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    Early Start Mathematics 11 – Intermediate Algebra = Indicates that the problem, section or title has been revised. Date: June 23, 2012

Course Content I. Foundations of Algebra a) Order of Operations and Integer Exponents b) Scientific Notation c) Rational Exponents and Radicals II. Equations and Inequalities a) Linear equations and Applications b) Interval Notation and Linear Inequalities c) And/Or problems d) Absolute Value III. Lines, Graphs and Applications a) Linear forms and graphs b) Applications: Interpret slope and y-intercept c) Equations of lines and Linear Applications d) Graphing Inequalities in the Plane e) Solving Systems of Equations IV. Factoring, Expressions, Equations, Variation a) Factoring techniques b) Factor to Solve, to Simplify c) Expressions and Equations d) Literal Equations and Variation V. Relations and Functions a) Domain and range b) Graphing Parabolas c) Algebra of functions d) Difference Quotient VI. Exponential and Logarithmic Properties, Equations and Applications a) Exponential graphs and equations b) Logarithmic properties and equations c) Exponential and Log Applications Calculator Policy: No calculator, phone, tablet, lap-top, etc. permitted Students in need of Elementary Algebra content can access ESM1 course materials at the same website this document is available.

 

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I. Foundations of Algebra a) Order of Operations

Order of Operations Agreement

Step 1: Grouping Symbols: [( )], , .ab value “inside-out”

Step 2: Exponents Step 3: Multiplication/division Left Right Step 4: Addition/subtraction Left Right

Example 1 Example 2 Example 3

211 2(2 5) 340 2 (5) 20 42 100 36

211 2( 3) 40 8(5) 20 42 64

11 2(9) 5(5) 20 16 8

11 18 25 20 24

7 5

i) Perform the operations; simplify completely.

1) ( 3) 9 ( 11) ( 6) 2) 6 ( 13) ( 5 5) 3) ( 8) 9 (20 11)3

4) 8 2[4 2 5] 5) 236 2(3 7) 6) 9 16 16

7) 2 210 8 8) 3 23 3 9) 2 26 10 10) 9 5 2 14 11) 1 7 4 12 12) 1 6 76 1 ( 1)

13) 36 3( 2) 14) 2 2 2 26 7 3 0 15) 2 1 34 4 4 16) 2 3 36(5 3 ) 17) 22 25 5 2 9 3 18) 45 ( 2)

19) 45( 2) 20) 12 3 64 36 21) 2[3 ( 4)] 6 Exponent Laws and Integer Exponents Laws of Exponents: )0( x

1) baba xxx 2) bab

a

xx

x 3) ( )a b abx x

ii) Simplify with exponent laws – express answers with positive exponents.

1) 4 6r r 2) 3 12w w 3) 5 89 ( 8 )a a 4) 4 8 37 ( 2 )a b a b 5) 3 58 ( 6 )x x x 6) 8 29 ( 4 )y y y 7) 5 4 510 (3 )ab a b 8) 8 7 55 ( 3 )p r p r

 

3 

9) 5 2( 6 )x y 10) 7 3(4 )ab 11) 4 5 323( )a b 12) 7 5 41

2( )p r 13) 5y y

14) 5 7b b 15)

3 6(5 )4x y xy 16) 3 4 5 68 (4 )a b a b 17)

3 6 5( )a b 18) 3 2 6( )a b 19) 4 7 33

2( )a b 20) 1 5 254( )a b

21)

3 5 2( 9 )a b 22) 4 1 2(8 )a b

23) 4 7 3(5 )a b 24) 10 4 3( 2 )p q 25)

3 5 4( )a b c 26) 3 7 6( )a b c 27) 4 7 6( )x y 28) 5 6 2 3( )p q r

29)

3 2 3( 7 )4a b a 30) 4 4 26 ( 11 )x x y 31) 0 2 6 7( 2 )(5 )x y x y 32) 1 5 1 512( 2 )( )r p r p

33)

3 2 0( 7 )a b 34) 4 2 0( 11 )x y 35) 6 7 04(5 )x y 36) 1 5 012( 2 )r p

37)

3 77 7 7 38)

2 85 5 5 39) 6 3(2 ) 40) 5 3(4 )

41)

8

2

r

r 42)

15

3

a

a 43)

4

12

m

m 44)

7

14

p

p

45)

8

2 9

12

18

r s

r s

46)

3

4 9

27

18

ab

a b

47)

2

6

6

8

m

m

48) 7

5

9

3

p

p

49)

