Summer Math Packet

In preparation for Honors Algebra 2, we have prepared a packet of concepts that students should know how to do. These concepts have been taught in previous math classes. This packet does not require you to use a calculator; in fact you should not use a calculator on some of the problem sets where it is noted. Honors Algebra 2 builds on the concepts in this packet. We start teaching Algebra 2 concepts on the first day of school. We expect you to know the concepts in the packet in order to help you be successful in Honors Algebra 2. If you are struggling with this packet, get help from a friend, parent, or tutor. If you can’t find someone to help you, there are tutors available. A list of tutors can be found by calling Reagan High School. Keep in mind these tutors may charge a fee. You will have a graded assignment within the first two weeks of school. The graded assignment will cover all the concepts in the packet, but will not be the exact same problems. The graded assignment will contain a calculator inactive part.

Vocabulary/Know the difference Review 1. Proportions vs. multiplying fractions

Proportion Multiplying Fractions

1282 =

x vs.

128

32 ⋅

2. x * x = ? vs. x + x = ? 3. order of operations: don’t be tricked by these common “mistaken identities”

a. 5 * 23 vs. (5 * 2)3 b. 5 + 3(x + 4) c. -82 vs. (-8)2

4. (x – 9)2 vs. (x + 9)2 vs. (x – 9)(x + 9) a. (x – 9)2 = b. (x + 9)2 = c. (x – 9)(x + 9) =

5. know the difference between a term, expression, equation, and inequality term – expression – equation – inequality –

6. know the difference between solve, evaluate and simplify

simplify – evaluate – solve –

7. know the difference between rational and irrational

rational – irrational –

8. know the > and < symbols by name:

9. know coefficient –

10. know factor –

11. reduce factors not individual terms

Fractions DO NOT USE A CALCULATOR

1) 94

32 +

2) 54

47 −

3) 61

43 +

4) 94

32 •

5) 54

47 •

6) 61

43 •

7) 94

32 ÷

8) 54

47 ÷

9) 61

43 ÷

Solve Equations and Inequalities Review DO NOT USE A CALCULATOR

1) 574

−=+x 2) 7

32

x+−=

Clear out fractions.

3) 78

14

34

116

a a a− + = +

Variables on both sides. 4) 3( x – 8 ) + 3( 2x + 4 ) = 15 5) 8 + 3( a – 3 ) = 4( a + 5 ) 6) 6 – 2x + 5x = 7 + 7x – 15 Solve Inequalities. When you multiply or divide both sides by a negative remember to flip the inequality. Get the variable on the left side. 7) ( )6 2 4x x− + > − + 8) 2 6 6x x− < − + Word Problems You must be able to write the equation or inequality first. Then solve for the variable. 9) The greater of two numbers is 6 more than 4 times the smaller. Their sum is 41. Find the numbers. 10) Find three consecutive integers whose sum is 105. 11) Find three consecutive even integers whose sum is 138. 12) The length of a rectangle is 2 feet more than its width. If its perimeter is 40 feet, find the length and the width. 13) The second angle in a triangle is 3° less than twice the first angle. The third angle measure 8° more than twice the first angle. Find each angle. 14) Jeffery has grades of 93 and 81 on the first two tests of the quarter. Progress reports will go home after the third test. If Jeffery does not have an A average on his progress report, he cannot go to the football game that week. Jeffery will have to make at least what grade on the third test to be allowed to go to the football game?

Linear Equations Review: Slope, Writing Linear Equations, Horizontal & Vertical Lines, Parallel & Perpendicular Lines Find the slope (rate of change) of the following problems. 1. ( )8,3 − , ( )1,5 −− 2. 3. 4. The cost of museum tickets is $48 for four people and $78 for 10 people. What is the cost per person? Write the equation of the line in slope-intercept form & standard form given the following.

5. ( )2,5 3m = 6. ( ) 54,2

7m = − 7. ( )( )2, 6 1, 2− − 8. m = –

21

, b = 2

9. (2, -3) m = 0 10. (4, -1) m = undefined 11. 12. The cost for 7 dance lessons is $82. The cost for 11 lessons is $122. Write a linear equation to find the total cost C for L dance lessons. Then use the equation to find the cost of 4 lessons.

