II III

I II III I II III IV




1) What is the additive inverse of the expression x + 5?

a) 5 b) 5 c) 5 d) 5x x x x


1I. II. III.


a) I II b) II + III c) II III d) I III

not

0 1 23) Find the value of 3 , when 3.x x x x

4) Determine the perimeter in simplest radical form of a square if the length of a side is 24.

5) If 5 and 5, determine the value of:

a) Subtract from . b) The product of and . c) Square the sum of and .

A x B x

A B A B A B


HW 3-8 continued

2

6) Express in simplest radical form: 2 5

7) What is 160

4 expressed in simplest radical form?

a) 4 10 b) 8 10 c) 2 10 d) 10


a) 3 1 4 b) 3 4 1

c) 3 1 4 d) 3 4 1

x x

x x x x

x x x x


a) 4 b) 4 c) 5 d) 5

x x x

x x x x

2 2 2

10) Which of the following are a difference of two squares:

: 4 : 8 : 1


not

I x II x III x

I II I II I II III


Unit 4 – Solving

Homework 1: Solving Linear Equations (A-CED.1, A-REI.1,3)

Notes: The goal of solving an equation for a given variable is to get that variable to have a coefficient of

one and be alone on one side of the equal sign.

Example: 2 3 4 3 2 7

Step 1: Try to remove any parenthesis, in this case with distribution 6 8 3 2 7

Step 2: Combine any like terms on the same side of the equal si

Solve for x in the equation x x

x x

gn 6 5 2 7

Step 3: Use the additive inverse property to move constant terms to one side and variable terms

to the other side, and combine like terms 6 5 2 7

5 5 6 2 12

2 2 4

x x

x x

x x

x x x

12

Step 4: Use the multiplicative inverse property to get the variable coefficient to equal 1, in other words

divide both sides by the variable coefficient 4 / 4 12 / 4 3

Step 5: Check your

x x

solution by substituting your answer into the original equation

2 3 3 4 3 2 3 7 13 13

1) Solve and check each equation:

a) 6b – 20 = 2b b) 5x + 3 = 15 + 2x c) 6x – 4 = 20 – 2x

d) 3x + x – 2 = 7 – 2x e) 18 – x = 4x + 3 f) 4 + 4a = 11a – 6

g) 4(2x + 6) = 40 h) 3x + 2( 50 – x ) = 110 i) 8 – 4( x – 1 ) = 2 + 3(4 – x)

2) Determine the number if:

a) Six less than a number is equal to two more than triple that number. Determine the number.

b) Double the sum of a number and 5 is 36. Determine the number.


HW 4-1 continued

Notes: When solving an equation with an expression squared, first solve for the expression squared then

square root both sides. This results in two possible equations (plus or minus), solve both equations and

state both solutions. Be sure to check both solutions because extraneous (false) solutions are possible.

2

2 2 2 2

2

2 2 2

Example 1: 3 4 8

3 4 8 3 12 4 4 2 2 and 2

Example 2: 2 1 10

2 1 10 2 9 2 9 2 3 2 3 and 2 3

5 and

Solve for x in the equation x

x x x x x x x

Solve for x in the equation x

x x x x x x

x x

1

3) Solve and check each equation:

a) 2 4 40x b) 2

3 4 21x c) 2

2 1 5 27x

4) Ted is asked to solve the equation 2

3 2 1 4 31x .

Ted’s solution to the problem starts as:

2

2

3 2 1 27

2 1 9

x

x

A correct next step in the solution of the problem is

2 2a) 4 1 9 b) 4 1 3 c) 2 1 3 d) 2 1 9x x x x

5) Explain why 2 2

3 5 does not equal 3 15 .x x


Unit 4 – Solving

Homework 2: Fractional and Decimal Equations (A-CED.1, A-REI.1,3)

Review 1) Solve x and check your answer:

a) 4x – 3 = 17 b) 5( 2x – 5) = 6x +7 c) 6(x + 2) + 3(2x – 3) = 51

Note: Solving an equation with decimals is the same procedure as with integers. Use a calculator for

assistance with the algebra. Check your answer the same way as before.

2) Solve each and check.

a) 0.4 12 16x b) 0.5 3.5 5.5a c) 0.06 3 4.8y

d) 0.4 10 3.2x e) 2.5 2 .5 11c c f) .6 5 2 .8h h

g) 5 .5 3 4 0x x h) .5 .25 1 2.5y

Note: When solving an equation with fractions, one helpful method is call “Eliminating Denominators”

This method requires you to first multiply out any denominators which remove the fractions from the

equation. Then you solve and check using normal algebra methods.

2Example: Solve for : 3 7

5

Step 1: Mutlipy each term by the denominator 5

25 3 5 7 5

5

Step 2: This results in an equation without fractions that can be solved with normal methods

2 15 35 2 50 2

x x

x

x x x

5

If multiple denominators are in the equation you can multiply each one out one at a time or multiply by a

least common multiple to remove all at once.


HW 4-2 continued

3) Solve each and check.

a) 3

3 152

x b) 3

2 15

x c) 5 2

3

x

d) 1

5 2 42

x x e) 2

6 43

xx f)

4 63 8

5 5x x

g) 3 2

52 3

x x h)

3 3

5 2

x i)

2 4 5

8 5

x x

4) Fred is four more than half of Bills age. If Bill is 12 years old, how hold is Fred?

5) Ted thinks to solve the equation .25 7x he should multiply both sides by 4. Will Ted’s procedure

give him the correct answer? Explain why Ted’s method does or does not work.

6) Solve for x in the equation: 2

12

2x x x


Unit 4 – Solving

Homework 3: Literal Equations (A-SSE.1, A-CED.1,4, A-REI.1,3)

REVIEW 1) Solve for x: 1

6 42

x x

Notes: Literal equations are equations with several variables, although they look more complicated and

different than regular equations, they are solved using the same methods. 2

2

2

Example: Solve for : 3

Step 1: Add to both sides 3

Step 2: Divide 3 to both sides3

Step 3: Square Root both sides3

a a b c

b a c b

c ba

c ba

The goal of a literal equation is to solve for a specific variable, meaning get that variable alone on one

side of equal sign and all other terms on the other side. Don’t forget to combine like terms when you

have them.

2) Solve for x in each of the following:

a) 3 6xyz yz b) 3 13x b b c) bx ab bc

d) 3 2 24x a a e) 3 4 2x a x a f) 1

4 62

ax b

3) For the given equation:

a) Solve for b) Solve for c) Solve for

V LWH

L W H


HW 4-3 continued

4) The formula for the perimeter of a rectangle is 2 2 .

a) Write a formula that can be used to find the width of a rectangle in terms of the perimeter and length.

b) If a rectangle has a perimeter o

P L W

f 24 and a length of 7, determine and state the width of the rectangle.

15) The formula for the area of a triangle is .

2

a) Write a formula that can be used to find the height of a triangle in terms of the area and base.

b) If a triangle has an area of 10 and a base of

A bh

4, determine and state the height of the triangle.

26) The formula for the area of a circle is , the radius, , of the circle may be expressed as

a) b) c) d) 2

A r r

A A AA

217) The formula for the volume of a square based pyramid is , the side of the square base, , of

3

the pyramid may be expressed as

3 3a) b) c) d)

3 3

V s h s

V V V V

h h h h

9

8) The formula to convert from fahrenheit to celsius is 32. 5

What is the formula to convert from celsius to fahrenheit?

F C F C


Unit 4 – Solving

Homework 4: Writing Equation Applications (A-SSE.1, A-CED.1, A-REI.1,3, A-APR.1)

REVIEW 1) The Pythagorean theorem is 2 2 2 ,a b c where a and b represent the length of the legs of

a right triangle and c represents the hypotenuse. Write a formula to solve for a in terms of b and c.

Notes: The perimeter formula for a rectangle is 2 2 .P L W Translate the given information into an

expression that can be substituted into the perimeter formula. After the substitution only one variable

remains and can be solved. Example: The length of a rectangle is five more than its width and the

perimeter of the rectangle is 26. Determine and state the length and width of the rectangle.

Formula : 2 2 , Given/Translated Information: 5 & 26

Substitution into formula: 26 2( 5) 2

Solved 26 2 10 2

26 4 10

16 4

4 and 4 5 9

P L W L W P

W W

W W

W

W

Width Length

2) The length of a rectangle is four more than the width and the perimeter of the rectangle is 20.

Determine and state the length and width of the rectangle.

3) The width of a rectangle is six less than the length and the perimeter of the rectangle is 28.

Determine and state the length and width of the rectangle.

4) The length of a rectangle is three more than double the width and the perimeter of the rectangle is 36.

Determine and state the area of the rectangle.

5) The perimeter of a rectangle is fourteen more than the length and the width of the rectangle is 2.

Determine and state the perimeter of the rectangle.


HW 4-4 continued

Notes: Money problems are solved using the same procedure as for perimeter.

Create expressions to represent the number of each coins involved. For pennies times .01, nickels .05,

dimes .10 and quarters .25. Example: Justin has two more nickels than quarters in his pocket for a total

of $1.30. How many nickels and quarters does Justin have?

Given/Translated Information: 2 & .05 .25 1.30

Substitution into formula: .05 2 .25 1.30

Solved .05 .10 .25 1.30

.30 .10 1.30

.30 1.20

# 4 and # 4 2 6

N Q N Q

Q Q

Q Q

Q

Q

of Quarters of Nickels

6) Jonny has four more dimes than nickels in his pocket for a total of $1.15. Determine and state how

many dimes Jonny has in his pocket.

7) Teddy has the same number of dimes and nickels in his wallet for a total of $1.50. How many nickels

does Teddy have in his wallet?

8) Bob has one less than triple the amount of nickels as he has quarters for a total of $1.15. Determine

and state how many nickels Bob has.

9) Meaghan has two less dimes as she does nickels and one more than double the amount of quarters as

she does nickels for a total of $2.00. How many nickels, dimes, and quarters does Meaghan have?


Unit 4 – Solving

Homework 5: Consecutive Integers (A-SSE.1, A-APR.1, A-CED.1, A-REI.3)

REVIEW 1) The length of a rectangle is three less than double the width and the perimeter of the

rectangle is 26. Determine and state the length of the rectangle.

Notes: Consecutive integers are integers that follow each other in order such as 8,9,10,11

When solving consecutive integer problems assign the first integer the variable x, and since the next

consecutive integer is one more than the previous the second consecutive integer is 1,x The third is

one more than the second so its 2x which is one more than 1.x

Example: Find three consecutive integers such that the sum is 24.

1 2 3 24 1 , 1 2 , 2 3

1 2 24

3 3 24 3 21 7

1 7, 2 8, 3 9

Translates to st nd rd let x st x nd x rd

Substitute x x x

Solve x x x

st nd rd

2) For each of the following find:

a) two consecutive integers such that their sum is 11.

b) three consecutive integers such that their sum is 12.

c) three consecutive integers such that twice the sum of the first and second is one more than three times

the third.

d) three consecutive integers such that sum of the second and third is two less than three times the first.


HW 4-5 continued

Notes: For consecutive even or odd integers such as 6,8,10 or 5,7,9 we can still assign the first integer

the variable x, but since the second is two more than the first it is 2x and the third is 4,x which is

two more than 2.x This is the same for even and odd. Odd integers are two more than the previous

just like even integers. A common mistake is for student to use , 1, 3x x x for odd which is incorrect

because if x is odd then 1x would be even. So stick with , 2, 4, ...x x x for consecutive even and

for consecutive odd integers.

Example: Find three consecutive integers such that the sum is 33.

1 2 3 33 1 , 2 2 , 4 3

2 4 33

3 6 33 3 27 9

1 9, 2 11, 3 13

odd

Translates to st odd nd odd rd odd let x st x nd x rd

Substitute x x x

Solve x x x

st nd rd

3) For each of the following find:

a) two consecutive even integers such that their sum is 10.

b) two consecutive odd integers such that their sum is 16.

c) three consecutive even integers such that their sum is 24.

d) three consecutive odd integers such that the sum of the first and third is three more than the second.


HW 4-5 continued

4) Determine and state four consecutive integers such that the fourth less than double the second is the

third decreased by three.

5) Determine and state four consecutive even integers such that the product of the first and fourth is the

second squared.

6) If the expression 2 1 represents an even integer. Which expression would represent the next

consecutive even integer?

a) 2 3 c) 3 1

b) 2 1 d) 2 2

x

x x

x x

7) If the expression 5 represents an odd integer. Which expression would represent the next

consecutive odd integer?

a) 6 c) 7

b) 2 5 d) 2 7

x

x x

x x


HW 4-5 continued

2

2 2

2 3



a) 3 2 c) 3 3

b) 4 1 d) 3 1

x

x x

x x

7) Katy thinks if represents an odd integer then would also represent an odd integer

as long as is an even number. Explain why you agree or disagree with Katy.

a a b

b

8) If represents an integer, does 2 1 represents an even or an odd integer?

Explain your answer.

x x


Unit 4 – Solving

Homework 6: Factoring to Solve Equations (A-SSE.2,3, A-CED.1, A-REI.1,3,4)

REVIEW 1) Factor each of the following expressions completely: 2 3 2 3a) 2 8 10 b) 12 c) 3 75x x x x x x x

3 2

: :

Example: Solve the equation 15 2

We can recognize that factoring must be used because there are multiple terms with that are

not like terms,

Notes Some equations require factoring to be solved

x x x

x

3 2

2

therefore we can not solve for (get alone) without factoring.

Before factoring the equation should be set equal to zero: 2 15 0

Then factor completely: 2 15 0 5 3 0

Then set ea

x x

x x x

x x x x x x

ch factor equal to zero and solve: 0, 5 0, 3 0

0, 5, 3

x x x

x x x

2) Solve for x in each of the following equations:

2 2 2a) 6 8 0 b) 4 0 c) 3 4x x x x x

2 2 3 2d) 9 0 e) 24 2 f) 7 +10 0 x x x x x x

3 2 3 2 2g) 5 5 0 h) 12 4 3 i) 2 9 4 0 x x x x x x x x


HW 4-6 continued

2) Solve for x in each of the following equations:

3 2 2a) b) 4 1 0 c) 10 25x x x x x

2 3 2 3d) 4 20 3 20 e) 2 8 3 0 f) 5 20 x x x x x x x x

3) Determine three positive consecutive even integers such that eight more than the sum of the first and

third is the second squared.

4) Determine three consecutive integers such that the first squared added to the second squared is five

times the third.


Unit 4 – Solving

Homework 7: Sets, Inequalities, and Notation (A-CED.1, A-REI.1,3)

REVIEW 1) Determine and state three consecutive integers such that sum of the second and third is two

less than triple the first.

Notes: A Set is a collection of unique items called elements that do not repeat. A set can be expressed by

a roster such as {5,6,7,8,9} which is the set of integers from 5 to 9, inclusive.

Inequality Notation

> "greater than" , < "less than" , "greater than or equal to" , "less than or equal to"

A set can be expressed with inequality notation which states a value is greater than, less than, or equal to

another value. Example: 5 " is greater than 5" this means the value of could be any valuex x x greater

than 5 such as 6,6.2,8,21,..

Example: 2 6 "2 is less than is less than 6" this means the value of could be any valuex x x

between 2 and 6 such as 3,4,4.2,5. If it were

Interval Notation is an alternative to expressing a set as an inequality.

Parentheses ( ) are used to mean not included or open and brackets [ ] are used to mean included or

closed. Example: 2 6 in interval notation is 2,6 , 5 2 in interval notation is 5,2x x

5 9 in interval notation is [5,9), 4 in interval notation is 4,x x

Examples of Inequality/Set Notation Graphs:

3, 3,x 4, ,4x 2 5, 2,5x 3 6, 3,6x

2) Sketch a graph for each of the sets and described them using both inequality and interval notation:

a) Numbers greater than 4 and less than 13. b) Numbers greater than 6.

c) Numbers greater than 2 and less than or equal to 9. d) Numbers less or equal to 3.

2 6 the numbers 2 and 6 would be included in the set. x


HW 4-7 continued

3) Which of the following is not a correct way to represent the set of integers

between 7 and 11, inclusive where x is an integer?

a) {7,8,9,10,11} b) 7 11 c) 6 12 d) 7,11x x

4) If y is an integer, what is the solution set of -3 ≤ y < 1?

a) { -3, -2, -1, 0, 1} b) { -3, -2, -1, 0} c) { -2, -1, 0, 1} d) { -2, -1, 0}

5) Which of the following is in the set (4,8]?

a) 4 b) 3 c) 8 d) 9

6) Which of the following is in the set 4,6 ?

a) 4 b) 3 c) 4 d) 0

not

7) Which of the following does contain 5 as an element in the set?

a) ( ,5] b) 5,11 c) 5 d) 5

not

x x

8) Which of following is a way to represent the set of real numbers less than or equal to 7?

a) ,7 b) ,7 c) 7 x d) ( ,7]

9) Jill thinks the set 4,2 contains the set 4,2 . Explain why you agree or disagree with Jill.

10)Tom says the smallest number in the set 1,4 is 0. Maria tells Tom that he is correct only if the

set is integers only. Explain why you agree or disagree with Maria or Tom.


Unit 4 – Solving

Homework 8: Solving Inequalities (A-CED.1, A-REI.1,3)

REVIEW 1) List all the integers that are elements in each of the following sets:

a) (1,5] b) 2,1 c) 2,7

Notes: Solving inequalities is the same as solving equations except that when you multiply or divide the

inequality by a negative you flip the inequality sign.