2 6

2 6

21

28

a b

a b

50)

12 4

2 4

24

32

a b

a b

51)

2

6

m m

m

52)

7

2 5

p

p p

53)

5

18

8 8

8

54)

16

2 7

10

10 10 55)

2 3

6 4

( )m n

m n

56) 7

2 5 3( )

p r

p r

57)

25 3

4

a b

a

58)

35

5

4x

x y

59)

45 3

1

r p

r p

60) 35 3

8

a b

a b

iii) True or False 1)

3 3( ) 1a 2) 2 2 1 2 2(5 5 ) 5 5 3) 3 3 2 6( ) 4x x x 4) 08 1

5) 4

4

13

3x

x 6)

2 3( 2 ) 64 7) 3

326v

v

8) 2 3 1 1(5 3 )

2

9) 3 1 2 63 3 3 3 10)

210 10 10n n n 11) 4 4

2 16

x x

12) 2 2 29 4 5

 

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b. Scientific Notation

i) Write each number in Scientific Notation, rounded to the nearest hundredth: . 10

1) 54,680 2) 512,900 3) 7,987,000 4) 4,322 5) 82,675,000 6) 786.5 7) 0.0614 8) 0.002348 9) 0.00094321 10) 3.0354824 11) 0.00008726 12) 0.000003857 ii) Convert to standard form.

1) 32.58 10 2) 45.29 10 3) 14.981 10 4) 38.6 10 5) 06.45 10 6) 47.81 10 7) 2461.5 10 8) 55.98 10 iii) Perform the operations. Write answers in Scientific Notation, rounded to nearest hundredth.

1) 8 8(4.62 10 ) (2.75 10 ) 2) 5 5(3.97 10 ) (2.6 10 ) 3) 7 7(7.64 10 ) (5.68 10 )

4) 12 12(5.68 10 ) (4.9 10 ) 5) 4 15(2.45 10 )(3 10 ) 6) 4 9(2 10 )(4.71 10 )

7) 8 2(2.8 10 ) 8) 4 2(1.9 10 ) 9) 15 2(2.07 10 )

10) 7 2(3.06 10 ) 11) 5

5

8.2 10

4 10

12) 12

3

7.62 10

6 10

13) 2

5

8.91 10

3 10

14) 12

4

1.526 10

0.7 10

15) 7

20

5.769 10

0.9 10

iv) Simplify completely; no decimal answers.

1) 210 2) 310 3) 36 3 4) 32 4 5) 2 15 3 6) 3 22 3 7) 28 8) 28 9) 2( 8) 10) 2( 8)

11) 3( 4) 12) 34 13) 34 14) 3( 4) 15) 2

10

7

16) 3

3

4

17) 4

2

3

18) 32

6

19) 2

2

6 20) 4

6

9

9

21) 11

8

5

5

22) 1

3

6

4 2

23)

1

2

6 10

5

24) 1 24 4 25) 2 32 2

 

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26) 1 26 3 27) 1 215 3 28) 3 12 3 12 29) 2 13 2 12 30) 3 26 2 4

31) 420(5 7) 32) 2(3 2 4) 33) 0 5 7 23 (3 3 ) 34) 33 2 35) 24 3

36) 8

6

2 2

2

37) 0 1 26 6 6 38) 3120 7 9 39) 2 36 16 2 40) 0 55 0

c) Rational Exponents and Radicals i) Write an equivalent expression with rational exponents.

1) 10 2) 3 9 3) 34 6 4) 268 5 5) 5(18.7)

6) 347 y 7) 37x 8) 4 216yx 9) 6 1012cab 10) 3 22 )(

4

xx

ii) Write an equivalent expression using radical notation (positive exponents, only).

1) 3

411 2) 2

520 3) 1

38 12 4) 1

42 5 5) 2

7( 26)

6) 2

726

7) 1

529 8)

143 9)

1 14 48 5 10)

1 12 22 3

11) 3 32x y

12)

12 5(6 )x y

13) 12 2( 25)x

14) 23 32( 8)a 15)

34 2( 25)x iii) Simplify. Assume all variables > 0.

1) 100

49 2) 3

8

125 3) 225 10 1a a 4) 123 64h

5) 7 714 )( ba 6) 8 124 6 n 7) 8 8)014.0( 8) 2

327

9) 2

327 10) 2

327

11) 2

3( 27) 12)

2327

13) 3

481

16

14) 31 210 4

15) 4 2

3 36 6 16)

10 1 29 6 9 9

17) 110 52(49 )n n

18) 3

23 2(19 )

19)

52

43

x

x

20) 23 0 25 ( )a b a b

 

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iv) Perform the operations - simplify.