Day Temperature ( F� )

1 60 2 62 3 64 4 66

13. Write the equation for a vertical line that goes through the point (2, 4). 14. Write the equation for a horizontal line that goes through the point (-1, 3). Solve the equation for y if necessary, and find the slope. Then, find the slope of a line parallel and perpendicular to the original line. Slope Parallel Perpendicular 15. 2x + 6y = 8 16. x = 3 17. y = -2 18. Write the equation for the line parallel to the given line and through the given point. Write the answer in slope intercept and standard forms. 4x – 3y = 9 and (3, –1) 19. Write the equation for the line perpendicular to the given line and through the given point. Write the answer in slope intercept and standard forms. 4x – 3y = 9 and (8, –3) 20. Given a line through (–2, 4) and (8, –1), find the equation of the line perpendicular to that line through the midpoint of those points. Write the answer in slope intercept and standard forms.

Graphing Linear Equations and Inequalities DO NOT USE A CALCULATOR Graph each of the following lines.

1) slope:34

− , through ( )5, 1− − 2) slope: 2, through ( )3, 4− 3) slope:14

, y-intercept: – 5

4) slope: 3− , x-intercept: 4 5) 2

43

y x= − 6) 5 2y x= +

7) 5

63

y x= − − 8) 3 5y x= − + 9) 27

y x=

10) 5 2x y− = 11) 6 3 12x y+ = − 12) 4 2 0x y+ =

13) 3 15x y+ = 14) 3 7 21x y+ = 15) 3y = 16) 4y = − 17) no slope; through (2, 5) 18) slope: 0 through (3, –7) Write each of the following in slope-intercept form: ( y mx b= + ) 19) A computer technician charges $75 for a consultation plus $35 per hour. 20) The population of Pine Bluff is 6791 and is decreasing at the rate of 7 per year. 21) A video store charges $10 for a rental card plus a $2 per rental. Graph each inequality. Remember to use either a solid or dotted line, then SHADE. 1) y < x – 5 2) x > 3y 3) x < 4

Systems of Equations Review IF THE LINES INTERSECT ONCE, THE ANSWER IS THE ORDERED PAIR. IF THE LINES DO NOT INTERSECT (PARALLEL), THE ANSWER IS ∅. IF THE LINES ALWAYS TOUCH (ARE THE SAME LINE), THE ANSWER IS INFINITLY MANY SOLUTIONS.

1) 62

421

−=

+−=

xy

xy 2)

0253=+−

=−yx

yx

3) 5

843=

=+y

yx 4)

2425

−==−

x

yx

5) 1

2−=+

=+yx

yx 6)

1864932

=+=+

yx

yx

Solve using Substitution or Elimination.

7) x y

y x

+ == +

42 1

8) x y

y x

= −= −

14 2

9) y x

y x

= −− =2 5

3 5 10)

yx

yx

222−=

−=

11) x y

x y

= −− =3 4

2 7 12)

x y

x y

− =+ = −

62

13) 642

32=+

−=−−yx

yx 14)

1534023

=−=−

yx

yx

Laws of Exponents Review

(Remember: baba xxx +=⋅ ,

abba xx =)( , ba

b

a

xxx −= , 10 =x and a

a

xx

1=−)

(negative exponents should always be simplified) 1) ( )4 7 33 5a b a b− 2) ( )( )( )2 2 2 43 5y y z yz− − 3) ( )( )( )3 2 2 2 3 4a b b c a c− − − 4) 04

5) ( )( )23 42 cc 6) ( )( )( )2 43 2 5x x x− − 7) ( )( )( )abbcac −−− 8) 04−

9) 106

208

2540

baba −−

10) 710

1210

yxyx

11) 313

63

1422

−−−

yxyx

12) 0)4(−

13) 12 2

5 7

a ba b

14) 1015

28

2128

baba

−

−

15) 3 15

13 6

279

x yx y

− 16) 04x

17) ( )73x 18) ( )423x 19) ( )642x y− 20) ( )342x−

21) ( )96a 22) ( )384x 23) ( )35 74x y− 24) 50 )2( x−

Polynomial and FOIL Review Simplify. 1) 25 (6 3 2)x x x+ − 2) ( )4 2 24 3 5ab ab a b− 3) 2 26 (2 7 1)x x x− + −