Example: Solve for : 7 2 15

2 8 subtracted 7 on both sides

4 divided 2 on both sides and flip the inequality sign from less than to greather than

x x

x

x

2) Find the solution set of each. Express the solution in set notation and graphically on a number line.

a) 3 5 16x b) 4 3 21x x

c) 5 2 3 4x x d) 5 3 2 30x x

e) 3 2 2 2x x f) 4 1 9x

3) Determine the smallest integer that makes 2 7 5 15 true.x x x


HW 4-8 continued

1

4) Determine the largest integer that makes 6 5 4 true.2

x x

5) For the following inequality:

a) Solve for : 5 4 2 1 2 8

b) If is a number in the interval 5, 1 , state all integers that satisfy the given inequality.

Explain how you determined these values.

x x x x x

x

2

6) Solve for and graph the solution set on a number line: 4 3 1x x x x

7) George thinks that 5 and 4 have the same integer solution set.

Explain why you agree or disagree with George.

x x

8) Solve for in terms of , and :

4

x a b c

x b a x bx c


Unit 4 – Solving

Homework 9: Compound Inequalities (A-CED.1, A-REI.1,3)

Review:

1) Find the solution set of each. Express the solution in set notation and graphically on a number line.

a) 4 2(5 3)x x b) 10 20 10(3 4)x x

Notes: Compound inequalities may have multiple parts with multiple inequality symbols. The goal with

solving these compound inequalities is the same as before, to get the variable alone. If the variable is

between two inequalities signs it is an intersection of sets, an “and” inequality.

Example: Solve for : 3 2 1 9, this can be read 2 1 is greater than 3 AND less than 9.

Get alone in the middle, any operation must be done to all three parts.

Step 1: Add 1 to the left, right, and

x x x

x

middle 3 2 1 9

+1 +1 +1

4 2 10Step 2: Divide 2 to the left, right, and middle

2 2 2

Solution has alone in the middl

x

x

x

e: 2 5x

If the inequality has two separate inequalities with an “or” it is a union of sets.

Example: Solve for : 2 6 OR 3 2

Solve for in both inequalities

Step 1: 2 6 becomes 3

Step 2: 3 2 becomes 1

Solution can be written as 3 or 1, or it can be written as , 3 1,

x x x

x

x x

x x

x x

2) Solve each compound inequality for 𝑥 and graph the solution on a number line.

a) 4 2 12 8 4 16x or x b) 4 2 4x

c) 3 4 7x d) 2 1 11 4 3 5 2x or x x


HW 4-9 continued

3) The solution set for 3 2 5 11 is

a) 1,3 b) 1,8 c) 1,3 d) 4,8

x

14) The solution set for 3 5 is

2

a) 5,9 b) 0,2 c) 5,9 d) .5,1.5

x

5) The solution set for 6 2(4 3 ) 14 is

11 7 1 1 1a) , b) 1, c) , 1 d) ,1

3 3 3 3 3

x

6) For the following inequality:

1a) Solve for and sketch a graph of the results for: 5 3 10 or 4 2.5 6

2

b) If is a number in the interval 6,1 , state all integers that satisfy the given inequa

x x x x

x

lity.

Explain how you determined these values.

7) State whether the following statements is sometimes, always or never true and justify your answer.

a) If x < y, then x + a < y + a b) If x < y, then x – a > y – a

c) If x < y then ax < ay d) If , thenx y

x ya a


Unit 4 – Solving

Homework 10: Writing Inequalities (A-CED.1, A-REI.1,3)

REVIEW 1) If is an integer, ,which is the solution set of 2 2?

a) { 2, 1,0,1,2} b) { 2, 1,0,1} c) { 1,0,1, 2} d) { 1,0,1}

x x

2) Frank’s mother said the cost of his lunch of a hamburger, h, French fries, f, and a soda, s, together

must be less than $7. Write an inequality to represent this relationship.

a) 7 b) 7 c) 7 d) 7h s h f s h s f h f s

3) A certain rectangle has a perimeter of at least 60. Given l represents the length of the rectangle

and w represents the width; select the inequality which represents this situation.

a) 2 2 60 b) 2 2 60 c) 60 d) 60l w l w l w l w

4) A company manufactures two types of shoes, one expensive and one cheap. The company decides

that to make a profit they must manufacture at least 150 pairs of shoes. The expensive shoes cost $85 a

pair and the cheap shoes cost $50. If e represents the number of expensive shoes produced

and c represents the number of cheap shoes produced, then which inequality represents this situation?

a) 150 b) 150 c) 85 50 150 d) 85 50 150e c e c e c e c

5) Six more than twice a number y is at least four times the number. Which of the following inequalities

best represents this information?

a) 6 2 4 b) 6 2 4 c) 2 6 4 d) 2( 6) 4y y y y y y

6) If a, b, c and d are real numbers, , , and, ,c d e b b a e c then which of the following has the

greatest value.

a) a b) b c) c d) d


HW 4-10 continued

7) Six more than 4 times a whole number is less than 60. Find the maximum value of the number.

8) Jane weighs 3 times as much as Barbara. The sum of their weights is less than 160 pounds. Find the

greatest possible weight for each girl if their weights are whole numbers.

9) Three times a number increases by 8 is at most 40 more than the number. Find the greatest value of

the number.

10) If a + b is less than c + d, and d + e is less than a+ b, then e is

a) less than d b) less than c c) greater than d d) equal to c

11) Determine the set of integers that satisfies the statement “one less than double a number is greater

than five and also less than 11”.

12) Determine an integer that is not being described below:

Three more than half a number is less than four or double the sum of the number and three is greater than

or equal to 12.


Unit 4 – Solving


1) Olivia solved the linear equation as follows:

She made an error between lines

a) 1 and 2 b) 2 and 3

c) 3 and 4 d) 4 and 5

1

2) Solve algebraically for : 1.5 0.252

x x

3) Find three consecutive integers such that the sum of the first and third is five less than triple the third.

24) Solve 2 7 4 for all values of .x x x

5) Patrick needs a 70 to pass his math test he took yesterday. He knows he did the 20-point written part

completely correct. The remainder of the test was 5-point multiple choice questions. Which inequality

represents the number of multiple choice questions, M, Patrick needs to get correct to pass the exam?

a) 20 70 b) 5 20 70 c) 5 20 70 d) 5 20 70M M M M

2

2 2 2 2


consecutive even integer?

a) 3 7 b) 3 6 c) 4 5 d) 3 3

x

x x x x

4( 3) 2 15x [Line1] 4( 3) 2 15

[Line2] 4(x 3) 17

[Line3] 4 3 17

[Line4] 4 14

14 1[Line5] 3

4 2

x

x

x

x or


HW 4-11 continued

7) Which inequality is represented in the graph below?

a) 4 2 b) 4 2 c) 4 2 d) 4 2x x x x

8) Solve each compound inequality for 𝑥 and graph the solution on a number line.

a) 6 4 14 6 4 14x or x b) 2 5 9x

9) A formula used for calculating velocity is 21.

2v at What is a expressed in terms of v and t?

2 2

2 2a) b) c) d)

2

v v v va a a a

t tt t

10) Determine three consecutive odd integers such that the product of the first and second is one less than

four times the third.


Unit 4 – Solving


221) If 2 and 3 what is the value of .x y xy x y


1I. 5 2 II. 5 2 III. IV.


a) I + II b) I II c) III + IV d) III IV

not

3) Determine the perimeter of a rectangle if the length is 80 and width is 3 5.

3

2

3 3 3 3


4 8 1 1a) b) c) d)

4 8

x

x x x x

2

2 2 2 2 2 2 2 2

5) The expression 2 is equivalent to:

a) b) c) 2 d) 4

a b ab

a b a b a b ab a b ab


HW 4-12 continued

6) The expression 5 2 subtracted from 7 3 is equivalent to:

a) 2 5 b) 8 5 c) 9 8 d) 5 8

a x x a

a x a x x a x a

7) The formula for the axis of symmetry of a parabola is , the formula solved for is2

2 2a) b) c) d)

2 2

bx a

a

b b b b

x x x x

8) The length and width of a rectangle are two consecutive even integers and the perimeter of the

rectangle is 20. Determine and state the length and width of the rectangle.

9) Caleb has nickels, dimes, and quarters in his piggybank. Which inequality represents the amount of

nickels, dimes, and quarters he has in his piggybank?

a) 0.05 0.10 0.25 0.40 b) 0.05 0.10 0.25 0.40

c) 0.05 0.10 0.25 0.40 d) 0.05 0.10 0.25 0.40

n d q n d q

n d q n d q

10) Determine and state the solution set of 3 5 2( 3) 7.x


Unit 5 – Linear Equations

Homework 1: Functions (A-REI.10, F-IF.1)

Definitions:

Relation: A set of ordered pairs, a set of pairs of input and output values.

Domain: The set of input values, the set of x values when referring to equations and coordinates.

Range: The set of output values, the set of y values when referring to equations and coordinates.

Function: A relation where no elements in the domain repeat, in other words no input has more than

one output. It is a correspondence between two sets( called the domain and range) such that to each

element of the domain, there is assigned exactly one element of the range.

1) Given the following relations, identity the domain, identify the range, and determine if the relation is a

function.

a) 1,2 , 3,4 , 5,6 , 7,8 b) 1,5 , 2,2 , 5,1 , 3,3 c) 0,4 , 3,4 , 5,6 , 8,8

d) 1,2 , 1,3 , 1,4 , 1,5 e) 1,2 2,2 3,2 4,2 f) 1,2 , 2,4 , 5,2 , 4,4

g) h) i)

j) The set of coordinates for 3

for values of 0 3.

y x

x x

2k) The set of coordinates for +1

for values of 2 1.

y x

x x


HW 5-1 continued

Notes: Anything graphed on the coordinate plane is a relation because all graphs consist of coordinate

(x,y) coordinate pairs. To determine if a graph is a function we use the Vertical Line Test.

Given the graph of a relation, if you can draw a vertical line that will intersect the graph more than

once, then the graphed relation is not a function. The reason this works is because if a vertical line

intersects a graph at two or more points than those points must have the same x value, and since the x

value represents the domain, that would mean an element in the domain repeats and this is not a

function.

Example 1: Given the relation

graphed on the axes shown on the

left, a vertical line can be drawn that

will intersect the one time revealing

coordinates with the same x value

as shown on the right.

Therefore this relation is NOT a function.

The domain can be determined by finding the points

on the graph that are furthest left 1,2 and right 7,6 as

shown. The Domain of this relation is 1 7x

The range can be determined by finding the points furthest up

7,6 and down 5, 1 as shown. The range of this

relation is 1 6y

Example 2: Given the relation graphed below, a vertical line cannot be drawn that will intersect the graph

more than once; therefore the relation IS a function.

The functions left most point is 2,3 and continues going to the right

to infinity. Therefore, the domain is 2x .

The functions lowest point is 4,1 and continues going up to infinity.

Therefore, the range is 1y .


HW 5-1 continued

2) Given the following relation graphs, identity the domain, identify the range, and determine if the

relation is a function.

a) b)

c) d)

e) f)


HW 5-1 continued

3) Given the following relation graphs, identity the domain, identify the range, and determine if the

relation is a function.

a) b)

c) d)

e) f)



Homework 2: Evaluating Functions (A-CED.1, A-REI.3,10, F-IF.1,2)

Notes: It is helpful to think of functions as rules or machines where you put something in (input, domain,

x) and you get something out (output, range, y). Instead of writing the equation as " "y , we write it in

function notation. Function notation replaces the “y=” with “f(x)=” pronounced “f of x” to express

equations. However they mean exactly the same thing.

Example: 2 3 can be expressed as 2 3 because .y x f x x f x y

One reason function notation is better than y notation is that it better distinguishes separate functions.

Instead of three functions all y we can use , , .f x g x and h x

For the f x notation, the x is a placeholder for input so it can be thought of as f input and for our

example 2 3 can be thought of as 2 3.y x f input input Meaning the rule for the

function f here is that any input gets multiplied by two and then subtracted by 3.

Example: For the function 2 3. Evaluate 4 :

4 2 4 3 8 3 5. The input value of 4 gives an output value of 5

This means the point 4,5 is on the graph of also known as 2 3.

f x x f

f

f x y x

1) For the function 2 3, algebraically evaluate each of the following:

a) 3 b) 5 c) 4 d) 3

f x x

f f f f

22) For the function 1, algebraically evaluate each of the following:

a) 3 b) 5 c) 4 d) 4

g x x

g g g g

23) For the function 3 , algebraically evaluate each of the following:

a) 1 b) 2 c) 1 d) 4

h x x x

h h h h


HW 5-2 continued

Notes: Another nice feature of the f x function notation is that expressions can be input into the

. Example: For the function 2 3. Evaluate 5 :

5 2 5 3 2 10 3 2 7

function f x x f x

f x x x x


a) 2 b) 3 c) 2 d) 2 3

f x x

f f f x f x

25) For the function 2, algebraically evaluate each of the following:

a) 4 b) 3 c) 3 2 d) 5

g x x

g g x g x g x


a) 2 b) 1 c) 3 d) 4

h x x x

h h g x g x


HW 5-2 continued

7) The graph of f x is shown below:

Evaluate each of the following:

a) 4 b) 5 c) 3

d) 0 e) 3

f f f

f f f

f) Determine the domain and range of f x g) Determine the value of where 1.x f x

8) The graph of g x is shown below:

Evaluate each of the following:

a) 0 b) 2 c) 3

d) 1 e) 6

g g g

g g g

f) Determine the domain and range of g x

g) Determine the value of where 5.x g x



Homework 3: What is a Line? (A-SSE.1, A-REI.10)

Notes: A linear equation (straight line) is an equation of degree one, meaning the largest exponent is

one. Two write a linear equation two things must be known - the slope or steepness of the line and

location. Location can be where the line intercepts the y-axis, known as the y-intercept or any known

point on the line. Linear equations have two common forms.

Slope-Intercept Form – Using slope and y-intercept, the linear equation is solved for y: y mx b

" " represents slope and " " represents th e interceptm b y

Point – Slope Form – Using slope and any point on the line: 1 1y y m x x

1 1" " represents the slope and , represents a point on the linem x y

Vertical and Horizontal lines only have one variable because they only cross one axis. Horizontal lines

only cross the y axis so have the form , where is the number where is crosses the y axis.y c c Vertical

lines only cross the x-axis so have the form , where is the number where it crosses the x axis.x c c

Slope can be thought of as the directions to get from one point on the line to another. Slope is expressed

as a ratio or fraction comparing the change in y to the change in x. Often referred to as rise over run.

change in y risem

change in x run

Example: To get from point A to point B you can travel up 3 and

over to the right 2, therefore the change in the y dimension is 3

and change in the x dimension is 2 so the slope, m, is 3

.2

Positive Slopes Negative Slopes Zero Slopes Undefined Slopes

(Inclined left to right) (Declined left to right) (Horizontal/Flat) (Vertical)

3 3

2 2

uppm

over

2 2

2 3

downm

over

0 00

4 4

upm

3 3

0 0

upm Undefined

over


HW 5-3 continued

1) Determine the slope for each line segment shown graphed below:

AB

CD

EF

GH

m

m

m

m

IJ

KL

MN

OP

m

m

m

m

2) Determine the slope for each linear equation given below:

a) 3 c) 5 3 2

1 2b) 4 d) 4 1

2 3

y x y x

y x x y


HW 5-3 continued

3) Rewrite the given linear equations in slope-intercept form and determine the slope:

1a) 3 b) 5 c) 2 6

2y x y x y x

d) 4y x e) 2 7 f) 3 5x y x y

2 1g) 2 h) 4

3 3y x x y i) 2 6 8y x

j) 3 6 3 k) 2 9 3 l) 2 4y x x y x y

4) What is the slope of the linear equation 4?3

a) 3 c) 3

1 1b) d)

3 3

yx

m m

m m


HW 5-3 continued

5) What is the slope of the linear equation 3?2

2 2 3 3a) b) c) d)

3 3 2 2

yx y

m m m m

16) Which of the following linear equations does have a slope of ?

2

1a) b) 1 3 c) 2 6 d) 2 8

2 2

not

xy y x y x x y

7) Given a positive slope, explain which is steeper. When the change in is greater than the change in

or when the change in is greater than the change in ?

x y

y x

8) Given the following three linear equations:

1: : :

Which two linear equations have the same slope?

a) & b) & c) & d) None of them have the same slope

xI y b II ay b x III y b x b

a a

I II I III II III



Homework 4: The Slope Formula (F-IF.4,6)

REVIEW 1) Determine and state the slope of the following lines:

a) 3 b) 2 4 3 1f x x x y x c)

Notes: The slope of a line can be determined by looking at a linear equation in standard or point-slope

form. It can also be determined from a graph by counting the change in y and the change in x from two

points on the line. Slope can also be determined from any two points on a line using the slope formula

which calculates the change in y and change in x algebraically.

The Slope Formula

2 12 1 2 1

2 1

, calculates the change in and calculates the change in y y

m y y y x x xx x

1 1

Example: Algebraically determine and state the slope of a line containing the points 2,3 and 1,5 .

, represents the first point which is the point to the left, this can be determined by which pointx y

1 1 2 2

2 1

2 1

has the lower value. So 2,3 is , and 1,5 is the second point , .

5 3 2 2, the slope of the line that contains these points is

1 2 3 3

x x y x y

y ym m

x x

2) Determine in simplest form the slope between the two given points:

a) 2,1 and 6,3 b) 0,2 and 3, 4 c) 2,3 and 5,3

d) 3,5 and 0,4 e) 1,2 and 1, 4 f) 2, 2 and 1,0

g) 6,0 and 0, 6 h) 4, 4 and 3,1 i) 2, 2 and 5,5


HW 5-4 continued

1

3) If the point 2,5 is on a line with a slope of , determine the next point to the right on the line.2

Use the slope formula to check your point and the given point have the given slope.

2

4) If the point 3,4 is on a line with a slope of , determine the next point to the right on the line.3


5) If the point 4,0 is on a line with a slope of 3, determine the previous point to the left on the line.


