lakota west high school

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LAKOTA WEST HIGH SCHOOL

HONORS ALGEBRA II – EXPECTATIONS (2017-2018)

Upon entering Honors Algebra II class at Lakota West HS it will be expected that you to have an excellent

understanding of certain topics covered in CP Algebra I and Honors Geometry. We will not spend time in

Honors Algebra II relearning these topics; instead, we will use the properties to work on problems that are

more complicated. A TI-83 plus or TI-84 is recommended but not required for this assignment, however, it

is a requirement for the Honors Algebra II class. Using the graphing calculator on certain problems will

allow you to become more familiar with its potential before the first day of school.

It would be advisable to complete the packet during the summer and not wait until the very last minute,

because there will be homework problems assigned on the first day of class. You will also have a quiz on

Friday, August 18th over this packet.

See you next year and enjoy the summer. Mrs. Green and Mrs. DeRossett

Number Sense

You should be able to identify Properties of Addition and Multiplication and be familiar with the

various number systems.

Notation:

N: Stands for the set of natural (counting) numbers 1,2,3,........

W: Stands for the set of whole numbers 0,1,2,3,........

Z: Stands for the set of integers ...... 3, 2, 1,0,1,2,3.......

R: Stands for the set of real numbers (all #”s including rational and irrational)

: Stands for the element or element of

Rational numbers are those which can be written in the form a

b, where b 0.

Irrational numbers are those which cannot be written in the form a

b. Decimals that are non-ending and non-

repeating are also irrational.

The set of real numbers includes all rational and irrational numbers.

Know basic properties: Commutative Properties, Associative Properties, Distributive Property,

Additive/Multiplicative Identity and Inverses, Reflexive Property, Symmetric Property and Transitive

Property.

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Put a check mark for the number sets that contain each number.

State the property illustrated.

1. 7x 9x 8 7x 9x 8 1._____________________________

2. 7n 2n 7 2 n 2. _____________________________

3. 3x *2y 3*2*x * y 3. _____________________________

4. 1

*4y 1y4

4. _____________________________

5. 4n 0 4n 5. _____________________________

Order of Operations

PEMDAS

P = Parenthesis (grouping symbols – may be brackets, braces, etc.)

E = Exponents

M,D = Multiplication and Division from left to right

A,S = Addition and Subtraction from left to right

W Z Q I R

4

15

0

3

4

2

W-whole numbers

Z-integers

Q-rational numbers

I-irrational numbers

R-real numbers

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Ex a). 1

6 3*2

Ex b). 23 2 4*2 6 3 1

1

6 6

3 2 4*4 6 2

1

6*6

3 2 16 6 2

1 3 18 12

3 6 18

Simplify:

1. 22 3 3 4

2.

2

2

3 4 5

2 4 6

2 03. 9a + 3a 2 a 4 a 2

4. 2

4 8 6 4 * 3 2*8 4

Evaluating and Simplifying Expressions

You should be able to evaluate and simplify expressions following order of operations.

Evaluate each expression (#1-3) at the given values. Simplify each expression #4-6

1) 2x 5y when x 2 and y = 8 2) 2

2

3ab 1 when a 2 and b 2

3a b 1

3) 37x 8x when x 1 4) 9 x 3 4 x 1

5) 10 8 5 x 2 6) 2 3 x 4 10

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UNDERSTANDING THE MAIN IDEAS

Laws of exponents *Same bases

1. x y x yb b b (Add the exponents.) 3 4 3 12 2 2 2 2 2 2 2

2. If x

x y

y

bb 0, b

b

(Subtract the exponents.) 4

1 4 3

3

2 2 2 2 22 2 2

2 2 2 2

3. x yb b if and only if x = y If x 72 2 , then x = 7

provided b 0, 1, or -1 (Set the exponents equal)

*Same exponents

1. x x xab a b

3 3 3 32 5 2 5 or 10 8 125

2. If

x x

x

a ab 0,

b b

3 33

3

5 5 125or 2.5

2 2 8

3. x xa b if and only if a b If 5 52t 7 , then 2t 7

(provided x 0,a 0, and b 0 )

*Power of a power: y

x xyb b 2

5 5 2 103 3 3

*Zero exponent: If b 0, then 0b 1 0

3x 1

*Negative exponent:

If x 0 and b 0, then x

x

1b

b

5

5

1 12

2 32

*Rational exponent ( b 0;b 1;p,q integers; q 0 ) 1

aab b 1

5 55532 32 2 2

p

pa paab b b

44

45532 32 2 16

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Exponents

You should be able to simplify using the exponent laws:

Examples:

a). 2 5 72x 3x y 6x y b). 3

5 2 3 15 6 93x y z 27x y z

c). 2 2 3 2 2

4 4 2

2x 9xy 18x y 6x

3x y 3xy y d).

