1-1 Sets of Numbers - Esperanza Summer School Algebra 2 - …€¦ · · 2013-06-231-30 Holt...

Name ________________________________________ Date __________________ Class__________________

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1-6 Holt Algebra 2

Reteach Sets of Numbers

As you move from left to right on a number line, the numbers increase. Use a number line to help you order real numbers. Order from least to greatest:

111, 2.6, , , 2.354.2 2

π− −

Use a calculator to approximate 11 and 2π− as decimals:

11 ≈ 3.32 and 2π− ≈ −1.57.

Plot each point on a number line.

Read the numbers from left to right on the number line.

From least to greatest, the order is 12.6, , , 2.354, 11.2 2π− −

Order the given numbers from least to greatest. Use a number line to help you.

1. π, 1.9− , 223

, −0.456, and 3

π ≈ 3.14, 223

≈ 2.67, and 3 ≈ 1.73

____________________________

2. −1.75, 1, 15

, 1.55, and 5−

15

= ____________

5− ≈ ____________

____________________________

3. 6, 2.63, 4.36, 2 3,− − and 16

−

6 ≈ ____________

2 3 ≈ ____________

16

− ≈ ____________

____________________________

LESSON

1-1

Name ________________________________________ Date __________________ Class__________________


1-7 Holt Algebra 2

Reteach Sets of Numbers (continued)

You can represent the same set in different ways.

Number line: Words: The set of numbers greater than or equal to −2 and less than 0 OR greater than or equal to 1 Interval Notation: [−2, 0) or [1, ∞) Set-Builder Notation: {x | −2 ≤ x < 0 or x ≥ 1} This set cannot be described in roster notation because you cannot list the real numbers in the intervals shown on the number line. The roster notation of , the set of natural numbers, is {1, 2, 3, ...}. The set-builder notation of is {x|x ∈ }.

Rewrite each set using the indicated notation. 4. the set of integers, or ; roster notation {…, −3, −2,_______________________________} 5. {0, 4, 8, 12, 16, …}; words ______________________________ 6. −5 ≤ x ≤ 12; interval notation [ _______________________________ 7. {x|x < 0}; interval notation (−∞ _______________________________ 8.

set-builder notation {x| _______________________________

LESSON

1-1

∞ means infinity and –∞ means negative infinity.

Brackets [ ] include the endpoints. Parentheses ( ) do not include endpoints.

Read this as “x such that”

Name ________________________________________ Date __________________ Class__________________


1-14 Holt Algebra 2

Reteach Properties of Real Numbers

Find the additive inverse of each number. 1. 20 2. –36 3. –7.9 20 + ( ________ ) = 0 –36 + ( ________ ) = 0 –7.9 + ( ________ ) = 0

4. 23

5. 3 6. 34

−

________________________ _________________________ ________________________

Identify the property of addition demonstrated by each equation.

7. x + 0 = x _____________________________

8. (2 + m) + 5n = 2 + (m + 5n) _____________________________

9. 4r + 6s = 6s + 4r _____________________________

10. π + (−π) = 0 _____________________________

11. c + (−2) = −2 + c _____________________________

12. 1 = 0 + 1 _____________________________

LESSON

1-2

Properties of Addition Examples

Additive Identity 0 is the additive identity.

4 + 0 = 4 n + 0 = 0 + n = n

Additive Inverse The sum of a number and its opposite is 0.

8 + (–8) = 0 n + (–n) = 0

Closure Property The sum of any two real numbers is a real number.

3 + 5 = 8 a + b ∈

Commutative Property The order does not change the sum.

6 + 12 = 12 + 6 a + b = b + a

Associative Property The grouping does not change the sum.

(2 + 5) + 9 = 2 + (9 + 5) (a + b) + c = a + (b + c)

The additive inverse of 8 is –8.The additive inverse of –8 is 8.

Name ________________________________________ Date __________________ Class__________________


1-15 Holt Algebra 2

Reteach Properties of Real Numbers (continued)

Find the multiplicative inverse of each number.

13. 25 14. 35

− 15. 18

−

25 ⋅ ( ________ ) = 1 35

− ⋅ ( ________ ) = 1 18

− ⋅ ( ________ ) = 1

16. 13

17. −4 18. π

________________________ _________________________ ________________________

IIdentify the property of multiplication demonstrated by each equation.