1

6) If the points 4,2 and 2, have slope of , algebraically determine and state the value of .2

y y

7) If the points 1, 6 and ,3 have slope of 3, algebraically determine and state the value of .x x

8) If the slope between the points , and , is positive and ,

explain which has the greater value or .

a b c d d b

a c



Homework 5: Intercepts (A-REI.10, F-IF.4)

REVIEW 1) Determine and state the slope between the points 4,2 and 5, 1 .

Notes: Intercepts are where a line or curve intercepts something such as an axis. The x-intercept is

where a line or curve intercepts the x-axis. Every point on the x-axis has a y-value of 0 as shown.

We can substitute zero in for the y-value and then solve for x. Example: Determine and state the x-

intercept for 2 6y x

2 6, substitude 0 becomes, 0 2 6, solving for with adding 6 and dividing 2

becomes 3 , 3 or 3,0 is the intercept.

y x y x x

x x

2) Determine and state the x-intercept for each of the following lines:

a) 3 b) 8 2 c) 2 6 y x y x y x

1d) 5

2y x e) 3 2 9 f) 7x y x

2g) 2 h) 4 ( 3) i) 3 5

3 2

xy x y x y x


HW 5-5 continued

Notes: The y-intercept is where a line or curve intercepts the y-axis. Every

point on the y-axis has a x-value of 0 as shown.

We can substitute zero in for the y-value and then solve for x.

Example: Determine and state the y-intercept for 2 6y x

2 6, substitude 0 becomes, 2 0 6, solving for since 2

times 0 equals 0, becomes 6 , 6 or 0, 6 is the intercept.

y x x y y

y y

When the equation is in slope intercept form y mx b , the y-intercept is the

constant number after x, but substitution works regardless of the equations form.

3) Determine and state the y-intercept for each of the following lines:

a) 3 b) 8 2 c) 2 6 y x y x y x

1

d) 5 32

y x e) 3 2 8 f) y 2x y

4) Describe the slope of a line if the x and y intercepts are both positive.

5) Ben says “A line with a negative slope can intercept one or two axes.” Explain why Ben is correct.



Homework 6: The Point-Slope Formula (A-CED.1, A-REI.3)

REVIEW 1) Determine x and y intercepts of the linear function 1

3.2

f x x

Notes: Point – Slope Form – Using slope and any point on the line: 1 1y y m x x

1 1" " represents slope and , represents a point on the linem x y .

1 1 1 1

1Example: Determine the equation of a line through the point 6,5 with a slope of .

2

16, 5,

2

1 15 6 5 6 .

2 2

1Writen in standard formfirst distribute the slope to get 5

2

y y m x x x y m

y x y x

y x

3,

1then solve for to get 8

2y y x

2) Write the equation of a line in both point slope and standard form with the given point and slope:

a) 5,2 , 3m b) 1, 4 , 2m c) 3, 7 , 1m

1

d) 4,2 , 4

m 1

e) 6, 1 , 3

m

2

f) 0, 2 , 3

m

3

g) 5,0 , 5

m

h) 0,0 , 1m i) 4, 5 , 0m


HW 5-6 continued

Example: Determine the equation of a line through the points 2, 3 and 1,3 .

3 3 6First find the slope of the line: 2

1 2 3

Then use either point with the point slope formula: 3 2 1 2 1

m

y x y x

3) Write the equation of a line in both point slope and standard form with the two given points:

a) 4, 1 , 1,2 b) 4,2 , 2,5

c) 1,5 , 2, 4 d) 0, 3 , 3, 1

e) 5, 3 , 1, 3 f) 4,2 , 4,7

4) Line L contains the points 2,2 , 6,0 , and 2, .k Determine and state the value of k.

5) Determine the equation of a line through the points ,2 and 1,2 1 .a a a a



Homework 7: Graphing Lines from Slope and Intercepts (A-REI.10, F-IF.4,7)

REVIEW 1) Write the equation of a line in both point slope and standard form with a given point of 6,0

2and a slope of .

3m

2) Given the following slopes, graph each of the following lines with a intercept of 2.

a) 0

1b)

3

c) 1

d) 4

e) undefined

y

m

m

m

m

m

3) Given the following slopes, graph each of the following lines with a intercept of 1.

5a)

2

b) 1

2c)

3

1d)

2

e) 0

y

m

m

m

m

m


HW 5-7 continued

4) Graph the following lines given their slope and y-intercept on the graph provided:

a) 1, 1m b

1b) , 2

4m b

c) 0, 4m b

5) Graph the following lines given their slope and y-intercept on the graph provided

3a) , 2

4m b

b) 2, 6m b

c) 0, 0m b


HW 5-7 continued

6) For each linear equation, state the slope, y-intercept, and then graph the lines on the graph provided.

1a) 3

3y x

b) 2 6x y

c) 6 2 2y x

d) 3 2 2y x x

7) For the given pairs of points:

i) Graph the line containing the points

ii) Determine the slope of the line

iii) Determine the intercept of the line

iv) Determine the equation of the line

a) 4, 1 and

y

4,3

b) 4,4 and 2, 5


Student Notes



Homework 8: Modeling Linear Functions (A-SSE.1, A-CED.1, A-REI.3, F-IF.2, F-LE.5)

REVIEW 1) Write the equation of line the line passes through the points 3, 3 and 6,3 .

2) The school astronomy club spends money to produce t-shirts and then sells the t-shirts to students in

the school. If the clubs profit, P s , for selling s shirts is represented by the equation

20 150,P s s then what do the values of 20 and 150 represent?

3) A balloon is released from atop a building. The balloon then increases in altitude at a constant rate. If

the height of the balloon, h, after m minutes is represented by the equation 55 72 ,h m then what do

the values of 55 and 72 represent?

4) Logan plants a flower in a garden that grows 0.8 cm a day. The height of the plant is modeled by the

equation 0.8 5.y x In this equation:

a) what does x represent the number of? b) what does 5 represent?

5) Patrick is choosing between two cell phone plans. Plan A has a set up fee of $30 and charges $5 for

every gigabytes of data used a month. Plan B is free to set up and charges $8 for every gigabytes of data

used a month.

a) If the plans were shown graphically, which plan would have a steeper slope?

b) If Patrick plans to use at most 5 gigabytes of data a month, which plan is cheaper?


HW 5-8 continued

6) A used car is bought for $8,500, and it is estimated to cost $280 a month for gas and insurance. Write

an equation that will represent the total cost of the car after, t, after m months.

7) Jonny currently weighs 180 pounds and plans to lose 2 pounds a week through diet and exercise.

a) Write an equation that will predict Jonny’s weight, y, after x weeks.

Using that equation, predict:

b) Jonny’s weight after 11 weeks. c) how many weeks until Jonny weighs less than 150 pounds

8) Jill keeps a piggy bank in her room where she is saving her money. For the past few weeks, she has

added the same amount into her piggy bank each week and recorded the amount inside. The table below

shows her data.

Jill plans to continue to add the same amount each week into her piggy bank. Write an equation to

represent the amount of money Jill will have in her piggy bank after w weeks.

9) Olive is on vacation in New York City. One day she decides to rent a bike.

Power Cyclers charges $20 plus $3.50 per mile.

Manhattan Cyclers charges $14 plus $5 per mile.

a) Write a cost equation for each bike rental in terms

of the number of miles.

b) Graph both cost equations.

c) For what trip distances is Power Cyclers cheaper?

Justify your answer.

Weeks (w) 1 2 3 4

Amount (a) $26.00 $27.50 $29.00 $30.50



Homework 9: Average Rate of Change (F-IF.6)

REVIEW 1) Ted currently weighs 220 pounds and plans to lose 3 pounds a week through diet and

exercise. Write an equation that will predict Ted’s weight, y, after x weeks.

Use the graph of f x shown below that represents a joggers speed during a 14 minute run to answer

questions #2-5.

2) Which best describes what the jogger was doing during the 0-2 minute interval of the run?

a) Running at a constant rate b) Accelerating

c) Decelerating d) Taking a rest







5) Which of the following is false about the joggers run?

a) The domain is 0 14x

b) The range is 0 7y

c) The jogger ran the fastest during the 5-8 minute interval

d) The jogger accelerates the quickest during the 0-2 minute interval


HW 5-9 continued

Notes: Average Rate of Change describes the average rate at which one quantity is changing with

respect to something else changing. For example, miles per hour which is calculated by dividing the

number of miles driven by the number of hours driven.

Since average rate of change is a calculation of the change in one thing divided by the change in

another, this is essentially slope which is the change in y divided by the change in x.

2 11 1 2 2

2 1

The slope between the points , and , is

y ychange in yx y x y m

change in x x x

In function notation remember that 1 1 1 1 so the point , can be written as , .f x y x y x f x

1 2

2 11 1 2 2

2 1

The Average Rate of Change over the interval , which is

from , to , is (The slope formula in function notation)

x x

f x f xx f x x f x A x

x x

Example: For the jogger on the previous page, the average rate of change over the interval [8,11] is the

slope between the points 8, 8 and 11, 11 which is 8,7 and 11,4 .f f

11 8 4 7 3

Which is = = 1.11 8 11 8 3

f fA x

#6-9 Use the graph of g x shown on the right that represents the distance, y, Logan hiked in x hours.

6) What was Logan’s average rate of change over the

hourly interval [4,8]?

7) What was Logan’s average rate of change over the

hourly interval [8,12]?

8) Which time interval had the greatest rate of change?

a) [0,2] b) [2,4] c) [4,8] d) [10,12]

9) What was Logan doing between the 8th

and 10th

hour?


HW 5-9 continued

10. The graph below represents the temperature in Fahrenheit, f, of a pizza taken from a refrigerator and

placed in an oven after m minutes

During which time interval did the temperature of the pizza show the greatest average rate of change?

a) [2,3] b) [3,4] c) [4,6] d) [6,8]

11). Which function below has the greatest average rate of change over the interval [2,8]?

Justify your answer.

a) b) 3

12

g x x c)


Student Notes




1) What is the slope of the linear equation 3 4 7 ?

a) 4 b) 3 c) 1 d) 1

y x x

m m m m

2

2) If the points 2,7 and 1, have a slope of , algebraically determine and state the value of .3

y y

3) Determine x and y intercepts of the linear equation 2 6.x y

4) Write the equation of line the line passes through the points 2,4 and 6,0 .

5) Mr. Grover starts with 30 students in his algebra class, if he sends 2 students to detention every x

minutes. Write an equation that represents the number of students remaining in his class, y, after x

minutes.

26) For the function 3 , algebraically evaluate each of the following:

a) 5 b) 4 c) 6 d)

g x x x

g g g x g x


HW 5-10 continued

7) Use the given information to graph each line and state the equation of the lines in standard form.

1

a) A line with the equation: 3 22

y x b) A line with points 1, 2 and 2,7

#8-10 Use the graph of f x is shown right:

8) Determine and state if f x is a function, its domain, and its range.

9) Evaluate each of the following:

a) 4 b) 5 c) 2 d) Two values of such 1f f f f x f x

10) Determine the average rate of change for the function over the interval 5, 1 .




1) Solve algebraically for x: 3 1 5 12 6 7x x x

2) Find the value of 2 29 16x y when

1 1and

3 4x y

120 2

13) Find the value of , when 4.x x x

x

24) If 3, express as a polynomial in simplest form:A x A A

2

5) Express as a polynomial in simplest radical form: 4 2 x


HW 5-11 continued

6) The basketball team scored 38 points at half time, if they score only two point baskets during the

second half. Which inequality could be used to solve for the number of two point baskets,b, they

need to score to end the game with at least 75 points?

a) 2 38 75 b) 2 38 75 c) 2 38 75 d) 2 38 75b b b b

2

2 2 2 2

7) If the expression 3 represents an odd integer. Which expression would represent the next


a) 5 b) 2 c) 1 d) 4

x x

x x x x x x x x

1

8) If the point 6, 2 is on a line with a slope of , determine the next point to the left on the line.3


9) Write the equation of line the line passes through the points 2,5 and 6,1 .

310) Solve 4 0 for all values of .x x x

.


Unit 6 – Solving Systems

Homework 1: Linear Systems Graphically (A-REI.6,10,11)

REVIEW 1) Tom brings $15 in quarters to the arcade to play video games. The games each cost 25

cents to play. Write an equation to represent the amount of money, y, Tom will have remaining after he

plays x arcade games.

2) Solve the following system of equations graphically. State the coordinates of the solution.

a) 4

2

y x

y x

b) 3 1

33

2

y x

y x

c) 3

2

y

x

d) 5

2 2

y x

y x


HW 6-1 continued


a) 4

2

y x

y x

b) 6

10

y x

x y

c) 2 1

4

y x

x

d) 3 6y x

y x


HW 6-1 continued


5

a) 1 2 , 3 2 42

y x y x

1

b) 3 2 1 , 3 33

y x y x

5) Determine the point of intersection for a line that passes through the points 2,2 and 4, 1 ,

and line 2 3 6 0.y x


Student Notes



Homework 2: Linear Systems Algebraically: Substitution (A-SSE.1, A-CED.2, A-REI.5,6)

REVIEW 1) Write the equation of a line that contains the points of 3, 1 and 6, 4 .

Notes: A linear system can be solved algebraically by using substitution. The goal is to have one of the

systems solved for a variable, then substitute its equivalence into the other equation. If done correctly

the resulting equation will have only one variable which can be solved, then used to find the second

variable.

Example: Solve the linear system algebraically using substitution:

2 3 2

2 10

The first step is solving for a variable in either equation. This can be done by solving for either or

in either e

x y

x y

x y

quation. In this example the easiest variable to solve for is in the bottom equation

becuase it has a coefficient of 1. When solved for the new bottom equation becomes 10 2 .

Since equals 10

y

y y x

y

2 , in the other equation can be substituted for with 10 2 as shown:

2 3 2 becomes 2 3 10 2 2, this equation now only has one variable, , which can be solved.

2 30 6 2 4 30 2 4 28 7, this

x y x

x y x x x

x x x x x

solution can then be substituted into

an equation to solve for the other variable, .

If 7 then 2 3 2 can become 2 7 3 2 14 3 2 3 12 4

The solution to the system is 7, 4 .

This can be easily

y

x x y y y y y

check by verifying this point works in both original equations:

2 3 2 2 7 3 4 2 14 12 2 2 2

2 10 2 7 4 10 14 4 10 10 10

x y

x y


HW 6-2 continued

2) Solve the following system of equations algebraically using the substitution method.

a)

6

y x

x y

b) 6

4 3 27

y x

y x

c) 2 3 1

2 2

x y

x y

d) 5 2

8 3

x y

x y

2e)

3

5 34

y x

y x

f) 1

19 4

y x

y x


HW 6-2 continued


a) 4 2

4

x y

y x

b) 2 5 11

4 1

x y

x y

c) 2 11

2 13

x y

x y

d) 9 3 6

3 2 5

x y

x y

e) 3 2 12

2 5 11

x y

x y

f) 2 5 1

3 2 11

x y

x y


HW 6-2 continued


a) 3 5 4

6

x y

y x x y

b) 3 4 6

5 2 3 2

x y

x y x y

5) Lori solved a linear system using substitution, her work is shown below.

2 12 12 2

3 4 7 3 4 12 2 7 3 48 8 7 5 48 7 5 55

11 2 11 12 22 12 34, Solution 11,34

x y y x

x y x x x x x x

x y y y

Lori made a mistake and got an incorrect solution, explain what Lori did wrong.



Homework 3: Linear Systems Algebraically: Elimination (A-SSE.1, A-CED.2, A-REI.6)

REVIEW 1) Solve the following linear system algebraically using substitution:

6

10

y x

x y

Notes: The substitution method is not the only way to solve a linear system algebraically. Another

method is to use elimination. Elimination is where you add the two equations together and the result is

an equation with only one variable which can then be solved. Elimination requires the two equations be

arranged in the same way and that one of the columns contains an additive inverse pair. To get the

additive inverse pair you will often need to change one or both of the equations through multiplication

Example: Solve the linear system algebraically using elimination:

3 1

2 4 4

The first step is aligning up the terms, typically the first columns are the terms, the second the terms,

then equal

y x

y x

x y

to the third column of the constants. Correctly rearranged and aligned the system becomes:

3 1

4 2 4

Neither variable column contains an additive inverse pair so we much change one or both of the

x y

x y

equations

using multiplication. The bottom term is 2 , we can make the top term, , into 2 by multiplying

the top equation by 2. This is not the only option, but the easiest for this example.

y y y y y

The top becomes:

2 3 1 6 2 2

Dont forget when you multiply an equation to distribute the multiplication to every term on both sides.

Now aligned the column contains an additive inverse and we c

x y x y

y

an now add the two equaitons together:

6 2 2

4 2 4

The sum is 2 2, which can be solved to become 1, this can then be used to solve for

If 1 then 3 1 becomes 3 1 1 3 1 4 4

The s

x y

x y

x x y

x x y y y y y

olution to the system is 1,4 .

This can be easily check by verifying this point works in both original equations:

3 1 3 1 4 1 3 4 1 1 1

4 2 4 4 1 2 4 4 4 8 4 4 4

x y

x y


HW 6-3 continued

2) Solve the following system of equations algebraically using elimination.

a) 12

4

x y

x y

b) 3 10

4 11

y x

x y

c) 2 6

3 10 2

x y

y x

d) 3 12

2 7

x y

x y

e) 5 4 7

5 6 3

x y

x y

f) 2 3 13

4 2 2

x y

x y


HW 6-3 continued


a) 5 3 2

4 6 20

x y

x y

b) 2 5 1

3 2 11

x y

x y

c) 5 3 39

3

x y

x y

d) 12 21 9

10 6 10

x y

x y

e) 12 5 3

8 2 8

x y

x y

f ) 2 10 3

4 5 18

y x

x y


HW 6-3 continued


g) 3 6 12

5 7 3

x y

x y

h) 9 6 3

3 5 2

x y

x y

5) If , and , , , , , and are all greater than zero.a b c d e f a b c d e f Which of the following would

create an additive inverse that would allow for the elimination method to occur for the following system?

a) b) c) d)

hx ky c

mx ny d

h hx ky c h hx ky c m hx ky c m hx ky c

m mx ny d m mx ny d h mx ny d h mx ny d



Homework 4: Modeling Linear Systems (A-CED.2,3, F-LE.5)

REVIEW 1) Solve the following linear system algebraically using elimination:

2 12

3 7 4

y x

x y

2) The sum of two numbers is 32 and their difference is 4. What are the numbers?

a) Create a system of two linear equations to represent this problem.

b) What is the solution to the system?