4) ( )( )3 8x x+ + 5) ( )( )2 4 2 3x x+ − 6) ( )( )34 5 6x x− −

7) ( )25 2x +

Radical & Rational Exponent Review Simplify Radicals 1. 100 2. 36 3. 121− 4. 49− 5. 8 6. 50 7. 45 8. 28 9. 80− 10. 450

11. 400 12. 983 13. 236x 14. 27x 15. 218a

16. yx 220 17. 2100a 18. 272a 19. zyx 10620 20. 6201275 zyx

21. ( )125+x 22. 5x 23. 35 x Rational Exponents DO NOT USE A CALCULATOR

See these as examples of rational exponents: 1225 = 25 3

1

125 = 3 125 1481 = 4 81

Simplify the following. 24. 21

16 25. 31

27 26. 41

256 27. 21

36−

Write as a rational exponent: 28. 121 29. 3 64 30. 4 16

Function Notation Review

State whether each set is a function. Answer yes or no. Find the domain and the range.

1) {(2, 5), (5, 6), (2, -6), (3, 8)} Domain: ______ Range: ______ 2) {(1, -2), (8, -4), (-3, 8), (-1, 2)} Domain: ______ Range: ______ Use the vertical line test to determine whether each graph is the graph of a function. Answer yes or no.

3) 4) 5) 6) Use ( ) 2 3f x x= − and ( ) 4 1g x x= − to find each value.

7) ( )3f − 8) ( )7g − 9) 43

f � �� �� �

10) ( )5 8f − +

11) ( )3f c 12) ( )7g w − 13) ( )2 3f m + 14) ( )2 3g x− −� �

15) The temperature of the atmosphere decreases about 5oF for every 1000 feet increase in altitude. Thus, if

the temperature at ground level is 77oF, the temperature t at a given altitude is found by using the equation77 .005t h= − , where h is the height in feet.

a) Write the equation in function notation where t is a function of h [ f(x) is meant as f is a function of x].

b) Find t(100) and explain its meaning in this problem.

16) The function ( ) 160 1.5g x x= + models the weight gain of a basketball player as he starts a workout

program where g is the weight in pounds after x weeks. a) Explain the meaning of 160 in the context of this problem. b) Explain the meaning of 1.5 in the context of this problem. c) Evaluate ( )6g and explain its meaning.

Solving Quadratic Equations 1) 9)3)(5( =+− xx 2) 20)1)(8( −=+− xx 3) 72)1( =+xx 4) 524)5( 2 =+− xx 5) 9)23( 2 =+x 6) xxx 15562 22 +=+ 7) 0)2)(5( =−+ xx

ANSWER

KEYS

Vocabulary/Know the difference Review KEY 1. Proportions vs. multiplying fractions

(cross multiplying for a proportion vs. multiplying numerators and multiplying denominators when multiplying fractions)

Proportion Multiplying Fractions

1282 =

x vs.

128

32 ⋅

24 = 8x 94

3616

)123()82( ==

⋅⋅

3 = x 2. x * x = x2 vs. x + x = 2x 3. order of operations: don’t be tricked by these common “mistaken identities”

a. 5 * 23 vs. (5 * 2)3 b. 5 + 3(x + 4) c. -82 vs. (-8)2 5 * 8 = 40 103 = 1000 5 + 3x + 12 not 8(x + 4) -1 * 64 = 64 (-8)(-8) = 64

4. (x – 9)2 vs. (x + 9)2 vs. (x – 9)(x + 9)

a. (x – 9)2 = x2 – 9x – 9x + 81 = x2 -18x + 81 b. (x + 9)2 = x2 + 9x + 9x + 81 = x2 +18x + 81 c. (x – 9)(x + 9) = x2 + 9x – 9x – 81 = x2 – 81