19. 6(2x + y) = 6(2x) + 6y _________________________

20. 9(4.2) = (4.2)9 _________________________

21. 4(3 10) (4 3) 10= i _________________________

LESSON

1-2

Properties of Multiplication Examples

Multiplicative Identity 1 is the multiplicative identity. −12 ⋅ 1 = −12

n ⋅ 1 = 1 ⋅ n = n

Multiplicative Inverse The product of a number and its reciprocal is 1.

2 3 13 2

⎛ ⎞− − =⎜ ⎟⎝ ⎠i

1 1, 0n nn

= ≠i

Closure Property The product of any two real numbers is a real number.

8(−0.5) = −4 ab ∈

Commutative Property The order does not change the product.

1 13 (3)6 6

⎛ ⎞ =⎜ ⎟⎝ ⎠

ab = ba

Associative Property The grouping does not change the product.

(3 ⋅ 7)4 = 3(7 ⋅ 4) (ab)c = a(bc)

Distributive Property −2(4 + 7) = −2(4) + (−2)(7) a(b + c) = ab + ac

The multiplicative inverse

of 23

− is 32

− .

Name ________________________________________ Date __________________ Class__________________


1-22 Holt Algebra 2

Reteach Square Roots

Use properties of square roots to simplify expressions with square roots.

Product Property: for a > 0 and b > 0, ab a b= i

200 100 2 100 2 10 2= = =i 27 3 27 3 81 9= = =i i

Quotient Property: for a > 0 and b > 0, a ab b

=

25 25 549 749

= = 108 108 36 633

= = =

Simplify each expression.

1. 20 2. 63 3. 80

4 5i 9 ________i 16 ________i

4 ________i 9 ________i 16 ________i

________________________ _________________________ ________________________

4. 3 12i 5. 6425

6. 2008

________ ________i ________________

________________

________________________ _________________________ ________________________

7. 6 24i 8. 4487

9. 49100

________________________ _________________________ ________________________

LESSON

1-3

Multiply under the radical. Look for a perfect square factor.

Evaluate perfect square factors. Divide under the radical.

Name ________________________________________ Date __________________ Class__________________


1-23 Holt Algebra 2

Reteach Square Roots (continued)

Rationalize the denominator to eliminate radicals from the denominator.

3 5 23 5 3 5 2 3 102 22 2 2

= = =i

i

Combine like radical terms to add or subtract square roots.

3 2 8+

3 2 4 2+ ⋅

3 2 2 2+

(3 2) 2+

5 2

Simplify by rationalizing each denominator.

10. 45

11. 16

12. 3 28

4 55 5i 1 ________

6i 3 2 ________

8i

________________________ _________________________ ________________________

Add or subtract.

13. 12 7 4 7− 14. 75 27− 15. 6 5 45+

(12 4) 7− 25 3 9 3−i i 6 5 ________+

________________________ _________________________ ________________________

LESSON

1-3

Multiply by 1: 2 12

= Think: 2 2 2⋅ =

Try to use 2 as a factor.

Both terms contain 2 . Combine like terms.

Name ________________________________________ Date __________________ Class__________________


1-30 Holt Algebra 2

Reteach Simplifying Algebraic Expressions

To evaluate an algebraic expression you substitute numbers for variables. Then follow the order of operations. Here is a sentence that can help you remember the order of operations. Evaluate x − 2xy + y2 for x = 4 and y = 6. 4 − 2(4)(6) + (6)2 Substitute 4 for x and 6 for y. 4 − 2(4)(6) + 36 Evaluate exponents: 62 = 36. 4 − 48 + 36 Multiply from left to right. −8 Add and subtract from left to right.

Evaluate each expression for the given values of the variables.

1. a2 + 2ab2 − 3a for a = 5 and b = 2

52 + 2(5)(2)2 − _______________ Substitute 5 for a and 2 for b.

25 + _______________________ Evaluate exponents.