3) The difference between two numbers is 24 and their sum is 48. Find the two numbers

a) Create a system of two linear equations to represent this problem.

b) What is the solution to the system?

4) Jack and Jamie went to the store to buy some school supplies. Jack bought 3 pencils and 1 pen for

$6.50. Jamie bought 1 pencil and 2 pens for $5.50. How much do pencils and pens cost at the store?


HW 6-4 continued

5) At the candy shop you can buy three pieces of candy and four pieces of gum for $2.50. You can buy

two pieces of candy and two pieces of gum for $1.50. How much is gum at the candy shop?

6) George and Tim together weigh 210 pounds. The difference between three times George’s weight

and twice Tim’s weight is 30 pounds. Find the weight of each.

7) Joe’s farm stand sold 37 apples and oranges. Apples sell for $1.25 each and oranges sell for $1.50

each. If Joe made $51.50 that day, how many apples did Joe sell?

8) There are 242 admissions tickets sold. Three times the number of 50 cent tickets is 12 more than four

times the number of 75 cent tickets. Find the number of 50 cent tickets sold.



Homework 5: Graphing Inequalities (A-REI.12)

REVIEW 1) Find the solution set of 5 3 1x x and express the solution in set notation and

graphically on a number line.

Notes: An inequality with one variable is graphed on a one-dimensional number line like the previous

problem. An inequality with two variables such as x and y, is graphed on a two-dimensional set of axes.

Graphing two variable inequalities is similar to graphing lines in regards to, solving for y, slope,

and y-intercept. The difference is that for an inequality the line could be solid or dotted and one side of

the line is shaded to represent the solution set of the inequality.

The inequality type determines if the line is solid or dotted.

Inequalities with < and > are dotted because the points on the line ARE NOT in the solution set.

Inequalities with and are solid because the points on the line ARE in the solution set.

To determine which side of the inequality should be shaded, you could use the y-axis as a reference for

example if the inequality symbol is < or then the solution is the side of the graph that contains the y-

axis that is less than the inequalities y-intercept. The opposite applies for > and .

Another option is the use of a test points.

1Example: Graph the inequality 2

2

1Notice the slope and the -intercept 2

2

Since the symbol is and NOT equal to,

the graph should start with a dotted line.

y x

m y b

greater than

Then shade the side of the graph that contains the y-axis that is greater than the graph’s y-intercept of 2.

The solution set for 1

22

y x is all the points in the

shaded region such as 6,0 , 3, 2 , and 0,5 ,

but no points on the dotted line.


HW 6-5 continued

2) Graph each inequality on the given set of axes:

a) 1

32

y x b) 3 2y x

c) 2 1y x d) 2y x

e) 1

23

y x

f) 1y x


HW 6-5 continued

3) Graph each inequality on the given set of axes:

a) 2 2y x b) 0x y

c) 3y d) 1x

e) 3 6x y f) 1 2x y


HW 6-5 continued

4) George says that the point 2,2 is in the solution set of the inequality 2 6,

Kathy says the point 3,3 is in the solution set.

a) Graph the inquality and the points to determine if George, Kathy

y x

, neither, or both

of them are correct.

b) Prove algebraically that the points 2,2 and 3,3 are or are not in the solution set.

c) If the inequality sign were to change from greater than to less than,

would either point still

remain in the solution set? Explain why.


Unit 6 – Inequality Systems

Homework 6: Inequality Systems Graphically (A-REI.12)

REVIEW 1) Determine algebraically if the point 1, 3 is in the solution set of 2 3.x y x

Notes: The solution set of an inequality system is their over lapping shaded region.

2) The graph below shows the inequality system for 1

2 and 3 2.2

y x y x Determine if the

following points are in the solution set for the system:

c) 0, 3 d) 4,0

e) 2,4 f) 10,3

g) If 0 , the point ,a b b a

3) Shown to the right is the system

of equations 1

3 and 2 1.2

y x y x

If the equal signs were both replaced by less than signs

which point would be in the solution set?

a) A b) B c) C d) D

a) 5,2 b) 5, 2


HW 6-6 continued

4) Graph the inequality systems on the set of axes and state the coordinate of a point in the solution set:

a)

11

3

2 3

y x

y x

b)

3 2

11

2

y x

y x

c) 1

2

y x

y x

d)

2 1

4

y x

x


HW 6-6 continued

5) Graph the inequality systems on the set of axes and state the coordinate of a point in the solution set:

a) 1

1

y

y x

b)

2

3

y x

y x

c)

22

3

4

y x

x y

d) 2 6

2

x y

y x


HW 6-6 continued

6) Graph the inequality systems on the set of axes and state the coordinate of a point in the solution set

e) 4

3

y x

y

x

f)

1

2 10

0

y x

y x

x

7) A quadrilateral region is defined by the following system of inequalities:

1

5, 2 4, 3 4, 62

y y x y x y x

a) Sketch the region.

b) Determine the vertices of the quadrilateral.

c) State the cooridnate of a point in the region.


Unit 6 – Inequality Systems

Homework 7: Inequality System Modeling (A-SSE.1, A-CED.2,3, A-REI.12)

REVIEW 1) Graph the system of inequalities.

13 and 3 4

2y x y x

a) Is 0,0 a solution? Explain.

b) Is 2,2 a solution? Explain.

c) Determine algebraically if 0, 4 is a solution

to the system.

2) The sum of the two ages of boys represented by and is at most 25. Which inequality represents the

boys age sum constraint?

a) 25 b) 25 c) 25 d) 25

x y

x y x y x y x y

3) A farm stand has 150 pounds of apples and 75 pounds of bananas to sell each day. If represents

the pounds of apples and represents the pounds of bananas sold each day. Which inequality

repres

x

y

ents the farm stands daily constraint?

a) 225 b) 225 c) 225 d) 225 x y x y x y x y

4) A farm stand sells at least $200 worth of apples and bananas each day. Apples sell for $1 per

pound and bananas sell for $0.50 per pound. If represents the pounds of apples and represents

the

x y

pounds of bananas sold in a day. Which inequality represents the farm stands daily constraint?

a) 0.5 200 b) 0.5 200 c) 0.5 200 d) 0.5 200 x y x y x y x y


HW 6-7 continued

5) Mr. Grover’s Algebra test has 10 multiple choice questions and 5 free response questions for a total of

15 questions. The multiple-choice questions are each worth 5 points and the free response are each

worth 10 points for 100 total possible points.

a) Write a system of linear inequality that can be used to find the possible combinations of correct

multiple choice questions, x, and correct free response questions, y, that a student could get on Mr.

Grover’s Algebra test.

b) Graph the solution of this system of inequalities on the set of axes below. Label the solution region

with an S.

c) A 70 is required to pass Mr. Grover’s test, Bob claims he can pass if he gets 7 multiple choice

questions and 3 free response questions correct. Explain whether he is correct or incorrect, based on the

graph drawn.


HW 6-7 continued

6) A school chorus concert is held in the school auditorium which has 120 seats. Student tickets to the

concert cost $3.00 and adult tickets cost $5.00 each. The school’s goal is to sell $400 worth of tickets for

the concert.

a) Write a system of linear inequalities that can be used to find the possible combinations of student

tickets, x, and adult tickets, y, that would satisfy the school’s goal.

b) Graph the solution of this system of inequalities on the set of axes below.

Label the solution with an S.

c) Michelle claims that selling 40 student tickets and 50 adult tickets will result in meeting the school’s

goal. Explain whether she is correct or incorrect, based on the graph drawn.


HW 6-7 continued

7) James has at least 30 coins in his piggy bank. The coins are all nickels and dimes for a total of at most

$3.00.

a) Write a system of linear inequality that can be used to find the possible combinations of the number of

nickels, x, and the number of dimes, y, that James has in his piggy bank.

b) Graph the solution of this system of inequalities on the set of axes below. Label the solution region

with an S.

c) Based on the graph drawn, determine a combination of nickels and dimes that James could have in his

piggy bank.




1) Solve the following linear system algebraically:

3 15

2 2

x y

x y

Solve the following systems graphically. State a coordinate that is in the solution.

2) 2 4 , 5y x y x 3

3) 2 4 , 2 44

y x y x

4) Logan and Troy go to the snack shack to buy lunch. Logan buys two burgers and two hotdogs for

$17.50. Troy buys one burger and two hotdogs for $12.00. How much are burgers and hotdogs each?


HW 6-8 continued

5) The school play sells regular tickets for $5, and student tickets for $3. The play sold 100 tickets

combined and made $430 on ticket sales. Determine and state the number of student tickets sold.

6-8) The school store sells shirts and hats. The store sells at most 20 shirts and hats a day combined. If

shirts sell for $10 and hats for $5, then the stores goal is to make at least $80 a day.

6) Write a system of linear inequality that can be used to find the possible combinations of the number of

shirts sold, x, and the number of hats sold, y.

7) Graph the solution of this system of inequalities on the set of axes below. Label the solution region

with an S.

8) Based on the graph drawn, determine a possible combination of shirts and hats sold that would work

for the stores goal given they sell at most 20 combined.




1) If 2 3 8 and 2 3,x y x y then find the value of 3 .x y

2) Determine and state three consecutive odd integers such that the second subtracted from the third is

three more than the first.

3) A barbeque sold 26 hamburgers and hot dogs. Hamburgers sell for $3.50 each and hot dogs sell for

$2.00 each. If the barbeque made $70 on the sales, determine and state the number of hot dogs sold.

4 3 24) Which of the following is not a solution of 2 2 0

a) 1 b) 0 c) 1 d) 2

x x x x

5) A used car is bought for $4,500, and it is estimated to cost $220 a month for gas and insurance. Write

an equation that will represent the total cost of the car , t, after m months.


HW 6-9 continued

2

2

2


a) The expression 4 6 factors to 2 2 3


c) The expression 4 factors to 2 2

d) The expression 6 5 can be factored

false

x x

x x x x

x x x

x x

7) The solution set for 5 1 2 7 is

a) 3, 2 b) 2, 3 c) 2, 3 d) 3, 2

x

8) A certain rectangle has a perimeter of at least 55. Given l represents the length of the rectangle

and w represents the width, select the inequality which represents this situation.

a) 2 2 55 b) 2 2 55 c) 55 d) 55l w l w l w l w

9) Write the equation of a line in both point slope and standard form with a given point of 3,1

2and a slope of .

3m

2 110) For the function 2 , evaluate 2

a) 7.5 b) 1 c) 0 d) 1

f x x fx


Unit 7 – Quadratics

Homework 1: Quadratic Graphs (A-REI.10, F-IF.2,4,7, A-APR.3)

Notes: A Quadratic Function is a function of the form 2 , 0.f x ax bx c where a The highest

power of the function is 2. The graph of a quadratic is called a Parabola.

Example: Graph the quadratic 2 4 3 over the domain 5 1.f x x x x

First, we will create a table to find the set of

points on the function over the given domain.

The first column is the given domain.

The second column is the values inputted into

the function.

The third column is the resulting range values.

The fourth column is the input paired with the

output as coordinates on the function.

Notice the y values decreasing then change to

increasing after the point 2, 1 , the point is

called the Vertex, also known as Turning

Point.

The points 3,0 and 1,0 are the x-

intercept points known as Roots.

Once the set of points has been determined we can graph the points, and draw a smooth labeled curve

with arrows on the ends as shown below.

x 2 4 3f x x x f x ,x f x

5 2

5 5 4 5 3f 8 5,8

4 2

4 4 4 4 3f 3 4,3

3 2

3 3 4 3 3f 0 3,0

2 2

2 2 4 2 3f 1 2, 1

1 2

1 1 4 1 3f 0 1,0

0 2

0 0 4 0 3f 3 0,3

1 2

1 1 4 1 3f 8 1,8


HW 7-1 continued

1) For each quadratic function and given domain below determine the set of points on the function in the

given domain, graph the parabola, and state the coordinates of the vertex and roots.

a) 2 2 3 over the domain 4 2.f x x x x

b) 2 6 5 over the domain 0 6.f x x x x

c) 2 4 3 over the domain 1 5.f x x x x


HW 7-1 continued



a) 22 4 6 over the domain 2 4.f x x x x

b) 23 6 over the domain 2 4.f x x x x

c) 2 4 over the domain 3 3.f x x x


HW 7-1 continued



a) 4 2 over the domain 0 6.f x x x x

b) 2 1 2 over the domain 2 4.f x x x x

c) 2

1 1 over the domain 4 2.f x x x



Homework 2: The Axis of Symmetry (F-IF.2,4)

Notes: A parabola in the form 2y ax bx c is symmetrical about a line called the axis of symmetry,

which is a vertical line that splits the parabola down the middle in half. The axis of symmetry passes

through the vertex, and its equation is the x-value of the vertex.

The equation of the axis of symmetry can be found algebraically by using the formula: 2

bx

a

For the above quadratic 1 4,with a and b this would be

4 42

2 1 2x

Since the axis of symmetry is also the x value of the vertex coordinate, the vertex can be found by

inputting the axis of symmetry x value into the function to find the accompanying y value.

2

2 2 4 2 3 4 8 3 1f , the vertex of the parabola is 2, 1 .

The axis of symmetry can also be used to create an appropriate domain to graph the function since it is

in the middle. The above parabola is graphed over the domain of x values greater than 5 (3 less than

the axis of symmetry) and less than 1 (3 more than the axis of symmetry).

2, 2 3 5, 2 3 1, 5 1x Graph over the domain x


7-2 continued

1) For each quadratic function given: determine the equation of the axis of symmetry and the coordinates

of the vertex.

2) 6 3a f x x x 2) 8 1b f x x x 2) c f x x x

2) 2 4 5d f x x x 2) 4 4 2e f x x x 2) 3 6 3f f x x x

2) 4g f x x 2) 4 9h f x x 2) i f x x


7-2 continued

2) For each quadratic function: determine the equation of the axis of symmetry, the coordinates of the

vertex, a domain of seven values with the vertex in the middle, and graph the function over that domain.

a) 2 4 1f x x x

b) 2 6 6f x x x

c) 2 6 9f x x x


7-2 continued

3) For each quadratic function: determine the equation of the axis of symmetry, the coordinates of the

vertex, a domain of seven values with the vertex in the middle, and graph the function over that domain.

a) 2 3f x x

b) 212 2

2f x x x

c) 22 4 3f x x x



Homework 3: Transformations (F-IF.2, F-BF.3)

REVIEW 1) For the function 22 8 10,f x x x algebraically find the axis of symmetry and the

vertex.

Notes: Function transformations are changes to a graph by translating, reflecting, and dilating the

function. The rules shown below apply to all functions, not only quadratics.

Given the graph of f(x)

Shifts the graph of up units

Shifts the graph of down units

Shifts the graph of left units

Shifts the graph of right units

f x c f x c

f x c f x c

f x c f x c

f x c f x c

f x

Reflects the graph of over the

Reflects the graph of over the

For 1, stretch the graph of vertically by a factor of

For 0 1, compr

f x x axis

f x f x y axis

cf x c f x c

c

ess the graph of vertically

by a factor of

For 1, compress the graph of horizontally by a factor of

For 0 1, stretch the graph of horizontally by a factor of

f x

c

f cx c f x c

c f x c


7-3 continued

22) Compared the parent function , what transformation will we see for the graph of 2 ?

a) translated 2 units left b) translated 2 units right

c) translated 2 units up d) translated 2 units down

f x x f x

23) Compared the parent function , what transformation will we see for the graph of 2?

a) translated 2 units left b) translated 2 units right

c) translated 2 units up d) translated 2 units down

f x x f x

24) Compared the parent function , what transformation will we see for the graph of 1 2?

a) translated 1 unit left and up 2 b) translated 1 unit right and up 2

c) translated 2 units left and up 1 d

f x x f x

) translated 2 units right and down1

5) The vertex of the function is 3,5 . What is the vertex of the function 4 ?

a) 7,5 b) 3,1 c) 1,5 d) 3,9

f x f x

6) The roots of are 1 and 4. Which of the following functions will have the same roots?

a) 2 b) 2 c) d) 2

f x x x

f x f x f x f x

7) The vertex of is 4, 2 . Which of the following functions will have the same vertex?

a) 2 b) 2 c) d) 2

f x

f x f x f x f x

8) Caleb tells his class that the range of the function will change if it is transformed to 2 1,

but the domian will remain the same. Explain why you agree of disagree with Caleb.

f x f x


7-3 continued

9) The graph of is shown below. The function is defined at 2 1.

On the same set of axes where is graphed, graph the function .

f x g x g x f x

f x g x

210) Given the function 3 1:

a) Determine the equation of the function which is definied as 2 .

b) Determine the equations of the axis of symmetry for both the functions and .

c) Dete

f x x x

h x h x f x

f x g x

rmine the coordinates of the vertex for both the functions and .f x g x


Student Notes



Homework 4: Comparing Functions (A-REI.10, F-IF 2,4,6,9)

2Review 1) Which transformation of the function 7 2 will change the axis of symmetry?

a) 4 b) 5 c) d) 2

f x x x not

f x f x f x f x

2) The following quadratics , , or f x g x h x are given below.