5. know the difference between a term, expression, equation, and inequality

term – number, variable, or product of numbers and variables (ex: 2, x, or 2x)

expression – terms with mathematical symbols (ex: 2x, 2x – 3, x2 – 4x + 3, 2x )

equation – expressions set equal to one another (ex: x = 2, 4x + 3 = 12 – 5x) inequality – expressions not equal to one another (ex: x > 2, 4x + 3 ≤ 12 – 5x)

6. know the difference between solve, evaluate and simplify simplify – to rewrite an expression in simplest form possible where nothing else can be performed (includes no parentheses or negative exponents; all fractions have been reduced) evaluate – to find the value of (once the value has been found, the final result should be written in simplest form) solve – to work out the solution to the problem

7. know the difference between rational and irrational

rational – real number that can be written as a fraction (ex:0, 1, 31 , 121 , 0.25) [repeating or

terminating decimals] irrational – real number that can’t be written as a fraction [nonrepeating, nonterminating decimals] (ex: ...478192.2,,12 π )

8. know the symbols by name: > as “greater than” vs. < as “less than”

9. know coefficient – numerical factor of a monomial [number being multiplied by a variable]

10. know factor – two or more numbers that multiply to produce another number [2 and 5 are factors of

10]

11. reduce entire factors not parts of a factor (individual terms)

- you can reduce 12510

−−

xx

by rewriting the numerator in factored form first: 12510

−−

xx

=

( )5

12125 =

−−

xx

- you can’t reduce 12510

−−

xx

by trying to reduce just 12510

−−

xx

Fractions DO NOT USE A CALCULATOR

1) 94

32 + = 9

10

2) 54

47 − = 20

19

3) 61

43 + = 12

11

4) 94

32 • = 27

8

5) 54

47 • = 5

7

6) 61

43 • = 8

1

7) 94

32 ÷ = 2

3

8) 54

47 ÷ = 16

35

9) 61

43 ÷ = 2

9

4

Solve Equations and Inequalities Review DO NOT USE A CALCULATOR

1) 574

−=+x x=-48 2) 7

32

x+−= x=17

Clear out fractions.

3) 78

14

34

116

a a a− + = + a=1/2

Variables on both sides. 4) 3( x – 8 ) + 3( 2x + 4 ) = 15 3x = 5) 8 + 3( a – 3 ) = 4( a + 5 ) 21a = −

6) 6 – 2x + 5x = 7 + 7x – 15 72

x = −

Solve Inequalities. When you multiply or divide both sides by a negative remember to flip the inequality. Get the variable on the left side. 7) ( )6 2 4x x− + > − + 8) 2 6 6x x− < − +

10x > − 4x < -10 Word Problems You must be able to write the equation or inequality first. Then solve for the variable. 9) The greater of two numbers is 6 more than 4 times the smaller. Their sum is 41. Find the numbers. ( ) 4164 =++ xx 7 and 34 10) Find three consecutive integers whose sum is 105. ( ) ( ) 10521 =++++ xxx 34,35,36 11) Find three consecutive even integers whose sum is 138. ( ) ( ) 13842 =++++ xxx 44, 46, 48 12) The length of a rectangle is 2 feet more than its width. If its perimeter is 40 feet, find the length and the width. ( )22240 ++= ww Length: 11 Width: 9 13) The second angle in a triangle is 3° less than twice the first angle. The third angle measure 8° more than twice the first angle. Find each angle. ( ) ( ) 1808232 =++−+ xxx 35, 67, 78 14) Jeffery has grades of 93 and 81 on the first two tests of the quarter. Progress reports will go home after the third test. If Jeffery does not have an A average on his progress report, he cannot go to the football game that week. Jeffery will have to make at least what grade on the third test to be allowed to go to the football game?

105

9338193

≥

≥++

x

x

Linear Equations Review: KEY Slope, Writing Linear Equations, Horizontal & Vertical Lines, Parallel & Perpendicular Lines Find the slope (rate of change) of the following problems. 1. ( )8,3 − , ( )1,5 −− 2. 3.