2 2 22 4 2 4 3 6

1 3 2 4 4 8

2xy z 2x z 3y 9y

3x y 3y 2x z 4x z

Practice Problems

1. 26 2. 2

2 3 35a b c

3. 2 5

6

3x y

12x

4. 3 5 2

2 5

15x y z

3x y z

5. 2 3

3 3 24xy 2x y

6. 0

abcdef

7.

23

4

4

3

8. 22 29x y 4xy

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Operations on Polynomials

You should be able to add, subtract, and multiply polynomial expressions.

Examples:

a. 2 3 2 3 23x 2x 5 5x 6x 7x 9 5x 9x 5x 4

b. 2 24x 9 3x 2x 3 3x 2x 6

c. 2 22x 3 4x 2 8x 4x 12x 6 8x 8x 6

d. 2 22x 5 2x 5 2x 5 4x 20x 25

Please note, letter “d” is a binomial square whose product is always a trinomial square. Paying close

attention that it is NOT 24x 25 , the exponent rule does not apply over addition or subtraction.

Recognizing these two patterns will be very helpful to you throughout the higher levels of mathematics.

Practice Problems:

1. 2 2 25c 3c 9 3c 4c 2 7c 2c 1 2. 2

8n 3

3. 24P 2 3P 5 4. 3

x 1

5. 22x 3 4x 6x 9 6. 2d 9 3d 9

Solving Equations and Absolute Value Problems

You should be able to solve any linear equations, including absolute value equations

Examples:

a). b).

15 1 3 2

15 15 3 6

18 9

1

2

x x

x x

x

x

2 2 2 1

2 2 2 2

0

x x x

x x x

x

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Absolute Value Examples continued:

c). 4 6 2x d). 6 2 5 0x x

4 6 2 or 4 6 2

6 2 or 6 6

1 or 1

3

x x

x x

x x

6 0 and 2 5 0

6 and 2 5

5

2

x x

x x

x

Practice Problems - Solve:

1. 2 x 4 12 2. 1 3 3

y4 8 16 3.

2 34x 2 3x 8

3 4

4. 3x 5 8 5. 2 5x 1 6 6. 7 2x 13

7. x 5 x 4 0 8. 2x 3 5x 4 0

9. P = 2L + 2W, Solve for W 10. Q P Pr t, for P

11. x2 16 12. 1

3x 3

Inequalities

You should be able to solve linear inequalities and be able to graph them on a number line.

Examples:

a. 2x 3 7 b. 3x 5 5x 13

2x 4 8x 8

x 2 x 1

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Practice Problems

Solve and graph on a number line

1. x 5 7 2. x 3 2x 4 3. 2x 10 7 x 1

Compound Inequalities You should be able to solve and graph compound inequalities and absolute value inequalities

Disjunction is 2 or more statements that are joined by the word OR to make a compound statement.

Conjunction is 2 or more statements that are joined by the word AND to make a compound statement.

Absolute Value: If ax b c , then c ax b c (conjunction)

If ax b c , then ax b c or ax b c (disjunction)

Examples: Conjunction: Disjunction: Absolute Value:

a.

6 2x 2 8

4 2x 10

2 x 5

b.

3x 1 4 or 2x 5>7

3x<3 or 2x >12

x<1 or x>6

c.

5 2 3

3 5 2 3

5 5 1

11

5

x

x

x

x

Practice Problems: Solve and graph the following problems.

1. 10 6 2x<8 2. 8 1 3 x 2 13 3. 6 2x 20 or 8 x 0

4. 2x 3 5 or x 4 4 5. 3x 2 6 6. x 7 13 4

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a

b

c

Equations of Lines

Vocabulary:

Ordered pairs x, y

Vertical line x #

Horizontal line y #

Slope of a line – rate of change

Slope of a vertical line is undefined. Slope of horizontal line is 0

The slope, m, of the non-vertical line passing through the points

1, 1x y and 2, 2x y is found by 2 1

2 1

y ym

x x

Parallel lines have the same slope. 1 2m m

Perpendicular lines have slopes that are negative reciprocals of each other. 1 2m m 1

Slope intercept form of a line: y mx b (m slope, b y intercept)

Point-slope formula: 11 1

1

y - yy y m x x or m

x - x .