___________________________ Multiply from left to right.

___________________________ Add and subtract from left to right.

2. c2 − cd + 3d for c = 7 and d = 6

___________________________ Substitute for the variables.

___________________________ Evaluate exponents.


___________________________ Add and subtract from left to right.

3. 3(5 )

3m n

n− for m = 4 and n = 2

___________________________ Substitute for the variables.

___________________________ Evaluate exponents inside parentheses.

___________________________ Multiply inside parentheses.

___________________________ Subtract inside parentheses.


___________________________ Divide from left to right.

Please Excuse My Dear Aunt Sally

Parentheses Exponents Multiply Divide Add Subtract

LESSON

1-4

Name ________________________________________ Date __________________ Class__________________


1-31 Holt Algebra 2

Reteach Simplifying Algebraic Expressions (continued)

Add or subtract the coefficients of like terms to simplify an algebraic expression.

Like Terms: 3x2 and 4x2 5xy and −xy 3x2 + 5xy + 4x2 − xy + 2 3x2 + 4x2 + 5xy − xy + 2 Group like terms. 7x2 + 4xy + 2 Add or subtract like terms.

You can use the Distributive Property to simplify an algebraic expression. −2(a2 − ab) + 6ab + 2a2 −2a2 + 2ab + 6ab + 2a2 Distribute. −2a2 + 2a2 + 2ab + 6ab Group like terms. 8ab Add or subtract like terms.

Simplify each expression.

4. −6x + 3 − 2x + 4x ________________________________ Group like terms.

________________________________ Add or subtract like terms.

5. c(4c + d ) − c2+ cd ________________________________ Distribute.

________________________________ Group like terms.

________________________________ Add or subtract like terms.

6. 4a2+ 5ab − 4a2− 2ab − 7 7. 3(s − 4t) + 3s − t

_________________________________________ ________________________________________

LESSON

1-4

Coefficients of x2 : 3 and 4 Coefficients of xy : 5 and −1

Think: 3x2 + 4x2 = 7x2

5xy − 1xy = 4xy

−2(a2 − ab) = −2(a2) − 2(−ab) = −2a2 + 2ab

Think: −2a2 + 2a2 = 0

Name ________________________________________ Date __________________ Class__________________


1-38 Holt Algebra 2

Reteach Properties of Exponents

−4x5 = −4(x ⋅ x ⋅ x ⋅ x ⋅ x) −(4x5) = −(4x)(4x)(4x)(4x)(4x) (−4x)5 = (−4x)(−4x)(−4x)(−4x)(−4x) 4x3(y + 6)2 = 4(x)(x)(x)(y + 6)(y + 6)

Zero Exponent Property: a0 = 1; a is not zero 380 = 1

Negative Exponent Property: 1nna

a− = and ;

n na bb a

−⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

a is not zero.

44

1 1 133 3 3 3 813

− = = =i i i

3 32 5 5 5 5 1255 2 2 2 2 8

−⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

i i

Write each expression in expanded form. 1. −8c3 2. (3xy)4 3. a3(b − c)2

−8(c ⋅ __________) ___________________ ____________________

Evaluate each expression. 4. 6−1 5. 100 6. 12−2

________________________ _________________________ ________________________

7. (−4)−3 8. 21

7

−⎛ ⎞⎜ ⎟⎝ ⎠

9. 33

4

−⎛ ⎞⎜ ⎟⎝ ⎠

________________________ _________________________ ________________________

10. −50 11. 22

5−⎛ ⎞

⎜ ⎟⎝ ⎠

12. 21

3

−⎛ ⎞− ⎜ ⎟⎝ ⎠

________________________ _________________________ ________________________

LESSON

1-5

Write

Expanded Form Exponent FormRead

a ⋅ a a2 a squared

a ⋅ a ⋅ a a3 a cubed

a ⋅ a ⋅ a ⋅ a a4 a to the fourth power

a ⋅ a ⋅ … ⋅ a an a to the nth power

List the factors to expand exponential expressions.

Name ________________________________________ Date __________________ Class__________________


1-39 Holt Algebra 2

Reteach Properties of Exponents (continued)

Properties of Exponents (m and n are integers; a and b are nonzero real numbers.)

Same Base: am ⋅ an = am + n m

naa

= am − n (am)n = am n

Different Bases: (ab)m = ambm

m m

ma ab b

⎛ ⎞ =⎜ ⎟⎝ ⎠

Combine properties of exponents to simplify expressions with exponents. (2x5)4 c4d(c−3d

2) 24(x5)4 Distribute the exponent. c4c−3dd

2 Group like variables. 24x5 ⋅ 4 Multiply exponents. c4−3d1+2 Add exponents. 24x20 Simplify. cd3 Simplify. 16x20

5

4 33rsr s

3r1−4s5−3 Subtract exponents. 3r−3s2 Simplify.