2 4 5h x x x

a) Which function , , or ,f x g x h x has the greatest maximum (highest vertex)?

b) Which function , , or ,f x g x h x has the greatest average rate of change over

the interval [-1,1]?

x g x

1 1

0 4

1 5

2 4

3 1


7-4 continued

2 23) Given the functions 1 and 2 4 , answer the following as or . j x x k x x x true false

a) The function is narrower than .j x k x

b) The functions and have the same domain.j x k x

c) The functions and have the same range.j x k x

4) Using the same functions and above, answer the following as or ,

and justify your answer.

a) The functions and have the same axis of symmetry.

j x k x true false

j x k x

b) The functions and have the same vertex.j x k x

c) The function has a greater average rate of change over the interval [1,4] than .k x j x

d) 4 4 .k j

e) The function 2 2.k x j x



Homework 5: Quadratic Function Modeling (A-SSE.1,3, A-REI.4)

2Review 1) Determine the average rate of change for the function 3 5

over the interval 0,2 .

f x x x

2) Amy tossed a ball in the air in such a way that the path of the ball was modeled by the equation 2 6 .y x x In the equation, y represents the height of the ball in feet and x is time in seconds. At

what value of x does the ball hit the ground?

3) The length of a rectangle is six minus the width as shown below.

a) Create a function A w that can be used to determine the area of the rectangle from the width.

b) Determine the maximum area for the rectangle.

4) Future sale projections for a company for the next two years is modeled by the function

22 24 100s x x x where s x is in thousands of dollars and x is the number of months in the future.

a) Per the model, what is the minimum amount of sales expected?

b) How many months in the future is that minimum expected to occur?


7-5 continued

5) A diver’s position above the water is represented by the function 216 32 48h t t t where h t

represents the height above the water and t represents time in seconds.

a) What is the greatest height the diver attains during the dive?

b) How many seconds elapse until the diver enters the water?

6) A projects projected profit is represented by the graph shown below where y is the profit in thousands

of dollars and x is the number of months

of operation.

a) How many months will it take for the

company to achieve its maximum profit?

b) When is the first time the company

showed a profit? Explain your answer.

c) What could be a reason the function begins below the x-axis?

d) How long will it take the company to make a profit of $25,000

e) Determine the domain that will only result in a profit for the company, and find its corresponding

range of profit.

f) Over which interval is the profit increasing fastest?

a) [2,3] b) [3,4] c) [4,5]



Homework 6: Piecewise Functions (A-REI.10, F-IF.2)

Notes: A function does not need to be a single connected graph; it can be in several pieces. A Piecewise

Function is a function with several pieces; each piece is described by a separate function with a

restricted domain.

2

Example: The function is shown below.

1When is less than 2 the function is represented by ,

2

when is greater or equal to 2 and less than or equal to 2 it is 1,

when is greater than 2 i

f x

x x

x x x

x

2

t is represented by 2 11.

1,

22

The proper notation for this is 1, 2 2

22 11,

x

xx

f x x x x

xx

Remember that an open circle means “not included”, and a closed circle means “included”. The x value

of 2 is included in the 2 1x x part of the function, but not the 2 11x part.

A piecewise function may be continuous where the pieces are connected and you can trace the function

without picking up your pencil, or they may be discontinuous (containing holes or gaps) such as the

example above where to trace the function you would need to pick up your pencil off the paper at some

point.


2

7-6 continued

3, 2

1) For the function 5, 2 , evaluate the following:

2 1, 2

a) 2 b) 2 c) 5 d) 3 e) 0

x x

f x x

x x

f f f f f f f

2) The graph of is shown below:g x

2 2

2 2

Which of the following describes the function:

3 32, 2,a) b)

3 310, 7,

3 32, 2,c) d)

3 310, 10,

x xx xg x g x

x xx x

x xx xg x g x

x xx x

3) The equation the phone company uses to determine the cost of a long distance phone call ,

where is the minute length of the call in minutes is:

.05 , 0 10

2 10 4, 10

P m

m

m mP m

m m

a) Determine the difference in cost of a 12 minute long distance phone call versus an 8 minute long

distance call.

b) Determine how many minutes a call must be to total $14?


7-6 continued

4) On the set of axes below:

2

2 5, 3

a) Graph 4, 3 3

, 3

x x

f x x x

x x

b) On the same set of axes graph f x


7-6 continued

5) On the set of axes below

2 4 1, 01

a) Graph and 1133, 0

2

x x x

g x h x xx x

b) How many values of x satisfy the equation ?g x h x

Explain your answer, using evidence from your graphs.




21) Which of the following is the equation of the axis of symmetry for the function 2 6 1?

3 3 3 2a) b) c) d)

2 2 2 3

f x x x

x x y

2) A rocket shot into the air attains a height that can be described by the equation 216 240 ,y x x

where y is the height of the rocket in feet and x is the time in seconds after launch.

a) What is the maximum height of the rocket?

b) When will the rocket hit the ground?

3) The graph of is shown below:g x

Which of the following describes the function:

3 8, 1 3 6, 1

3, 1 3 3, 1 3a) b)

1 1, 3 , 3

3 3

1 3 8, 18, 1

3 3, 1 3c) d)

3, 1 3 1, 3

3 , 3 3

x x x x

x xg x g x

x x x x

x xx x

xg x g x

xx x

x x


7-7 continued

4) For the function 2 2 3,f x x x determine the equation of the axis of symmetry, the coordinates of

the vertex, a domain of seven values with the vertex in the middle, and graph the function over that

domain.

5) On the same set of axes, graph and label the

function 1 2. f x

26) Given the functions 3 1 and 3 1, which function has the greatest

average rate of change over the interval 2,5 ? Justify your answer.

g x x h x x

2

3, 17) For the function , evaluate the following:

4 1, 1

a) 2 b) 1 c) 2

x xp x

x x x

f f f f

8) Graph the function on the set of axes on the right.p x




1) Rhonda has $1.35 in nickels and dimes in her pocket. If she has six more dimes than nickels, which

equation can be used to determine x, the number of nickels she has?

a) 0.05 6 0.10 1.35 b) 0.05 0.10 6 1.35

c) 0.05 0.10 6 1.35 d) 0.15 6 1.35

x x x

x x x

2) If the formula for the perimeter of a rectangle is 2 2 , then can be expressed as

2 2 2a) b) c) d)

2 2 2 2

p l w w

l p p l p l p ww w w w

3) Peter begins his kindergarten year being able to spell 10 words. He is going to learn to spell 2 new

words every day.

a) Write an inequality that can be used to determine how many days, d, it takes Peter to be able to spell at

least 75 words.

b) Use that inequality to determine the minimum number of whole days it will take for him to be able to

spell at least 75 words.

24) Express the product of 2 7 10 and 5 in standard form.x x x

5) What is the equation of the line that passes through the points 3, 3 and 3, 3 ?

a) 3 b) 3 c) 3 d) y x y y x

2 266) What is the value of in the equation 3 ?

1 1a) 8 b) c) d) 8

8 8

xx x


7-8 continued

7) The accompanying table represents the number of hours a student worked and the amount of money a

student earned. Write an equation that represents the number of dollars, d, earned in terms of the

number of hours, h, worked.

8) Using the equation from #7, determine the number of dollars the student would earn for working 40

hours.

9) On the set of axes below, solve the system of equations graphically.

2 6

2 8

x y

y x

10) The function f has a domain of {1, 3, 5, 7} and a range of {2, 4, 6}.

Could f be represented by 1,4 , 3,6 , 5,2 , 7,2 ? Justify your answer.

Number of hours (h) Dollars earned (d)

8 $50.00

15 $93.75

19 $118.75

30 $187.50


Unit 8 – Solving Quadratics

Homework 1: Solving Factorable Quadratics (A-CED.1, A-REI.3,4, F-IF.8)

When a quadratic equation is set equal to zero and solved for x, the solution found are the roots (x-

intercepts) of the equation. These roots may be rational or irrational. They may also be a type of

complex imaginary number that you will learn about in algebra 2. In the case of rational roots, the

quadratic can be solved using factoring which are skills covered in Unit 4 Homework 6.

1) Use factoring to solve the following quadratic equations:

2 2 2a) 5 14 0 b) 49 0 c) 3 0x x x x x

2 2 2d) 21 10 e) 5 0 f) 7 6 0x x x x x x

2 2 2g) 8 15 h) 10 3 i) 4x x x x x x

2 2 2j) 2 10 12 0 k) 3 3 10 0 l) 3 2 1x x x x x x

2 2

m) 5 4 0 n) 2 3 12 o) 5 6 1 19x x x x

5 1 2 1 3p) q) r)

2 3 4 2 9 4 3 3

x x x

x x x x


8-1 continued

2) Determine three consecutive integers such that the square of the first is 45 more than the sum of the

second and third.

3) One positive integer is five more than the other and their product is 36. Determine the two integers.

4) The area of a rectangle is 40 units squared, and its length is six more units than its width. Determine

the length and the width of the rectangle.

5) A contractor needs 54 total square feet of brick to construct a rectangular walkway. If the width is

three feet less than the length, determine the dimensions of the walkway.

6) A family with a square piece of property with side length s has a rectangular swimming pool installed.

The length of the swimming pool is 4 yards less than the side length of the property and the width is 8

yards less than the side length of the property, as shown in the diagram. If the area of the swimming pool

is 60 square yards, determine the square yardage of the property.



Homework 2: Solving Quadratic Linear Systems (A-CED.2, A-REI.4,6,11)

A quadratic linear system can have at most two solutions. Shown below are the graphs of

2 4 3 1.f x x x and g x x

The solution to the system is the intersection

of the functions which are the points

1,0 4,3 .and

A quadratic linear system can be solved

graphically as shown or algebraically

using substitution and factoring shown below.

2

2

Set the functions equal to each other 4 3 1

Then set the equation equal to zero 5 4 0

Then factor and solve for the values 4 1 0

4 0, 1 0

4, 1

Then substitu hete t

x x x

x x

x x x

x x

x x

values back into either equation to find the corresponding values.

4 4 1 3 4,3 1 1 1 0 1,0

The solution to the system is the points 4,3 and 1,0 . To check the solutions you can substitute the

solu

x y

g g

tions into both functions to verify the points are a part of both functions.

1) Solve the following system graphically on the

provided set of axes:

2 2 3 2 1.h x x x and j x x


8-2 continued



2 4 3 3.k x x x and p x x

3) Solve the following systems algebraically:

2a) 6 7

1

y x x

y x

2b) 4 3

7 3

y x x

y x


8-2 continued



2 2 1 2 5.q x x x and r x x


2a) 4

11

2

y x

y x

2b) 6 8

4

y x x

y x


8-2 continued



2 6 5 4.s x x x and t x


2a) 2 8 3

2 3

y x x

y x

21 3b) 3

2 2

4

y x x

y x



Homework 3: The Quadratic Formula (A-CED.1, A-REI.3,4)

REVIEW 1) Solve the following systems algebraically:

2 6 5

2 5

y x x

x y

Notes: In homework 8-1 of this chapter we practiced solving quadratic equations by factoring. The

result was at most a pair of rational roots. Some quadratics have irrational roots (or even imaginary

which you will learn about in algebra 2). These irrational roots cannot be solved by factoring so we

need another method. One such method is the use of The Quadratic Formula:

2 4

2

b b acx

a

The a, b, and c terms come from 2 0ax bx c

Example: Solve the quadratic equation 22 6 3 0x x

First, we substitute the a, b, and c values into the formula:

26 6 4 2 3

2 2x

Then simplify using PEMDAS:6 36 24 6 12

4 4x

Then we simplify the radical (if possible): 6 12 6 2 3

4 4x

Lastly simplify through division (if possible:) 6 2 3 3 3

4 2x

The final answer can be separated into two terms, one with the plus and the other with the minus

3 3 3 3 and

2 2x x


8-3 continued

2) Solve each of the following for x, express the answer in simplest radical form:

2a) 3 2 2 0x x 2b) 11 0x x

2c) 3 1x x 2d) 2 3 2 0x x

2e) 2 1 4x x 2f) 6 3 0x x

2g) 5 8 1 0x x 2h) 5 4x x

2i) 2 10 1 0x x 2j) 6 5 4x x



Homework 4: Completing the Square (A-CED.1 A-REI.1,3,4, A-SSE.3, F-IF.8)

REVIEW 1) Solve for x in simplest radical form: 23 8 3x x

Notes: The quadratic formula is one way to solve for the roots of quadratic, another is the method called

Completing the Square:

Example: Solve the quadratic equation by completing the square: 2 6 3 0x x

The first step is set the equation equal to the c term: 2 6 3x x

Then add to both sides the term obtained from

2

2

b

which for this example is

26

92

2 6 9 3 9x x

Then the left can be factored into two of the same binomials so we can write it as a binomial squared and

simplify the right 2

3 3 6 3 6x x x

Then take the square root of both sides, note that when you take the square root you introduce the plus or

minus in front of the radical to account for the two possible answers 2

3 6 3 6x x

Lastly solve for x by moving the constant to the right side to get x alone on the left.

3 6 3 6x x

We write the solution in simplest radical form. In this example the radical cannot be simplified further.

Completing the square gets a little trickier when the a term does not equal 1, if 1a one method is to

divide each term by a so the coefficient of 2x becomes 1, then complete the square.

One way to check your solution is to use the quadratic formula because if it is done correctly will give

the same solution.

26 6 4 1 3 6 36 12 6 24 6 2 6

3 62 1 2 2 2

x


8-4 continued

2

2

2) Consider solving 6 5 0 by completing the square.

6 ____ 5 ____

What is the number that goes in the blanks?

a) 9 b) 9 c) 36 d) 6

x x

x x

2

2


4 ____ 1 ____


a) 16 b) 2 c) 4 d) 8

x x

x x

2

2


5 ____ 7 ____


5 25 25a) b) 25 c) d)

2 4 2

x x

x x

2

2

2 2


8 16 4 16

Which equation shows the next step for solving the equation by completing the square?

a) 8 20 b) 4 20 c) 8

x x

x x

x x x 2 2

20 d) 4 20 x

2

2

2


10 25 7 25

5 32

Which equation shows the next step for solving the equation by completing the square?

a) 5 32 b) 5 32

x x

x x

x

x x 2

c) 5 32 d) 5 32 x x


8-4 continued

7) Complete the square to solve for x in simplest radical form for each of the following quadratics:

2a) 6 4 0x x 2b) 8 13 0x x

2c) 4 8 0x x 2d) 10 7 0x x

2e) 6 2x x 2f) 41 4x x

2g) 2 12 10 0x x 2h) 4 12 7 0x x


2

2

2

2 2

8-4 continued

Use the following for question 8 - 11

Consider solving 0 by completing the square.

____ ____

8) Which term should go in the blanks?

a) b) 2 4

ax bx c

b cx x

a a

b cx x

a a

b b

a a

2 2

2 2

2 22 2

2 2

2

c) d) 2 4

9) Which equation shows the next step for solving the equation above by completing the square?

4 4a) b)

2 24 4

4c)

4

b b

a a

b ac b b ac bx x

a aa a

b ax

a

22 2

2

2 2

2 2

2

4d)

2 24


4 4a) b)

2 24 4

4 4c) d)

2 2 4

c b b ac bx

a aa

b ac b b ac bx x

a aa a

b ac b b acx x

a a a

2

2

2 2

2 2

4


4 4a) b)

2 2 2 2

4 4c) d)

2 2 4 2

b

a

b b ac b b acx x

a a a a

b b ac b b acx x

a a a a



Homework 5: Vertex Form (A-SSE.2,3, A-CED.1, A-REI.3, F-IF.8)

REVIEW 1) Solve for x in simplest radical form by completing the square: 2 10 15 0x x

Notes: The Vertex Form of a quadratic equation is 2

, y a x h k where a is a constant and the

vertex of the quadratic is the coordinates , . , h k Notice the x h this means we will see an opposite

sign in the expression for h than we will in the coordinates.

2

Example: The vertex of the quadratic 3 5 4 is 5, 4 y x

2) State the vertex for each of the following quadratics:

2

a) 3 2 y x 2

b) 2 4 7 y x 2

c) 6 1 y x

A quadratic in standard 2y ax bx c form, 1,when a can be converted into the vertex form by use

of the completing the square procedure.

2

2

22 2

Example: Convert 8 13 into vertex form.

First temporarily set the equation equal to zero: 8 13 0

Then begin the complete the square procedure as normal:

8 13 8 16 13 16 4 3

Stop c

y x x

x x

x x x x x

2

2

ompleting the square when you get to the binomial squared stage.

Then reset the equation back equal to zero: 4 3 0

Finally set the equation back equal to : 4 3

From this we can see the vertex of

x

y y x

the quadratic is the point 4, 3 .


8-5 continued

3) Convert each of the following quadratics into vertex form by use of completing the square and state

the coordinates of the vertex:

2a) 10 22y x x

2b) 8 17y x x

2c) 2 1y x x

2d) 4 8y x x

2e) 6 12y x x

2f) 14 43y x x

2g) 12 40y x x


8-5 continued

For quadratics 1,when a the procedure is a little more complex.

2

2

2

Example: Convert 3 12 13 into vertex form.

First temporarily set the equation equal to zero: 3 12 13 0

Then set equal to the term like normal: 3 12 13

Then factor out the term on the l

y x x

x x

c x x

a

2

2

eft: 3 4 13

Then complete the square inside the parenthesis, but the term that gets added to the left is

multiplied by the term: 3 4 4 13 12

Then complete the process like normal:

Factor left/s

x x

a x x

2 2 22

implify right Set equal to zero Set equal to

3 4 4 13 12 3 2 1 3 2 1 0 3 2 1

y

x x x x y x

4) Convert each of the following quadratics into vertex form by use of completing the square and state

the coordinates of the vertex:

2a) 2 12 17y x x

2b) 5 10 7y x x

2c) 8 21y x x

2d) 3 12 16y x x


8-5 continued

2The vertex of the parent quadratic function is 0,0 .