87−

2 23

4. The cost of museum tickets is $48 for four people and $78 for 10 people. What is the cost per person?

$5 per person Write the equation of the line in slope-intercept form & standard form given the following.

5. ( )2,5 3m = 6. ( ) 54,2

7m = − 7. ( )( )2, 6 1, 2− − 8. m = –

21

, b = 2

13 −= xy 7

3475 +−= xy 24 +−= xy 2

21 +−= xy

13 =− yx 3475 =+ yx 24 =+ yx 42 =+ yx

9. (2, -3) m = 0 10. (4, -1) m = undefined 11. 123 +−= xy

3−=y 4=x 223 =+ yx 12. The cost for 7 dance lessons is $82. The cost for 11 lessons is $122. Write a linear equation to find the total cost C for L dance lessons. Then use the equation to find the cost of 4 lessons.

52$1210 += LC

Day Temperature ( F� )

1 60 2 62 3 64 4 66

13. Write the equation for a vertical line that goes through the point (2, 4). x = 2 14. Write the equation for a horizontal line that goes through the point (-1, 3). y = 3 Solve the equation for y if necessary, and find the slope. Then, find the slope of a line parallel and perpendicular to the original line. Slope Parallel Perpendicular 15. 2x + 6y = 8

34

31 +−= xy 3

1− 31− 3

16. x = 3 undefined/none= undefined/none 0

17. y = -2 0 0 undefined/none 18. Write the equation for the line parallel to the given line and through the given point. Write the answer in slope intercept and standard forms. 4x – 3y = 9 and (3, –1)

534 −= xy 1534 =− yx

19. Write the equation for the line perpendicular to the given line and through the given point. Write the answer in slope intercept and standard forms. 4x – 3y = 9 and (8, –3)

343 +−= xy 1243 =+ yx

20. Given a line through (–2, 4) and (8, –1), find the equation of the line perpendicular to that line through the midpoint of those points. Write the answer in slope intercept and standard forms.

29

2 −= xy 924 =− yx

19) 35 75y x= + 20) 7 6791y x= − +

21) 2 10y x= +

Systems of Equations Review IF THE LINES INTERSECT ONCE, THE ANSWER IS THE ORDERED PAIR. IF THE LINES DO NOT INTERSECT (PARALLEL), THE ANSWER IS ∅. IF THE LINES ALWAYS TOUCH (ARE THE SAME LINE), THE ANSWER IS INFINITLY MANY SOLUTIONS.

1) 62

421

−=

+−=

xy

xy 2)

0253=+−

=−yx

yx

( )2,4 ( )1,2

3) 5

843=

=+y

yx 4)

2425

−==−

x

yx

( )5,4− ( )7,2 −−

5) 1

2−=+

=+yx

yx 6)

1864932

=+=+

yx

yx

∅ infinitely many

7) x y

y x

+ == +

42 1

8) x y

y x

= −= −

14 2

( )3,1 ( )2,1

9) y x

y x

= −− =2 5

3 5 10)

yx

yx

222−=

−=

( )3,4 ∅

11) x y

x y

= −− =3 4

2 7 12)

x y

x y

− =+ = −

62

( )3,5 ( )4,2 −

13) 642

32=+

−=−−yx

yx 14)

1534023

=−=−

yx

yx

infinitely many ( )45,30 −−

Laws of Exponents Review KEY

(Remember: baba xxx +=⋅ ,

abba xx =)( , ba

b

a

xxx −= , 10 =x and a

a

xx

1=−)

(negative exponents should always be simplified) 1) ( )4 7 33 5a b a b− 2) ( )( )( )2 2 2 43 5y y z yz− − 3) ( )( )( )3 2 2 2 3 4a b b c a c− − − 4) 04