1) 4 5( 8 5 )(2 10 )x y xy 2) 3( 3 6 )( 5 8 )a a 3) 2( 6 2 3)

4) 2(3 2 6) 5) 4 53 3( 8 6 )(2 9 )x y xy 6) 5 3 4 23 3( 3 25 )(4 10 )a b a b

7) 4

103332

2x

x yy

8)

4 10 10

3 32

5

40

a a b

b

9) 3 4 2( 3 5 )x x

10) 4 2 33( 2 9 )a a 11) 33 32 128 2 8 54 12) 3 3 3375 6 81 3

13) 3 (2 )x x y 14) 23 ( )x x y 15) 23 (2 ) ( )x x y x y

v) Simplify; rationalize denominators.

1) 15

5

12 5

3 45

y

y 2)

5

6 98

8 2

ab

a b 3)

7

5

32

2

x

xy 4)

58

14

r

rw

5) 2

30

5 18

cd

cd 6)

2 5

12

6

xy

x y 7)

3

8

2 8)

3

6

4

9) 3

12

12 10)

3

12

18 11)

2

23

x

xy 12)

53

xy

x y

13) 8

1 3 14)

20

5 3

15) 2 4

2

x

x

16) 2 6

3

x

x

17) 15 3

2 3 2 3 18)

5 2 2

6

19)

5 4

45 20 20)

2 4

75 27

21) 2

13

3

22) 2

32

2

23)

2

6 2

4 4

24)

2

1 3

2 2

vi) Given a rectangle’s width and length, find its perimeter, area and diagonal length. Simplify answers; use correct units.

1) width = 4 cm, length = 6 cm 2) width = 3 meters, length = 6 meters 3) width = 5 inches, length = 10 inches 4) width = 8 miles, length = 10 miles

 

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II. Equations and Inequalities a) Linear equations and Applications i) Solve or simplify – whichever is appropriate.

1) a) 5 2( 9) 10w w 2) a) (4 7)3 2 (11 3 )c c c 3) a) 3 35 59d d

b) 5 2( 9) 10w w b) (4 7)3 2 (11 3 )c c c b) 3 35 59d d

4) 292 100m 5)

21 35

36 48v 6)

100 25

49 21p

7) a) 17 3 547 9 7 6t t 8) a) 3 5 3 5

8 6 4 8h h 9) a) 712(8 ) 7n n

b) 17 3 547 9 7 6t t b) 3 5 3 5

8 6 4 8h h b) 712(8 ) ( 7)n n

10) a) 7

6 [20 8( 3)] 7a a a 11) a) 0.3 0.2 1.7 1.4a a 12) a) 0.3 0.2 1.7 1.4a a

b) 76 [20 8( 3)] 7a a a b) 0.3 0.2 1.7 1.4a a b) 0.3 0.2 1.7 1.4a a

13) a)

5 31 13 8 6 4a a 14) a) 1.7 1.3 0.36M M 15) a) 3.4 2 1.7 4.3n n

b) 5 31 1

3 8 6 4a a b) 1.7 1.3 0.36M M b) 3.4 2 1.7 4.3n n

16) 0.28 0.4y 17) 0.8 0.26n 18) 0.24 0.06 1 0.2p p

19) a) 3 14 6(8 2 ) 1y y 20) a)

4 7

12 8

nn

21) a)

4 3 7 7 3

15 10 6

a a

b) 3 14 6(8 2 ) 1y y b)

4 7

12 8

nn

b)

4 3 7 7 3

15 10 6

a a

ii) Applications 1) The smaller of two complementary angles is six degrees more than 40% of the measure of the larger angle. Find the measure of each angle. a) Express each in terms of x : smaller angle ___________ larger angle ____________ b) Equation (solve!): 2) The smaller of two supplementary angles is nine degrees less than three-fourths the measure of the larger angle. Find the measure of each angle. a) Express each in terms of x : smaller angle ___________ larger angle ____________ b) Equation (solve!): 3) The smallest angle of a triangle is 7 less than one-third the measure of the largest angle; the middle-sized angle is one-half the measure of the largest angle. Find the measure of the three angles. a) Express each in terms of x : smallest angle ___________ middle angle ____________ largest angle ____________ b) Equation (solve!):

 