5) What is the coordinate of the vertex of which is defined as after a transformation

of left 3 down 4?

a) 3,4 b) 3, 4 c) 3,4 d) 3

f x x

g x f x

2 2 2 2

, 4

6) What is the equation of which is defined as after a transformation of left 3 down 4?

a) 3 4 b) 3 4 c) 3 4 d) 3 4

g x f x

y x y x y x y x

227) To map the function onto the function 4 1 each point of must move:

a) right 4 and up 1 b) right 4 and down 1

c) left 4 and up 1

f x x h x x f x

d) left 4 and down 1

2

8) What is the axis of symmetry of the function 2 3 4

a) 3 b) 3 c) 3 d) 4

j x x

y x x x

29) Which of the following is about the function 2 3 4?

a) The vertex is the coordinates 3, 4 .

b) The vertex is located left 3 and down 4 from the origin.

c) The function is equivalent to 2

false k x x

l x 2 12 14.

d) The axis of symmetry is 3.

x x

y




2

2


8 ____ 3 ____


a) 16 b) 8 c) 16 d) 4

x x

x x

22) What is the coordinates of the vertex of 3 2 5

a) 2,5 b) 2, 5 c) 2,5 d) 2, 5

y x

2 2 2

3) Factor and solve the following quadratics:

a) 7 0 b) 4 12 0 c) 2 11 15x x x x x x

4) The area of a rectangle is 21 units squared, and it length is four more units than its width. Determine

the length and the width of the rectangle.

2 2

5) Use the quadratic formula to solve the following quadratics in simplest radical form:

a) 6 3 0 b) 2 4 5 0 x x x x


8-6 continued

2 2

6) Solve the following quadratics in simplest radical form by completing the square:

a) 12 22 0 b) 14 29 0 x x x x



2 6 5 2 5.j x x x and g x x

8) Solve the following system algebraically:

2 2 3

1

y x x

x y

2 2

9) Convert the following quadratics into vertex form:

a) 6 14 b) 2 20 52y x x y x x

2 210) Convert the quadratic 4 5 from vertex form to standard form.y x y ax bx c




1) Marie currently has a collection of 58 stamps. If she buys s stamps each week for w weeks, which

expression represents the total number of stamps she will have?

a) 58 b) 58 c) 58 d) 58sw sw s w s w

2) Which ordered pair is not in the solution of 1

4 and 3 1?2

y x y x

a) 4,2 b) 3,3 c) 5,3 d) 6,2

3) Which value of x is the solution of the equation 2

5?3 6

x x

a) 6 b) 10 c) 15 d) 30

4) The sum of two numbers is 47, and their difference is 15. What is the larger number?

a) 16 b) 31 c) 32 d) 36

5) If , the value of in terms of and can be expressed as:

1 1a) +1 b) c) d)

1

a ar b r a b r

b b b r b

r r r r b

6) At Genesee High School, the sophomore class has 60 more students than the freshman class. The

junior class has 50 fewer students than twice the students in the freshman class. The senior class is three

times as large as the freshman class. If there are a total of 1,424 students at Genesee High School, how

many students are in the freshman class?

a) 202 b) 205 c) 235 d) 236


8-7 continued

7) Which graph represents the solution of 3 9 6 ?y x

a) b)

c) d)

8) What is the equation of the line that passes through the point 4, 6 and has a slope of 3?

a) 3 6 b) 3 6 c) 3 10 d) 3 14y x y x y x y x

9) When 2 24 7 5 is subtracted from 9 2 3, the result is:x x x x

2 2 2 2a) 5 5 2 b) 5 9 8 c) 5 5 2 d) 5 9 8x x x x x x x x

10) Jane is given the graph of 216,

2y x shown below. She wants to find the zeros of the function but

is unable to read them exactly from the graph. Find the zeros in simplest radical form.


Unit 9 – Exponential, Absolute Value, and Other Functions

Homework 1: Exponential Functions (A-REI.3, F-IF.2,4,6,7, F-LE.1)

Notes: An Exponential Function has the form: ,x

f x a b where a is a coefficient, b is the base of

the exponent , b>o , b 1 , and x (the input/domain) is the exponent. A simple example is 2 ,xy this

looks very similar but is very different to the quadratic 2.y x In a quadratic, x is the base, and in an

exponential, x is the exponent (hence the name exponential).

There are two different types of Exponential Functions. Exponential Growth Functions and Exponential

Decay Functions.

Exponential Growth Exponential Decay

2

, 1

. , as ,

(as the graph goes right it goes up)

x

x

f x a b

f x

b

ex x y

1

2

, 0 1

. , as , 0

(as the graph goes right it goes down)

x

x

f x a b

f x

b

ex x y

Notice in both graphs, the curves get really close to the x-axis but never intersect it, this is known as an

asymptote. Exponential Functions (without a transformation) do not have an x-intercept, but they do

have a y-intercept. The Domain for both functions is all real numbers and the Range for both is all

numbers greater than zero.

1) Determine if the following exponential functions are growth or decay functions by examining the base

and the exponent:

1a) 5 b) c) 2 3

4

5d) 3 2 e) 4 .75 f)

4

1 3g) 4 h) i)

3 2

xxx

xx x

xx

f x g x h x

j x k x l x

m x n x p x

x


9-1 continued

2) For the function 4 , determine:xf x

a) If the function is growth or decay? b) the coordinates of the y-intercept.

c) the value of 3f d) the value of 2f e) the value of 1

2

f

3) For the function 1

g 2 , determine:2

x

x

a) If the function is growth or decay? b) the coordinates of the y-intercept.

c) the value of 1f d) the value of 2f e) the value of 3f

4) For the function 2 3 , determine which interval will have the greatest averagex

h x rate of change?

a) 0,1 b) 1,2 c) 2,3

5) For the previous question, explain how the knowledge that the function is exponential growth, and

that each interval is of consecutive integers, could be used to answer which interval has the greatest

average rate of change.



Homework 2: Exponential vs Linear (A-CED.1, A-REI.3, F-BF.1, F-LE.1,2)

Note: An Exponential Function can easily be identified from its equation or graph, but it can also be

identified from a table because in an exponential function each term (y-value) is a constant multiple of

the previous term.

Ex. To determine the exponential base, divide any y term by the previous: Here we can

determine the base to be 3 (each y term is the previous times 3). We can then use any

point in the table with substitution to determine the coefficient of the equation.

2 2

Using the found base 3 and the point from the table 2,18 ,

becomes 18 3 which can be solved for : 18 3 18 9 2

Now we can write the exponential grown equation as 2 3

x

x

b y a b

a a a a a

y

Note: A linear function table can be identified because each term (y-value) is a constant sum of the

previous term.

Ex. To determine the linear slope, subtract any y term by the previous: Here we can

determine the slope to be 2 (each y term is the previous plus 2). We can then use any

point in the table with the point slope formula to determine the equation.

1 1

Using the found slope 2 and the point from the table 3,11 ,

The point slope formula becomes 11 2 3

which simplified and solved for :

11 2 3 11 2 6 2 5

m

y y m x x y x

y

y x y x y x

1) For each table below, determine if the table represents a linear or exponential function. If linear,

determine the slope, if exponential, determine the base.

a) b)

c) d)


9-2 continued

2) Each day John records the height of a plant. Below is the data he has recorded.

a) Write an equation to represent ,H d the height of the plant on the dth day.

b) Using your equation, predict the height of John’s plant on the 9th

day.

3) Ted decides to start a training regimen with running. The first week he runs 20 minutes, then each

following week he runs double the time as the previous week. The table below shows his running log for

the three weeks.

a) Write an equation to represent ,T w the amount of time Ted runs on the wth week.

b) Using your equation, predict the amount of time Ted will run in the 7th

week.

4) Given the function ,f x represented by the table shown:

Determine if the table represents a linear or exponential function.

Explain how you know.



Homework 3: Percent Rate of Change (A-SSE.1,2,3, A-CED.1, A-REI.3, F-IF.2, F-LE.1,5)

Notes: An exponential growth function that increases by a Percent Rate of Change is represented by

1 ,t

f t P r where P represents the principal or initial value, r represents the percent rate of

change, and t represents time.

Ex 1. The value of a $1,000 investment made into a savings account that gains 5% interest annually with

no deposits or withdrawals can be modeled by the equation 1000 1 .05 ,t

f t note that r is .05 and

not 5. This is because 5% is equal to 5

100 which equals .05. The equation can be simplified inside the

parentheses to be 1000 1.05 .t

f t Since the base is greater than 1, the function is clearly a growth

function. This equation can now be used to determine the value of the account after t years:

Ex 2. Determine the value of a $1,000 investment with an annual interest rate of 5% after 6 years.

66 1000 1.05 $1,340.10f

The model changes to 1 ,t

f t P r for exponential decay functions that decrease by a percent rate

of change. This will yield a base less than 1.

1) Create an equation to represent the value of a saving account after t years with an initial investment of

$250 and an annual interest rate of 4%. Use this equation to determine the value of the account after 10

years if no other deposits or withdrawals are made.

2) A $40,000 car loses value at an annual rate of 12%. Create an equation to represent the present value

of the car after t years. Use this equation to determine the value of the car after 4 years.

3) A type of bacteria has a very high exponential growth rate at 80% every hour. If there are 10 bacteria

to start, determine how many there will be in 5 hours.


9-3 continued

4) The population of a certain Midwest City is modeled by the function 68,990 0.925 ,t

P t where t

is the number of years since 2010.

a) What is the population in the year 2019?

b) What is the rate at which the population is decreasing per year?

c) What percentage of the population remains each year compared to the previous year?

5) Given the function 500 1.025t

A t to represent the value in a savings account. Which of the

following is false?

a) The initial investment was $500 c) The account gains interest at rate of .025%

b) The account is gaining value each year d) In three years the accounts value will be $538.45

6) If t is time in years, the function 750 0.85t

A t

a) increases 15% each year c) decreases 15% each year

b) increases 85% each year d) decreases 85% each year

7) The breakdown of a sample of a chemical compound is represented by the function

200 0.75 ,t

p t where p t represents the number of milligrams of the substance

remaining after t years. In the function ,p t explain what 200 and 0.75 represent.

8) A savings accounts value is determined by the function 250 1.03 .t

V t Determine the y-intercept

of the function and explain what it represents.



Homework 4: Absolute Value Functions (A-REI.6,10,11, F-IF.2,4,6,7, F-BF.3)

Notes: The Absolute Value Function, ,f x x is a type of piecewise function which is continuous

and contains two linear parts with additive inverse slopes (same slope but one positive and one

negative).

Below is the graph of the absolute value parent function .f x x

The Domain of the function is all real numbers,

the Range, which could change with transformations, is 0, .

Remember the transformation rules from Unit 7 Homework 3:

Horizontal Translations, Vertical Translations, Reflection of the x-axis, Dilations

f x c f x c f x cf x

1) Label each graph below as one of the given functions: 3f x x , 2g x x , 4h x x ,

k x x , 2 1p x x , or 2 4r x x .

a) b) c)

d) e) f)


9-4 continued

2) For the two functions, 2 and g 3 2,f x x x x

a) Graph each function on the same set of axes given below.

b) For what values of does g ?x f x x

c) For what values of are g ?x f x x

d) Which function has the greatest average rate of change over [a,b] given 0 and 0?a b

Explain how you know.


9-4 continued

3) Below is the graph of ,h x on the same set of axes graph the transformation defined as 3 2.h x

4) Jason tells his friends that the graphs of and , y x d x a will have exactly one point of

intersection regardless of the values of d and a. Do you agree with Jason? Explain why or why not.


9-4 continued

5) For the two given functions, 2

2 4 and g 1,3

f x x x x

Solve graphically to determine all the values of for which g ?x f x x

6) The graphs of and ,y x c y a will intersect how many times:

a) when ?a c b) when ?a c c) when ?a c



Homework 5: Square Root, and Step Functions (A-REI.10, F-IF.2,4,7, F-BF.3, F-LE.1)

Notes: The Square Root Function, ,f x x shown below has a Domian of 0x because we cannot

take the square root of a negative number

(imaginary numbers will be introduced in Algebra 2),

the Range of this function is 0y because to be a

function the square root of a positive number can

only have one solution, a positive one.

1) Label each graph below as one of the given functions: 2f x x , 2g x x ,

4 3h x x , k x x , 3p x x , or 2 3 1r x x .

a) b) c)

d) e) f)

2) Below is the graph of ,t x on the same set of axes graph the transformation defined as 2 4.t x


9-5 continued

Notes: The Step Function, is a type of piecewise function which is discontinuous and contains

all constant (horizontal) linear

pieces with zero slopes.

Example:

3, 2

2, 2 3

5, 3

x

f x x

x

A famous step function is the Greatest Integer Function, f x x shown below, which returns the

largest integer less than or equal to x. Ex. 3 3, 3.4 3, 3.9 3,

4.9 5, 4.1 5, 3

3) Use the greatest integer function to determine the following values:

a) 7 6. b) 5 3. c) 3 2. d) 0 7. e) 2 f)

4) Graph the following piecewise function on the given graph:

4, 3

1, 3 2

3, 2

x

g x x

x

5) Which of the following describes a step function?

a) Sally gets her haircut once a month to the same length.

b) The date changes every 24 hours.

c) The temperature increases during the summer months then decreases in the fall.

d) A savings account gains 3% interest compounded annually.



Homework 6: More Piecewise Practice (A-REI.6,10,11, F-IF.2,6, F-BF.3)

2, 1

For the function 2, 1 2,

3, 2

1) Evaluate the following:

a) 2 b) 2 c) 4 d) 5

x x

f x x x

x

f f f f f

2) For the function above,f x determine the average rate of change for the function over the given

intervals

a) 5, 2 b) 1,2 c) 3,8

3) Graph the function ,f x on the set of axes provided below:


9-6 continued

4) Solve graphically for the two functions given below to determine all the values

of for which ?x j x k x

2

2 4 , 2

2, 2 1,

11 1,

4

x x

j x x x

xx

k x x

,5) Given the function which correctly shows the function 2 3?

, ,

2 3, 2 2 3,a) 2 3 c) 2 3

2 3, 2, 2 3, ,

2 3, 2b) 2 3 d)

2 3, 2,

x x ap x f x

x x a

x x a x x ap x p x

x x a x x a

x x ap x p x

x x a

, 22 3

, 2,

x x a

x x a



Homework 7: Arithmetic Sequences (A-CED.1, A-REI.3, F-IF.3, F-BF.1)

Notes: A Sequence is an ordered list of terms. Sequence subscript notation is used to represent each

term, 3 5 7 9 3 5na the nth term . In the sequence , , , , ... , the first term is and the second term is ,

1 2 3 5The notation for this would be a and a . Sequences may be represented by explicit formulas,

which are a formula in terms of the nth term, or as recursive formulas, which are formulas for a term in

terms of the previous term. The two most common types of sequences are Arithmetic and Geometric.

Geometric sequences will be covered in the next homework.

Arithmetic Sequences are linear sequences where each consecutive term has a constant difference. The

example 3 5 7 9 2, , , , ... is an arithmetic sequence with a constant difference, d, of . Meaning each

consecutive term is found by adding 2. The constant difference, d, is d anyterm previous term .

1) For each sequence below, state if the sequence is an arithmetic sequence or other. If arithmetic state

the constant difference, d, for the sequence.

a) 2 5 8 11, , , , ... b) 2 4 8 16, , , , ... c) 12 7 2 3, , , , ...

1 1 1 1d)

2 6 18 54, , , , ... e) 0 3 7 12, , , , ...

2 4 5f) 1

3 3 3, , , , ...

g) 2 4 6 8x, x, x, x, ... 2 3h) 2 2 2 2, x, x , x ,...

i) 3 3 5 5 7 7 9n , n , n , n , ... 2 2 2 2j) 3 4 2 5 3 6 4x y, x y, x y, x y, ...


9-7 continued

Notes: The explicit formula for the nth term of an arithmetic sequence is 1 1na a n d .

Example: Determine the 15th

term in the arithmetic sequence {5, 9, 13, …}

1 15

15 15 1

We can see the first term , 5 and the common difference, 4 and we want so 15

These terms substituted into the explicit arithmetic formula and simplified gives us:

5 15 1 4 5 14 4

a , d , a n .

a a a

5 155 56 61a

2) For each of the following arithmetic sequences, use the explicit arithmetic formula to find the

specified term.

a) Find the 7th

term in the sequence {1, 3, 5, 7, …}

b) Find the 12th

term in the sequence {0, 4x, 8x, 12x, ...}

c) Find the 10th


d) Find the 13th

term in the sequence 3 3 3 5 3 7 3, , , , ...

e) Find the 9th

term in the sequence 1 5 9 13

8 8 8 8, , , , ...


9-7 continued

3) What is a formula for the nth term of the following sequence? {4, 9, 14, 19, …}

a) 4 5 b) 4 5 c) 5 1 d) 4 5 n n n n

4) Which of the following statements about the following sequence is false? {11, 8, 5, 2, …}

a) The sequence is arithmetic b) 14 3na n c) The common difference is 3 d) 7 7a

5) If Jeremy runs for 15 minutes on the first day of this new workout routine. He plans to run an

additional 3 minutes every day. If Jeremy were to continue this pattern for 50 days, determine and

state the length of his run on the 50th

day.