41115 ba−

6515 zy 646 cba− 1

5) ( )( )23 42 cc 6) ( )( )( )2 43 2 5x x x− − 7) ( )( )( )abbcac −−− 8) 04−

58c

730x 222 cba− 1−

9) 106

208

2540

baba −−

10) 710

1210

yxyx

11) 313

63

1422

−−−

yxyx

12) 0)4(−

14

10

58ab−

5y 10

9

711

xy

1

13) 12 2

5 7

a ba b

14) 1015

28

2128

baba

−

−

15) 3 15

13 6

279

x yx y

− 16) 04x

5

7

ba

8

7

34ba

10

93xy− 4

17) ( )73x 18) ( )423x 19) ( )642x y− 20) ( )342x−

21x

881x 62464 yx

128x−

21) ( )96a 22) ( )384x 23) ( )35 74x y− 24) 50 )2( x−

54a

2464x 211564 yx− 32−

Polynomials and FOIL 1) 3 230 15 10x x x+ − 2) 2 6 3 512 20a b a b− 3) 4 3 212 24 6x x x− − + 4) 2 11 24x x+ + 5) 24 2 12x x+ − 6) 4 34 5 24 30x x x− − + 7) 225 20 4x x+ +

Radical & Rational Exponent Review Simplify Radicals 1. 100 2. 36 3. 121− 4. 49− 5. 8

10 6 11− not real 22 6. 50 7. 45 8. 28 9. 80− 10. 450

25 53 72 54− 215

11. 400 12. 983 13. 236x 14. 27x 15. 218a

20 221 x6 7x 23a

16. yx 220 17. 2100a 18. 272a 19. zyx 10620 20. 6201275 zyx

yx 52 a10 26a zyx 52 53 35 3106 zyx

21. ( )125+x 22. 5x 23. 35 x

6)5( +x xx2

xx5 Rational Exponents

See these as examples of rational exponents:

1225 = 25 3

1

125 = 3 125

1481 =

4 81

Simplify the following. 24. 21

16 4 25. 31

27 3 26. 41

256 4 27. 21

36− 6−

Write as a rational exponent: 28. 121 21

121 29. 3 64 31

64 30. 4 16 41

16

Function Notation Review KEY

State whether each set is a function. Answer yes or no. Find the domain and the range.

1) {(2, 5), (5, 6), (2, -6), (3, 8)} no Domain:{ }5,3,2 Range: { }8,6,5,6− 2) {(1, -2), (8, -4), (-3, 8), (-1, 2)} yes Domain: { }8,1,1,3 −− Range: { }8,2,2,4 −− Use the vertical line test to determine whether each graph is the graph of a function. Answer yes or no.

3) no 4) yes 5) yes 6) no Use ( ) 2 3f x x= − and ( ) 4 1g x x= − to find each value.

7) ( )3f − 8) ( )7g − 9) 43

f � �� �� �

10) ( )5 8f − +

6 29− 911−

30

11) ( )3f c 12) ( )7g w − 13) ( )2 3f m + 14) ( )2 3g x− −� �

39 2 −c 294 −w 6124 2 ++ mm 88 +− x 15) The temperature of the atmosphere decreases about 5oF for every 1000 feet increase in altitude. Thus, if

the temperature at ground level is 77oF, the temperature t at a given altitude is found by using the equation77 .005t h= − , where h is the height in feet.

a) Write the equation in function notation where t is a function of h. [f(x) is meant as f is a function of x]

hht 005.77)( −=

b) Find t(100) and explain its meaning in this problem. 5.76

16) The function ( ) 160 1.5g x x= + models the weight gain of a basketball player as he starts a workout program where g is the weight in pounds after x weeks.

a) Explain the meaning of 160 in the context of this problem. Starting/initial weight (y-intercept) b) Explain the meaning of 1.5 in the context of this problem. # of pounds added per week (slope) c) Evaluate ( )6g and explain its meaning. 169; weight 6 weeks after starting

Solve Quadratic Equations ANSWERS 1) 9)3)(5( =+− xx 2) 20)1)(8( −=+− xx

4,6 −=x 4,3=x 3) 72)1( =+xx 4) 524)5( 2 =+− xx

8,9−=x 3,9 −=x 5) 9)23( 2 =+x 6) xxx 15562 22 +=+

31

,35−=x 7,8=x

7) 0)2)(5( =−+ xx

2,5−=x