Standard form: Ax By C where A, B, and C are integers.

(no fractions or decimals)

Graph linear equations using an x-y chart, x-y intercepts or slope intercept form.

Slope

You should be able to find the slope of a line given the graph.

A line which is increasing from left to right has a positive slope and a line decreasing from left to

right has a negative slope.

Example: Given the points 3, 2 and 4, 9 Solution: 9 2 7

m 14 3 7

Find the slope of the following lines.

am

bm

cm

Practice Problems:

1. 5, 4 3,9 2. 9, 8 9, 3 3. 4,7 4,3

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Cartesian Coordinate Plane – Know the axes, and quadrants (use Roman Numerals)

Plot the following ordered pairs and determine which quadrant or axis the point lies on.

Equation of Lines - You should be very familiar with Slope-Intercept Form, Standard Form,

Point-Slope Formula and be able to use all interchangeably.

Slope intercept form is y mx b . The m is the slope and b is the y intercept written as 0,b

Example: Find the slope and the y intercept of the line whose equation is y 2x 3

2m , y-

1 intercept 0, 3

Find the slope and y intercept of the following lines.

1. y 5x 6 2. y 0 3. x 3 4. 2y 4 6x

m=_________ m=_________ m=_________ m=_________

y-int. =_________ y-int. =_________ y-int. =________ y-int. =_________

Standard form

The standard form of a linear equation is Ax + By = C, where A,B,C, are all integers (no fractions or

decimals) and A is positive

Example: Put the following equation in standard form.

8x 10 5y 8x 5y 10 , therefore A 8, B 5 and C = 10

1. A -3,4

2. B 0,8

3. C -5, - 6

4. D 7,0

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Put the following equations in standard form and identify A, B, and C.

1. 4x 8 y 2. 3

6 3x y2

3. 3 2x 8 6y

________________ ________________ _______________

A = _________ A = _________ A = _________

B = _________ B = _________ B = _________

C = _________ C = _________ C = _________

4. Use Point Slope Formula to write the equation of a line that passes through the points

2,3 and 1, 4 . Put the equation in Standard Form.

5. Use Point Slope Formula to write the equation of a line that passes through the points 3, 2

and parallel to 5 3 9x y . Put the equation in Slope-Intercept Form.

Graphing

You should be able to graph an equation of a line using various methods.

Graphing using an x-y chart

Graphing using x and y intercepts

Graphing using the slope – intercept form

Examples:

a. x y 6 some points that satisfy the equation are 3,3 , 2,4 , 6,0

b. 2x 5y 5 x intercept is 5

,02

; y intercept is 0, 1

c. 8x 2y 6 rewrite in slope – intercept form y 4x 3

Slope is -4 and y-intercept is 0, 3

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Graph using the x-y chart.

1. x 2y 8

2. 2y x 5

Graph using x and y – intercepts

The y-intercept is the point where the graph intersects the y axis. x 0 .

It is the ordered pair, 0, y

The x-intercept is the point where the graph intersects the x axis. y 0 .

It is the ordered pair, x,0

Example: Find the y – intercept and x – intercept of the following equation and graph.

y 2x 4

To find the y - intercept, substitute 0 in for x and solve.

y 2x 4 y 2 0 4 y 4 Therefore, the y intercept is 0,4

To find the x intercept, substitute 0 in for y and solve

y 2x 4 0 2x 4 4 2x

x 2 Therefore, the x-intercept is 2,0

X Y

X Y

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Find the x and y – intercepts and graph the equations.

1. 3x 4y 12 2. x 3 y

y intercept y intercept

x intercept x intercept

Graph using slope – intercept form.

Example: y 2x 3

2m

1 ; y intercept 0,3

Practice Problems:

1. y 7 x 2. 2x 9 3y

m _______ m _______

y intercept ______ y intercept ______

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Systems of Equations You should be able to solve two linear equations with two unknowns by using substitution method

and linear combination method. Example: Substitution Method:

2x 4y 3 0

Solve one of the equations for x or y and substitute into the other equation

5x y 10

y 5x 10 ; 2x 4 5x 10 3 0 ; 2x 20x 40 3 0 ; 22x 43 ; 43

x22

43y 5 10

22

;

5y

22 The intersection point then is

43 5,

22 22

Example: Linear Combination:

2x 4y 3

5x y 10

2x 4y 3 2x 4y 3 22x 43

4 5x y 10 20x 4y 40 43

x22

then solve for y

The intersection point then is 43 5

,22 22

Practice Problems:

Solve the following system of equations by using substitution method.