6) An arithmetic sequence has a third term of 3, and a sixth term of 15, determine:

a) the common difference b) the first term c) the 20th

term

7) An arithmetic sequence has a second term of 0, and a fourth term of 16 , determine:

a) the common difference b) the first term c) the 7th

term

8) An arithmetic sequence has 4 4 6a x , , and 6 6 10a x , determine:

a) the common difference b) the first term c) the 3rd

term



Homework 8: Geometric Sequences (A-CED.1, A-REI.3, F-IF.3, F-BF.1)

Geometric Sequences are exponential sequences where each consecutive term has a constant ratio.

The example 3 6 12 24 2, , , , ... is a geometric sequence with a ratio, r, of . Meaning each consecutive

term is found by multiplying the previous term by 2. The ratio, r, is anyterm

r .previous term

1) For each sequence below, state if the sequence is geometric or other. If geometric state the ratio, r, for

the sequence.

a) 2 8 32 128, , , , ... b) 2 4 8 16, , , , ... c) 0 7 14 21, , , , ...

1 1 1 1d)

2 6 18 54, , , , ... e) 1 5 25 125, , , , ...

2 3 4f) 1

5 5 5, , , , ...

2 3 4g) 2 2 2 2x, x , x , x , ... h) 2 1 3 2 4 3 5 4x , x , x , x ,...

i) 3 2 6 4 12 8 24n , n , n , n , ... 2

2

1j)

3 3

x y y, , , , ...

y x x

2 3 4k) 2 4 6x, x , x , x , ... 2 2 3 2 2 3l) a b, a b , a ab ba b , ...


9-8 continued

Notes: The explicit formula for the nth term of a geometric sequence is 1

1

n

na a r

.

Example: Determine the 8th

term in the arithmetic sequence {5, 15, 45, …}

1 8

8 8 8 8

8 1 7

We can see the first term , 5 and the ratio, 3 and we want so 8

These terms substituted into the explicit geometric formula and simplified gives us:

5 3 5 3 5 2187 10 935

a , r , a n .

a a a a ,

2) For each of the following geometric sequences, use the explicit geometric formula to find the specified

term.

a) Find the 7th


b) Find the 8th

term in the sequence 5 5

10 52 4

, , , , ...

c) Find the 6th

term in the sequence 2 10 50, , , ...

d) Find the 7th

term in the sequence 5 2 5 4 5 8 5, , , , ...

e) Find the 5th

term in the sequence 5 5 10

2 3 9, , , ...


9-8 continued

3) What is a formula for the nth term of the sequence shown? {2, 8, 32, 128, …}

4) Which of the following statements about the following sequence is false?

a) The sequence is geometric b)

13

22

n

na

c) The ratio is d) 5

81

8a

5) What is a ratio for the geometric sequence? 3 5 2 7 32 10 20 40x, x y, x y , x y , ...

6) A geometric sequence has a fourth term of 125, and a fifth term of 625, determine:

a) the ratio b) the first term c) the 3rd

term

7) A geometric sequence has a second term of 15, and a fourth term of 135, determine:

a) the ratio b) the first term c) the 7th

term

8) A geometric sequence has a second term of 10 , and a fifth term of 80 , determine:

a) the ratio b) the first term c) the 4th

term

11

a) 4 b) 4 2 c) 4 2 d) 2 42

n n nn

9 272 3

2 4

, , , , ...

2

3

22 2 22

a) b) 2 c) d) 2x

x y xy x yy



Homework 9: Recursive Sequences (A-CED.1, F-IF.3, F-BF.1)

Recursive Sequences - Equations that have a starting term, but every term in the sequence is based on the

previous term. You are given either:

The starting value of 1a

The recursive equation for na as a function of

1na ( 1na is the previous term)

The starting value of 1a

The recursive equation for 1na as a function

of na ( na is the previous term)

Recursive Sequences can be arithmetic, geometric, or something else completely.

Example: The sequence 3 7 11 15 19, , , , , ... is an arithmetic sequence, but can be expressed as the

recursive sequence: 1

1

3

4n n

a

a a

Read this sequence as the first term is 3 and each term is the previous

plus 4.

Recursive sequences can also be expressed in function notation where 1 1na f n , a f

1 and 1na f n

1) Express each of the following sequences as a recursive sequence:

a) 3 5 7 9, , , , ... b) 2 8 32 128, , , , ... c) 0 7 14 21, , , , ...

1 1 1 1d)

2 6 18 54, , , , ... e) 1 5 25 125, , , , ... f) 2 4 16 256, , , , ...

g) 5 6 7 8x, x, x, x, ... h) 2 3 6 9 18 27 54x , x , x , x , ...


HW 9-9 continued

2) For each recursive sequence given, determine the first four terms:

1 1a) 4 and 3n na a a 1 1b) 2 and 3n na a a

c) 1 0 and 1 2f f n f n d) 1 5 and 4 1f f n f n

2

1 1e) 2 and 1n na a a f) 1 1 and 1 2 5f f n f n

3) Given the recursive sequence 1 15 and 4n na a a ,

a) Determine and state the first five terms of the sequence.

b) Determine an explicit arithmetic formula to represent the sequence.

c) Determine and state the value of 12a


HW 9-9 continued

4) Given the recursive sequence 1 12 and 5 nna a a ,

a) Determine and state the first five terms of the sequence.

b) Determine an explicit geometric formula to represent the sequence.

c) Determine and state the value of 12a

5) If 1 5 and 3 1 then 4

a) 8

b) 14

c) 135

d) 17

f f n f n , f

6) If 1 2 and 4 1 then 3

a) 32

b) 32

c) 128

d) 10

f f n f n , f


HW 9-9 continued

7) Which recursively defined function has a first term equal to 7 and a common difference of 3

a) 1 3 and 1 7 c) 1 7 and 1 3

b) 1 7 and 3 1 d) 1 3 and 7 1

f f n f n f f n f n

f f n f n f f n f n

8) Which recursively defined function represents the sequence 3, 5, 9, 17, ...?

a) 1 3 and 1 4

b) 1 3 and 1 2 1

c) 1 3 and 1 3 4

d) 1 3 and 1 2 1

f f n f n

f f n f n

f f n f n

f f n f n

9) A pattern is shown below

Which of the following is false about the above sequence?

a) The arithmetic explicit formula that represents the sequence is 2 1na n

b) The recursive formula that represents the sequence is 1 11 and 2n na a a

c) The sixth term in the sequence will have 9 hexagons

d) The common difference is 2




1) Which of the following is an example of exponential decay?

2) For the function 5 2 , determine which interval will have the greatest averagex

h x rate of change?

a) 0,1 b) 1,2 c) 0,2 ) [ 1,0]d

3) Given the table below:

a) Determine if the function is exponential or linear. Explain your answer.

b) Create an equation to represent the function.

4) Create an equation to represent the value of a saving account after t years with an initial investment of

$375 and an annual interest rate of 2.5%. Use this equation to determine the value of the account after 15

years if no other deposits or withdrawals are made.

5) A $55,000 car loses value at an annual rate of 15%. Create an equation to represent the present value

of the car after t years. Use this equation to determine the value of the car after 3 years.

6) Given the function 2000 0.83 ,t

A t which of the following is false?

a) The initial value is 2,000 c) The function has a decay rate of 83%

b) The function has a domain on fall real numbers d) The function will never become negative

a) 2 b) 0.5 c) 1 .013 d) 5 0.87

x x x x

f x f x f x f x

1

7) Given the functions 5 and g 1, which value of makes ?2

a) 4 b) 3 c) 4 d) 6

f x x x x x f x g x


9-10 continued

8) Which of the following describes a step function?

a) Jordan runs twice the distance each week.

b) Meghan earns $35 an hour at her job.

c) The cost of parking if parking meters only accept

quarters and charge 25 cents per hour.

d) The height of a ball thrown into the air over time.

9) Graph the given piecewise function below on the given set of axes.

10) At which value of x does the function ,w x become

discontinuous?

11) What is a formula for the nth term of the sequence shown? {3, 7, 11, 15, …}

12) Which of the following statements about the following recursive sequence is false?

a) The sequence is geometric b) 1

The explicit formula is 2 3n

f n

c) The fifth term will be positive and d) The third fourth term is 54

the sixth term will be negative

13) The common difference of the arithmetic sequence 2 2

a) b) c) d)

x y, x, y, x y, ... is

x y x y xy y x

For the given sequence 5 1 3 7 ..., , , ,

14) Write a recursive formula that represents the sequence.

15) Determine and state the 100th

term.

3 1, 0

2, 0 2,

22 3,

x x

w x x

xx

a) 3 4 b) 4 7 c) 4 1 d) 1 4

n n n n

1 2 and 3 1 f f n f n




21) If 3 2 which statement is true?

a) 1 2 b) 1 6 c) 2 3 d) 2 13

f x x ,

f f f f

2) What is the solution to 3 7 4 3?

a) 10 b) 10

10c) 10 d)

7

h h

h h

h h

23) Which expression is equivalent to 16 64

a) 4 8 4 8 c) 16 2 2

b) 4 2 2 d) 8 2 4 2 4

x

x x x x

x x x x

2

3 2 3 2

3 2 3 2

4) What is the product of 3 2 and 2 5?

a) 6 7 17 10 c) 6 7 17 10

b) 6 7 17 10 d) 6 7 17 10

x x x

x x x x x x

x x x x x x

5) Joshua bought x boxes of cookies to bring to a party. Each box contains 20 cookies. He decides to

keep two boxes for himself. He brings 100 cookies to the party. Which equation can be used to find

the number of boxes, x, Joshua bought?

26) What are the solutions to the equation 3 13 10?

3 3 2 2a) and 5 b) and 5 c) and 5 d) and 5

2 2 3 3

x x

a) 20 20 100 c) 40 100

b) 40 20 100 d) 20 40 100

x x

x x


9-11 continued

8) Is the sum of 3 5 and 2 5 rational or irrational? Explain your answer.

9) Solve the equation below for in terms of

7 12 2

x a.

ax a

10) The school book store sells pens and pencils. If Mary buys two pens and three pencils for $4.00 and

Jane buys four pencils and one pen for $3.25. Determine and state the cost of purchasing one pen and

one pencil together.

2

2

2

2

2

7) The function 3 12 8 can be written in vertex form as

a) 3 2 4

b) 3 2 4

c) 3 2 4

d) 3 6 28

f x x x

f x x

f x x

f x x

f x x


Unit 10 – Statistics

Homework 1: Center and Spread (S-ID.2)

Notes: Mean (Average) is the sum of all the data divided by the total number of data values. Median

(Middle) is the middle number after the data is ordered from least to greatest. If a set of data has two

middle numbers, the median is the mean of the two middle numbers. Mode (Most) is the most often

occurring number in the data. It is possible to have more than one mode. If finding mean, median, and

mode without a calculator, make sure to first arrange the data in order from least to greatest.

1) Determine the mean, median, and mode, to the nearest integer, for each of the following data sets:

a) 87, 98, 85, 90, 98, 78, 93, 87, 76, 98 b) 23, 25, 12, 25, 15, 20, 18

c) 90, 88, 94, 95, 81 d) 25, 28, 15, 32, 27, 23, 28, 22, 30, 21, 28, 15, 20

2) Find the mean, median and mode of the data presented in the given frequency table.

3) Which of the following statements is true about the data 25, 37, 45, 40, 37, 39?

a) median = mode c) mean < median

b) mean > median d) mode > mean

4) Using the data in the given frequency table, which of the following statements is true about the data?

a) median = mode c) mean < median

b) mean > median d) mode > mean

xi fi

65 8

75 4

85 5

95 3


HW 10-2 continued

5) If the mean of a set of data is 32 and the data includes 30, 35, 28, 25, 32, and x. Find the value of x.

6) If the median of a set of data is 50 and the data includes 25, 67, 75, 48, and x. Find the value of x.

7) The table below provides the average retail price (cents per kilowatt-hour) to residential customers

of the New England states in October of 2009.

a) Compute the mean and the median.

b) If you wanted to describe a typical price for electricity, would you use the mean or the median?

Explain your choice.

8) Using the scores 47, 45, 33, 67, 47, 55, 42 and x.

a) What is the value of x if it is the median of the data?

b) What is the value of x if it is the mean of the data?


HW 10-1 continued

Notes: The Standard Deviation measures how much variation or how spread out the data is relative to

the mean. A high standard deviation indicates a wide spread out data set. A low standard deviation

indicates the data is closely grouped to the mean. Standard deviation is a very useful way of comparing

two sets of data with similar central measurements but different spread.

9) Both of the following sets of data have the same mean. Without actually calculating the mean or the

standard deviation, which set would appear to have the smaller standard deviation? Explain why.

Set A: 40, 50, 55, 60, 45 Set B: 20, 85, 55, 50, 40

10) If Mr. Generous gives an additional 5-point bonus on a particular test:

a) What change (if any) would occur to the mean of that test?

b) What change (if any) would occur to the median?

c) What change (if any) would occur to the standard deviation?

11) Tanner and Robbie discovered that the means of their grades for the first semester in Mrs. Merrell’s

mathematics class are identical. They also noticed that the standard deviation of Tanner's scores is 20.7,

while the standard deviation of Robbie's scores is 2.7. Which statement must be true?

a) In general, Robbie's grades are lower than Tanner's grades

b) Robbie's grades are more consistent than Tanner's grades.

c) Robbie had more failing grades during the semester than Tanner had.

d) The median for Robbie's grades is lower than the median for Tanner's grades.


HW 10-1 continued

12) The following table shows Bart’s test grades in Mathematics for the year:

89, 92, 96, 77, 83, 86, 90, 89, 79, 86, 89

a) Determine the mean, median, mode, and standard deviation to the nearest integer.

b) Bart has another test on Friday. If he gets another grade of 92, determine which of the mean, median,

mode, and standard deviation would increase, decrease, or stay the same.

c) Bart believes if he does really well on his Friday test, he could raise his average by 2 points. Friday’s

test is a standard 100-point test. Is Bart correct, if so what grade would he need? Explain your

answer.

13) The two data sets below represent the number of goals scored by two soccer teams.

Team A: 1, 1, 4, 7, 2, 2, 3

Team B: 5, 2, 1, 5, 3, 4, 1

What set of statements about the mean and standard deviation is true?

a) mean A > mean B

standard deviation A > standard deviation B

b) mean A < mean B

standard deviation A < standard deviation B

c) mean A > mean B

standard deviation A < standard deviation B

d) mean A < mean B

standard deviation A > standard deviation B



Homework 2: Quartiles (S-ID.1)

Notes: Range is the difference between the maximum and minimum data values. The median (second

quartile) splits data into two equal halves, Quartiles splits the data into four equal parts. The

Interquartile Range is the difference between the third and first quartiles. The First Quartile is the

median of the lower half of the data and the Third Quartile is the median of the upper half.

Example: The data set {1, 2, 3, 4, 4, 6, 7, 7, 8, 10} has a minimum of 1 and maximum of 10, therefore the

range max – min is 9. The set has two middle numbers so the median is the mean of 4 and 6 which is 5.

Splitting the set in half we see 1,2,3,4,4 6,7,7,8,10 The first quartile is 3 and the third quartile is 7.

The interquartile range, Q3 – Q1 is 4.

The min, first quartile, median, third quartile, and max make up what is known as the Five Statistical

Summary numbers.

1) For each given data set, determine to the nearest tenth, the five statistical summary numbers, range,

and interquartile range.

a) 5, 10, 10, 12, 15, 18, 19, 20, 22, 25, 28 b) 123, 125, 112, 125, 115, 120, 118

c) 90, 88, 94, 95, 81, 90, 89 d) 25, 28, 15, 32, 27, 23, 28, 22, 30, 21, 28, 15

The Box and Whisker Plot (Box Plot) is a graphic representation of the five statistical summary

numbers. Using the data from the previous example

To draw a box and whisker plot you first need to draw a number line appropriate for the data

1 31 3 5 7 10min , Q , med , Q , max

Place dots above the number line for the five statistical summary numbers.

Draw a box with ends through the first and third quartiles.

A vertical line in the box for the median, and horizontal lines

from the box to the min and max.

Each of the four sections of the graph contain 25% of the data. Box and whisker plots are useful for

determining if data is symmetrical or skewed.

Symmetrical Skewed Left Skewed Right


HW 10-2 continued

2) The test scores from Mrs. Gray’s math class are shown below.

Construct a box-and-whisker plot to display these data.

3) Using the line provided, construct a box-and-whisker plot for the 12 scores below.

4) The data below are the calories in an ice cream bar.

a) Find Q1, median and Q3

b) Draw the box and whisker plot.

c) What is the interquartile range (IQR) for this distribution? What percent of the ice cream bars fall

within this interval?

d) Do you think the data distribution represented by the box plot is a skewed distribution? Why or why

not?

e) Estimate the typical number of calories in an ice cream bar. Explain why you chose this value.

72, 73, 66, 71, 82, 85, 95, 85, 86, 89, 91, 92

26, 32, 19, 65, 57, 16, 28, 42, 40, 21, 38, 10

342 377 319 353 295 234 294 286 377 1 82 310

439 1 11 201 1 82 1 97 209 1 47 1 90 1 51 1 31 1 51


HW 10-2 continued

5) The data set 5, 6, 7, 8, 9, 9, 9, 10, 12, 14, 17, 17, 18, 19, 19 represents the number of hours spent on the

Internet in a week by students in a mathematics class. Which box-and-whisker plot represents the

data?

a) c)

b) d)

6) Given the box and whisker plot below:

a) What is the value of the third quartile shown on the box-and-whisker plot below?

b) What is the interquartile range? c) What percent of the data is contained in the interval [6,12]?

7) The box-and-whisker plot below represents 20 students' scores on a recent English test.

a) What is the value of the upper quartile?

b) What do you think the box plot tells us about the students’ 20 scores on the English test?

c) Why might understanding the data behind this graph be important?