2

33sr

Record answer with positive exponents.

Simplify each expression. Assume all variables are nonzero. 13. (−5ab3)2 14. w2x5(w4 x3) 15. y4z3(y−1z)2 (−5)2a2b3 2 w2+4x5+3 y4y−2z3z2

________________________ _________________________ ________________________

16.3

263

s tst

17. 34

2ab

⎛ ⎞⎜ ⎟⎝ ⎠

18. −3x−2y3(9x−1y5)

3 1 1 263

s t− − 4 3

2 3ab

i

i −3(9)x−2−1y3+5

________________________ _________________________ ________________________

LESSON

1-5

To multiply, add exponents.

To divide, subtract exponents.

To raise to a power, multiply exponents.

Distribute the exponent.

Name ________________________________________ Date __________________ Class__________________


1-46 Holt Algebra 2

Reteach Relations and Functions

A relation pairs input values (x) and output values (y).

List domain and range elements from least to greatest.

Domain: {1996, 1998, 2000, 2002, 2004} Set of x-coordinates Range: {56, 82, 95, 136, 212} Set of y-coordinates The domain of a set of ordered pairs is the x-coordinates. The range is the y-coordinates. Each value is listed only once. For the graph at right: Domain: {−4, −2, 0, 2, 4}; Range: {0, 2, 3}

Give the domain and range for each relation. 1.

Domain: {2001, ______________________ Range: {25, ______________________

2. Domain: {−4, ____________________

Range: {−2, ____________________

LESSON

1-6

Domain Set of input values or x-coordinates

Range Set of output values or y-coordinates

Soccer Registration

Year 1996 1998 2000 2002 2004

Number of Players 56 82 95 136 212

Concert Ticket Price

Year 2001 2002 2003 2004 2005

Price ($) 25 28 35 42 46

Name ________________________________________ Date __________________ Class__________________


1-47 Holt Algebra 2

Reteach Relations and Functions (continued)

A function is a special type of relation. A function has only one output for each input.

Use the vertical-line test to decide whether a relation is a function.

Use the vertical-line test to determine whether each relation is a function. If not, identify two points a vertical line would pass through.

3.

4.

_________________________________________ ________________________________________

LESSON

1-6

Draw a vertical line. The line passes through no more than one point on the graph. This is a function.

Draw a vertical line. The line passes through two points on the graph, at (1, −1) and (1, 1). This is not a function.

Name ________________________________________ Date __________________ Class__________________


1-54 Holt Algebra 2

Reteach Function Notation

You can use function notation to write a function.

f(x) = 2x − 3

Evaluate f(0), 12

f ⎛ ⎞⎜ ⎟⎝ ⎠

, and f(−2) for f(x) = 2x2− x + 1.

f(0) = 2(0)2 − 0 + 1 = 1 21 1 1 1 1 1 12 1 2 1 1 1

2 2 2 4 2 2 2f ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + = − + = − + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

f (−2) = 2(−2)2 − (−2) + 1 = 2(4) + 2 + 1 = 8 + 2 + 1 = 11

For each function, evaluate f (0), ⎛ ⎞⎜ ⎟⎝ ⎠

32

f , and f (−1).

1. f (x) = 4x2 − 2

f (0) = 4(0)2 − 2 23 34 2

2 2f ⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

f (−1) = 4(−1)2− 2

________________________ _________________________ ________________________

2. f (x) = −2x + 10

f (0) = ______________ 32

f ⎛ ⎞⎜ ⎟⎝ ⎠

= ____________ f (−1) = ______________

3. f (x) = x2+ 6x

f (0) = ______________ 32

f ⎛ ⎞⎜ ⎟⎝ ⎠

= ____________ f (−1) = ______________

LESSON

1-7

Read: f of x equals 2x − 3.

Output f(x) Input x

Subsitute 0 for x in the function and evaluate.

Substitute 12

for x.

Substitute −2 for x.

Name ________________________________________ Date __________________ Class__________________


1-55 Holt Algebra 2

Reteach Function Notation (continued)

Plot ordered pairs on a coordinate plane to graph a function.

To graph f (x) = −2x + 3, make a table of values.

Graph each function on the coordinate plane given.

4.