1. y x 3 2. 3x y 5

4x y 13 y 3x 2

Solve the following system of equations by using the linear combination method.

3. 5x 2y 7 4. y 3x 5

3x y 13 4y 12x 20

5. What different types of solutions can you have to a system of linear equations?

Multiply one or both equations by

what you need so that when you add

the equations together one of the

variables cancels.

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Graphing and Transformations

You should know the basic parent graphs for a linear, quadratic, absolute value,

cubic, square root and exponential graph. You should also be able to describe the

type of transformation the parent graph goes through and be able to graph it.

You should know the concepts of domain and range.

Transformation Appearance in Function Transformation of Point

Vertical Translation (up/down) f x f x d x, y x, y d

Horizontal Translation (left/right) f x f x c x, y x c, y

Vertical Stretch/Shrink If 1a , then the graph is stretched vertically

If 1a , then the graph is shrunk vertically

f x a f x x, y x,a y

Horizontal Stretch/Shrink If 1b , then the graph is stretched horizontally

If 1b , then the graph is shrunk horizontally

f x f b x

xx, y , y

b

Reflection over x-axis f x f x x, y x, y

Reflection over y-axis f x f x x, y x, y

General Graphing Form: y a f b x c d

Stretches/Shrinks and reflections are done first and then translations

Practice – Sketch the following parent graphs; State Domain and Range:

1. f x x 2. f x x 3. 2f x x

4. 3f x x 5. xf x 2 6. f x x

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Practice – Describe the transformation of each equation below for 3f x x

6. 3f x x 2 7. 3

f x x 3

8. 3f x x 7 9. 3

f x x 5

10. 3

f x x 11. 3f x x

12. 3f x 2x 13. 3

1f x x

2

14. Graph 2

f x 3 x 2 4

vertex:

domain:

range:

15. Graph f x x 4 3

vertex:

domain:

range:

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Factoring You should be able to factor and solve equations by factoring.

Vocabulary:

Factor

Binomial

Trinomial

Polynomial

Degree of a polynomial

Difference of Squares

Binomial Square

Trinomial Square

Greatest Common Factor

Prime

Examples: Factor:

a). 4x 10 2 2x 5 b). 2 4 2 23x 9x y 3x 1 3x y

c). 2x 3x 4 x 4 x 1 d). 2x 5x 6 x 2 x 3

e). 26x 19x 10 3x 2 2x 5 f). 2 9 3 34x 2x 2x

25 5 5

g). 2216 40 25 4 5x x x ; this is a Trinomial Square that factors into a Binomial Square

h). Solve: 2x 6x 16 0 x 8 x 2 0 x 8, x 2 x = 2, 8

Practice problems - Factor:

1. 2t 12t 27 2. 2x 8x 15 3. 2 49y 36x

4. 2 5 310x y 25xy 5. 29x 13x 4 6. 214y 15 8y

7. 28x 33x 4 8. 24z 5z 6 9. 3 24y 32y 132y

10. 29 30 25y y 11.

2 216 72 81x xy y 12. 4 636 16x y

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Solve.

13. 23x 4 x 14. 29y 81 0 15. 3 23x 30x 21x

Solving Quadratic Equations

You should be familiar with Factoring, Complete the Square and the Quadratic Formula:

Quadratic Formula (QF): 2 4

2

b b acx

a

Complete the Square (CTS): Example:

Practice problems - Solve:

1. Solve by CTS a). 2 4 1 0x x b) 23 5 10 0x x

2. Solve by QF: 2 6 4 0x x

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Radicals

You should be able to work with and simplify all types of radicals.

Simplify radicals

Add and Subtract radicals – remember you need like radicals

Multiply and divide radicals – including rationalizing the denominator

Rational exponents: m

k mka a

Simplify:

1. 16 2. 64 3. 40

4. 72 5. 2 49 6. 3 3

7. 2 5 3 5 8. 3 6 3 9. 2

2 7

10. 4

9 11.

5

7 12.

2

1 5

13.

2

38 14. 3

416

lakota west high school

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