HW 10-2 continued

8) The box-and-whisker plot below represents the math test scores of 20 students.

a) What percentage of the test scores are less than 72?

b) How many of the 20 students scored below or equal to an 88?

c) Which interval contains the most test scores from this data: 60s, 70s, 80s, or 90s?

e) Which of the following cannot be determined from the box plot: {Range, Mean, Median, Mode}

9) Box plots A and B are shown below:

a) Which plot how the larger mean? b) Which plot has the larger interquartile range?

c) Which plot has the smaller standard deviation? Explain your answer.



Homework 3: Tables and Plots (S-ID.1,3)

Notes: Some common graphical representations of data, other than box and whisker plots, include Dot

Plots, Frequency Histograms, and Cumulative Frequency Histograms.

Example: Given the data set A = {0,0,1,2,2,2,4,5,5,5,5,6,7,7,7}

In a dot plot each data value is represented by A Dot Plot for the set A data

a dot. This a good visual way to see every value of the data.

It is also possible to see any missing values such as the

value 3 in this example.

Histograms divide the data into intervals and are usually produced from frequency tables as shown

below. It is not possible to see specific data values in a histogram, for example it is not possible to see

that there is no value 3 in this example. Histograms are a good way to visualize large amounts of data

A Frequency Histogram for set A data

Cumulative frequency histograms display the running total of the frequencies. The data can be displayed

in a cumulative frequency table. Cumulative Frequency Histogram for set A Data


10-3 continued

1) The accompanying histogram shows the heights of the students in Kyra’s health class.

a) What is the total number of students in the class?

b) How many students are shorter than 190 cm?

c) Which height interval contains the median of the

data?

d) Would you describe this histogram as representing a symmetrical or a skewed distribution? Explain

your answer.

2) The following cumulative frequency histogram shows the distances swimmers completed in a recent

swim test.

a) Based on the cumulative frequency histogram,

determine the number of swimmers who swam

between 200 and 249 yards.

b) Determine the number of swimmers who swam

between 150 and 199 yards.

c) Determine the number of swimmers who took the

swim test.

d) Would you describe this histogram as representing a

symmetrical or a skewed distribution? Explain your answer.


10-3 continued

3) Ms. Hopkins recorded her students' final exam scores in the frequency table below.

a) On the grid below, construct a frequency histogram based on the table.

b) Would you describe your graph as symmetrical or skewed? Explain your choice.

4) The following set of data represents the scores on

a mathematics quiz:

Complete the frequency table below and, on the accompanying grid, draw and label a frequency histogram of these

scores.

58, 79, 81, 99, 68, 92, 76, 84, 53, 57,

81, 91, 77, 50, 65, 57, 51, 72, 84, 89


10-3 continued

5) Jennifer asked members of her class which kind of movie they liked the best. The results are in the

table below.

a) Create a Dot Plot for the data above.

b) What do you think this graph is telling us about the class’s favorite movie?

c) Would you describe this dot plot as representing a symmetric or a skewed data distribution?

6) A sample of 20 colleges and universities with the following class sizes are shown below.

a) Create a Dot Plot for the data above.

b) What do you think this graph is telling us about the class size in most colleges?

c) Who might be interested in this data distribution?

14 20 20 20 20 23 25 30 30 30

35 35 35 40 40 42 50 50 80 80



Homework 4: Two Way Tables (S-ID.5)

Notes: Box Plots, Histograms, and Dot Plots are all ways to show one variable statistics, in each case

the data could also be shown in a frequency table. When two variables are being studied for a possible

relationship between them, a Two-Way Table (Two Way Frequency Table) can be used. A two-way

frequency table uses rows to one variable with totals on the right and columns for the second variable

with totals on the bottom. The sum of the row which equals the sum of the columns is the total number of

data values.

Example: A survey was conducted which asked 50 participants if they prefer vanilla or chocolate ice

cream. The results separated into gender are shown below.

Percent of students that are female

and prefer vanilla

714 100 14

50. %

The table shows gender as rows with 28 males, 22 females, and flavor as columns with 26 chocolates and

24 vanillas. The intersection of row and columns, such as female and vanilla is 7, can be read several

different ways: 7 out of 50 (14%) participants are female and prefer vanilla, 7 out of 22 (about 32%)

females prefer vanilla, or 7 out of 24 (about 29%) vanilla fans are female.

Sometimes two way tables are shown without the totals, in

these cases you will need to determine the totals by adding up

the values in each row and column before answering questions

about the data. The same two-way table as above is shown

below without totals.

1) A television network conducted a survey where they asked elementary, middle, and high school

students which type of television programs they prefer to watch. The results are shown below.

a) How many students took the survey?

b) What percent of the students who took the survey

are in middle school?

c) What percent of the high school students who took the survey prefer comedy?


10-4 continued

2) A random sample of 284 students was asked to evaluate a professor’s performance. The students were

also asked to supply their midterm grade.

Teacher evaluation

A B C

Positive 35 33 28

Neutral 25 46 35

Negative 20 22 40

a) What percent of students surveyed had an A?

b) What percent of students surveyed who received a B gave a negative rating?

c) Based on this sample, if 500 students take this professor next year, how many of those 500

students will rate the professor positive?

3) Students in a high school statistics class observed the students in one lunch period to determine if they

brought lunch from home or purchased lunch from the school cafeteria. Their results are shown below.

a) What percent of students observed purchased lunch?

b) What percent of students observed were juniors who purchased lunch?

c) What percent of sophomores observed brought lunch?

d) If a student were chosen at random, would it be more likely the student was a freshman or a senior?



Homework 5: Linear Correlation (S-ID.6,7,8,9, A-CED.1, A-REI.3)

Notes: A Scatter Plot is a type of graph that plots points (coordinates) from two variables to determine if

and what type of relationship exists between the two variables. Correlation measures the strength of the

relationship. The Correlation Coefficient (r) is a numerical value for correlation: 1 1r . A positive

correlation indicates a positively sloped trend that will have a positive r and the closer to 1 the stronger

the correlation. A negative correlation indicates a negatively sloped trend that will have a negative r

and the closer to 1 the stronger the correlation. No correlation will have an r value close to 0.

Strong Positive Weak Positive Strong Negative Weak Negative No Correlation

0 9r . 0 7r . 0 9r . 0 7r . 0 2r .

Since the points correlate in a linear fashion, a Linear Regression Equation or Line of Best Fit can be

found to represent the data relationship as an equation. The line of best fit will not go through every

point on the scatter plot (unless the 1 or r 1 r ), but because it is an approximate representation of

the data, it can be used to predict data points that weren’t given. Shown below are scatter plots with the

line of best fit drawn.

The correlation coefficient and line of best fit can be found from data by using a graphing calculator.

Not all scatter plots are best represented by linear equations. Other equation types (regressions) can be

used to represent bi variate data (2 variable data). Some of these other types will be covered in

homework 7.


10-5 continued

1) Match the correlation description on the left with the correct scatter plot on the right.

a) Strong positive correlation

b) No correlation

c) Strong negative correlation

d) Weak positive correlation

2) What is the correlation coefficient of the linear fit of the data shown below, to the nearest hundredth?

a) 1.12 b) 0.97 c) 0.81 d) 0 95.


10-5 continued

3) Given below is data for 9 people’s height and shoe size.

a) Find the linear correlation coefficient for the relationship between height and shoe size as expressed in

the given table rounded to the nearest hundredth.

b) Create a scatter plot for the data on the axis

provided.

c) Find the equation of the line of best fit, with the

slope and y-intercept rounded to the nearest tenth.

d) Using your equation for the line of best fit,

predict the shoe size ,to the nearest integer, of

someone with a height of 72

The results of a study comparing a person’s age and the number of text messages they send on a daily

basis is shown.

4) Which statement(s) can correctly be concluded?

I. There is a strong correlation shown.

II. There is a negative correlation shown.

III. Getting older causes a person to text less.

a) I only c) I and II

b) II only d) I, II, and III

5) If a linear model is applied to the above scatter plot for age/text messages sent per day, which

statement best describes the correlation coefficient?

a) It is close to 1 b) It is close to 0 c) It is close to 1 d) It is close to 0 5.

Height 60 61 62 63 64 65 66 67 68

Shoe Size 7 7 8 8 8.5 9 9 9.5 10


10-5 continued

6) A math teacher collected information for 7 randomly selected students in her class on how long they

studied for a test and compared it with the score they received on that test. The results are shown in

the accompanying table below.

a) Write the correlation coefficient for the line of best fit. Round your answer to the nearest hundredth.

b) Explain what the correlation coefficient suggests in the context of this problem?

c) Determine whether the data suggests a strong or weak association.

d) Write the equation for the line of best fit rounded to the nearest hundredth.

e) Use the line of best fit equation to predict how long a student studied if they received a 75 test grade.

Minutes

Studying

Test

Grade

15 83

27 85

6 63

13 82

25 92

30 95

21 93



Homework 6: Residuals (S-ID.6)

Notes: A Residual is the difference between the points on a scatter plot and the line of best fit (or curve).

Residuals are the distances from the points on the scatter plot to the line of best fit.

Example:

Scatter plot Line of best fit drawn Residuals shown

On the last graph with residuals shown, each original scatter plot point is shown with its vertical distance

to a point on the line of best fit. A few specific points are labeled A, B, C, and D for reference. Point A

is a distance of 1 unit above the line of best fit, point A has a residual of 1. Point B is two units below

the line so point B has a residual of 2 (residuals can be either positive or negative). Point C has a

residual of 0 since it is on the line and point D has a residual of 3.

A separate Residual Scatter Plot can be drawn with the residuals on the y-axis and the same x-axis as

the original scatter plot. Remember to think of this graph as the distance each point is above (positive) or

below (negative) the line of best fit.

When the line of best fit is a perfect fit like when found from

a calculator, the sum of the residuals equals zero.

Residual plots are a good way to determine if the line is a “good fit” for the data. When points are

randomly placed above and below the x-axis (like this example), this indicated an appropriate linear fit.

If the plots follow a pattern, a curve, or all to one side, this indicates the line is not a good fit for the data.

Good Fit Bad Fit Bad Fit


10-6 continued

1) The tables below represent residuals for two lines of best fit. Plot each residual on the provided set of

axes below, assess the fit of the lines for their residuals, and justify your answers.

a)

b)

2) After performing analyses on a set of data, Jamie examined the scatter plot of the residual values for

each analysis. Which scatter plot indicates the best linear fit for the data?

a) c)

b) d)


3) For the given table below: a) Draw a scatter plot on the set of axes.

b) Barbara thinks that the equation 2 2y x would be a good fit for the above data. Draw the equation

on the graph above. Do you agree it is a good fit? Justify your answer using information from the

scatter plot.

c) Complete the table below using Barbara’s predicted linear model to determine the predicted y-values

from the given x-values and graph the residual scatter plot on the set of axes provided below.

d) Based on the residual scatter plot, do you agree with Joshua that 2 2y x is a good fit for this data?

Explain why using the residual scatter plot.


10-6 continued

Notes: Not all regression models have to be linear and residuals are a way to test other types of models

to determine if a model is or is not a good fit.

4) For the given table below: a) Draw a scatter plot on the set of axes given.

b) Jonathan thinks that the equation 1

22

xy would be a good fit for the above data. Draw the equation

on the graph above. Do you agree it is a good fit? Explain why using the scatter plot above.

c) Complete the table below using Jonathan’s predicted exponential model to determine the predicted y-

values from the given x-values and graph the residual scatter plot on the set of axes provided below.

d) Based on the residual scatter plot, do you agree with Joshua that 1

22

xy is a good fit for this data?

Explain why using the residual scatter plot.



Homework 7: Regression Models (S-ID.6, A-CED.1, A-REI.3)

Notes: Some bivariate data is better described by nonlinear regressions (models) such as exponential

and quadratic. Residuals can be used to determine if a model is a good fit but most graphing calculators

also have utilities for this function. It is very helpful to first recognize the type of model the data

represents from the scatter plot.

1) For each scatter plot below, determine if a linear, exponential, or quadratic model best describes the

data.

a) b) c)

d) e) f)

2) Given the data below:

a) Graph the data on the given set of axes

b) Using a graphing calculator determine a linear regression

and an exponential regression equation, to the nearest tenth,

to represent the data.

c) Looking at the correlation coefficient, determine which model is a better fit.


10-7 continued

3) According to basic economics, if the demand for a product increases, then the price will decrease.

The following chart shows the number of items requested and the corresponding price.

a) Draw the scatter plot

b) What type of model (linear, quadratic, or exponential) would you use to describe the

relationship between demand and the price of the items?

c) Using a graphing calculator determine a linear regression and an exponential regression equation, to

the nearest tenth, to represent the data.

d) Determine and state the model which best fits the data.

e) Using the best fit model equation, what price would you predict for a demand of 30 items?

Demand Price

1 $105

4 $92

7 $80

9 $70

12 $60

14 $55




1) The scatter plot below shows the profit, by month, for a

new company for the first year of operation. Kate drew a line

of best fit, as shown in the diagram.

Using this line, what is the best estimate for profit in the 18th

month?

a) $35,000

b) $42,500

c) $37,750

d) $45,000

2) The residuals for a set of data represent the:

a) differences between consecutive x-values

b) vertical differences between data points and the line of best fit

c) data points that lie above the line of best fit

d) data points that lie below the line of best fit

3) The graph shows the residuals for a set of data with respect to a line of best fit. How could the line be

adjusted to improve the fit?

a) increase the slope of the line

b) decrease the slope of the line

c) increase the y-intercept of the line

d) do not adjust the line it is the best fit line


10-8 continued

4) The hours that John worked over the past 12 weeks are recorded in the table. For these hours find (to

the nearest tenth).

a) the mean

b) the median

c) the mode

d) the standard deviation

e) the range

f) the interquartile range

5) Megan and Bryce opened a new store called the Donut Pit. Their goal is to reach a profit of $20,000

in their 18th month of business. The table and scatter plot below represent the profit, P, in thousands of

dollars that they made during the first 12 months. Determine a linear regression equation to fit the data,

rounding values to the nearest tenth. Using this line of best fit, predict whether Megan and Bryce will

reach their goal in the 18th month of their business. Justify your answer.

Hours Frequency

30 1

35 2

37 5

40 3

42 1


10-8 continued

6) The following table compares the wing length (cm) of a particular species of bird,

with the age (in days) of the bird.

a) Find the correlation coefficient to the nearest hundredth.

Is this considered a high or low correlation?

b) Find the equation of the line of best fit, rounding all values to the

nearest tenth.

c) Use this equation to estimate, to the nearest thousandth, the wing

length of a bird that’s been alive for 24 days.

7) The data below represents the length and diameter of a particular bone of a certain animal.

a) Determine and state an exponential regression equation

for the data, rounded values to the nearest tenth.

b) Determine and state the length of a bone, to the nearest

millimeter if its diameter is 75 millimeters.

8) The table below represents a projection of future sales in the thousands

of dollars by months in business.

What type of model (linear, quadratic or exponential) would best describe

the relationship between months in business and sales? Justify your

response.

WING

LENGTH

(cm)

AGE

(days)

1.5 4.0

2.2 5.0

3.1 8.0

3.2 9.0

3.2 10.0

3.9 11.0

4.1 12.0

4.7 14.0

DIAMETER

(mm)

LENGTH

(mm)

17.6 159.4

26.0 206.2

31.9 236.4

38.9 269.7

45.8 300.5

51.4 324.1

58.5 352.2

64.3 376.9


10-8 continued

9) Given the box-and-whisker plot below:

a) What is the interquartile range? b) What is the value of the median?

c) Which of the following is false?

i) the first quartile is 30

ii) the range is 40

iii) there is a larger percentage of date from the minimum to the first quartile then from the third

quartile to the max

iv) half of the data is less than or equal to 30 and greater than or equal to 50

10) A survey was conducted which asked people what their favorite season of the year was. The results

separated by gender are displayed in the table below.

a) To the nearest percent, what percent of the responses favor the summer as their favorite season?

b) To the nearest percent, what percent of males favor the fall as their favorite season?

c) To the nearest percent, what percent of all the responses were females who favor spring?

d) If participant were chosen at random would it be more likely that they were a male who favored spring

or a female who favored fall?




2

2

1) Which expression is equivalent to 4 16?

a) 2 2 8 c) 2 4 2 4

b) 2 4 4 d) 4 2 2

not x

x x x

x x x x

2) Which expression is equivalent to 5 2 3 12 4 ?

a) 2 19 c) 22 19

b) 2 11 d) 2 19

a a

a a

a a

3) A family leaves home in their car on a road trip. The equation 450 55 can be used to represent

the distance, from their destination after hours. In this equation, the 55 represents the

D t

D, t

a) car's distance from the destination

b) speed of the car

c) distance between home and the destination

d) number of hours driving

2

4) What is the value of the function ?

a) b) c) d)

minimum y x h k

h k h k

5) When the function is multiplied by the value where 1 the graph of

the new function,

a) opens upward and is wider c) opens downward and is wider

b) opens upwar

f x x a, a ,

g x a x

d and is narrower d) opens downward and is narrower

2

2 2

2 2

6) Which equation is a correct step in the method of completing the square to

solve the equation 3 12 9 0

a) 2 13 c) 2 3

b) 2 1 d) 2 7

x x ?

x x

x x


10-9 continued

7) Caleb purchased four apples and five bananas from the farm stand for $1.50. Later

that day, Logan purchased six apples and four bananas from the same farm stand for $1.90.

Determine and state the cost per apple and banana from the farm stand.

8) Lucie is working on her math homework. She decides that the expression 20

5 is irrational

because both the numerator and denominator are irrational. Is Lucie correct? Explain your reasoning.

9) Given the table below which represents the time in a car and the distance traveled?

Time (hrs) 0 2 4 7

Distance (mi) 0 120 235 420

Determine the average rate of change between hour 2 and 7, including units.

10) Graph the inequality 3 7y x on the set of axes below.

State the coordinate of a point in its solution.


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