5. g(x) = 2x − 4

x 2x − 4 g (x)

0

1

2

3

4

LESSON

1-7

Values between given points are not defined. Do not connect the points.

x −2x + 3 f (x)

−2 −2(−2) + 3 7

−1 −2(−1) + 3 5

0 −2(0) + 3 3

1 −2(1) + 3 1

2 −2(2) + 3 −1

All points are defined. Connect the points.

Name ________________________________________ Date __________________ Class__________________


1-62 Holt Algebra 2

Reteach Exploring Transformations

A translation moves a point, figure, or function right, left, up, or down. Translate the function y = f (x) left 2 units.

A reflection flips a point, figure, or function across a line. Reflect the function y = f (x) across the x-axis.

Perform each transformation of y = f (x). 1. translation up 2 units 2. reflection across x-axis

LESSON

1-8

Horizontal Translation (right or left) Vertical Translation (up or down)

The x-coordinate changes. (x, y) → (x + h, y)

The y-coordinate changes. (x, y) → (x, y + k)

Move each point 2 units left. Connect the points. (x, y ) → (x − 2, y)

Reflection Across y-axis Reflection Across x-axis

The x-coordinate changes. (x, y ) → (−x, y)

The y-coordinate changes. (x, y ) → (x, −y)

Flip each point across the axis. Connect the points. (x, y ) → (x, −y)

Name ________________________________________ Date __________________ Class__________________


1-63 Holt Algebra 2

Reteach Exploring Transformations (continued)

In a stretch or a compression, the new figure has a different shape than the original. Perform a vertical stretch of the function y = f (x) by a factor of 2. In a vertical stretch (x, y) → (x, ay). In this case, a = 2.

Perform each transformation of y = f (x).

3. horizontal stretch by a factor of 2 4. vertical compression by a factor of 12

LESSON

1-8

Horizontal Stretch (away from y-axis)

The x-coordinate changes. (x, y) → (bx, y); | b | > 1

Vertical Stretch (away from x-axis)

The y-coordinate changes. (x, y) → (x, ay); | a | > 1

Horizontal Compression (toward the y-axis)

The x-coordinate changes. (x, y) → (bx, y); 0 < | b | < 1

Vertical Compression (toward the x-axis)

The y-coordinate changes. (x, y) → (x, ay); 0 < | a | < 1

Original Figure (solid line)

x 2y Stretched Figure(dashed line)

(−3, 3) −3 6 (−3, 6)

(−1, 1) −1 2 (−1, 2)

(0, 2) 0 4 (0, 4)

(1, 1) 1 2 (1, 2)

(3, 3) 3 6 (3, 6)

Name ________________________________________ Date __________________ Class__________________


1-70 Holt Algebra 2

Reteach Introduction to Parent Functions

Functions can be classified into families. In the families, similar-looking functions represent transformations of a parent function. The parent function of g(x) = x3 − 1 is cubic. g(x) = x3 − 1 is a transformation from the parent function. It is a vertical translation 1 unit down.

Identify the parent function for g from its function rule. Then describe what transformation of the parent function it represents. 1. g(x) = −x2 2. ( ) 2g x x= +

Parent function ____________________ Parent function ____________________

________________________________ _________________________________

LESSON

1-9

Parent Functions

Linear f (x) = x

Quadratic f (x) = x2

Cubic f (x) = x3

Square Root ( )f x x=

x has a power of 3.

Compare the graph to the graph of the parent function.

Name ________________________________________ Date __________________ Class__________________


1-71 Holt Algebra 2

Reteach Introduction to Parent Functions (continued)

You can graph data to identify a parent function.

The shape looks like the quadratic parent function f (x) = x2. It looks like a horizontal translation 2 units right.

Graph the data from each table. Describe the parent function and the transformation that best approximates the data set. 3. 4.

Parent function ____________________ Parent function ____________________

________________________________ _________________________________

________________________________ _________________________________

LESSON

1-9

x 2 1 3 0 4

y 0 1 1 4 4

Plot the points. Connect the points with a smooth curve. Compare shape to graphs of parent functions.

x −2 −1 0 1 2

y 1 −2 −3 −2 1

x −2 −1 0 1 2

y 2 1 0 −1 −2

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1-1 Sets of Numbers - Esperanza Summer School Algebra 2 - …€¦ · · 2013-06-231-30 Holt...

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