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4) Angle A of a triangle is three times the measurement of angle B; angle C is 40 more than angle B. Find the measures of the three angles. a) Express each in terms of x : angle A ___________ angle B ____________ angle C ____________ b) Equation (solve!): 5) Thirty ounces of a drink that is 12% juice is mixed with 50 ounces of a drink that is 20% juice. What per cent of the mixture is juice? 6) Ten grams of jewelry that is 24 karat gold is melted with 30 grams of jewelry that is 18 karat gold. Find the karat rating of the melted mixture. 7) Some amount of a 12% acid solution is mixed with 80 ml of a 4% acid solution to produce a 10% solution. Find the amount of the 12% acid needed to produce the desired outcome. a) Model this mixture problem - draw/label a diagram, complete an organized chart, etc. b) Write an equation – solve. 8) Sixty ounces of a 30% saline solution is mixed with some amount of an 8% saline solution to produce a 20% solution. Find the amount of the 8% solution is needed to produce the desired outcome. a) Model this mixture problem - draw/label a diagram, complete an organized chart, etc. b) Write an equation – solve. 9) A 14% HCL solution is mixed with a 6% HCL solution to a produce a 40 ml solution. How much of each concentration is needed to produce a mixture that is 11% HCL? a) Model this mixture problem - draw/label a diagram, complete an organized chart, etc. b) Write an equation – solve. 10) A fruit drink that is 30% juice is mixed with a drink that is 12% juice. How much of each drink is needed to produce a 90 ounce mixture that is 18% juice? a) Model this mixture problem - draw/label a diagram, complete an organized chart, etc. b) Write an equation – solve. b) Interval Notation and Linear Inequalities i) Write the inequalities in [Interval, Notation).

1) 2x 2) 15a 3) 100n 4) 15b 5) 5 2x 6) 5 20a 7) 300 0n 8) 0.5 1.8b 9) 3 2x or x 10) 4 10x or x 11) 0.45 2.17x or x 12) 0 2x or x

 

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ii) a) Translate each statement to an inequality. b) Graph the inequality. Label 3+ numbers. c) Write the inequality in [Interval, Notation). Statement Inequality Graph [Int, Not)

1) Heather has more than $30 in her purse.

2) Egor owns fewer than eight monkeys.

3) There are at least 12 people in the club.

4) The cost of the bicycle is at most $250.

5) My rabbit weighs less than 2.8 pounds.

6) The mass of the fox is greater than 3.4 kg.

7) A recipe calls for no more than 132 cups of milk.

8) Beatrice has more than 5 cats, but at most 8.

9) The temperature is between 75 and 80 .

10) Juan has at least $25, but less than $40. iii) Solve the inequalities. Express solutions in [Interval, Notation).

1) 3 1 7a a 2) 5 2 3 10n n n 3) 5 7 1w w 4) 2 6 3(4 8)c c 5) 150 a 6) 600 b 7) 3 1.3 2.6x 8) 2 5.2 3y 9) 3 5 1

4 3 6h 10) 3 712 5 10p 11) 3(2 6) 4 18d d 12) 15 (7 4)5r r

13) 133 20b 14)

146 75x 15)

1 72

6 8

m m 16)

5 1 21

9 12

a a

17) 0.6 8.52a 18) 0.4 7.4t 19) 0.7 5.01y y 20) 0.6 3n n 21) 2 3 2 14r 22) 2[ (3 2 )] 1n n 23) 3[ (4 1)2] 1x x 24) 11 2 7 1a 25) 2

59 3 1x 26) 3413 2 11y 27) 1 8 0.3 5a 28) 4 2 0.2 3b

 

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c) And/Or problems i) Solve each And/Or problem. Graph solution; label 3+ numbers. 1) 2) 3

83 1 5 0x and x 32 1 5 2 0y and y

3) 4) 2 1 3 2 5a or a 3

25 5 3n or n

ii) Solve each And/Or problem. Express solution in [Interval, Notation). 1) 5 4 6 2 1 7x and x 2) 1

37 4 4 2y and y 3) 1 14 26 2 5a and a

4) 2 6 5 11m and m 5) 4

3 12 3 21p and p 6) 538 1 1 10x or x

7) 4 1 3 2 11a or a 8) 4 5 10 0.2 4c c or c 9) 2 3 4 3 1y or y

d) Absolute Value i) Solve each .ab value problem.

1) 32 11 1x 2)

7 15

4

y 3) 15 4a a 4) 2 7 5 2n n

ii) Solve each .ab value problem. Express solution in [Interval, Notation).

1) 7 2

13

y 2) 9 5 4x 3) 1

43 1b 4) 3 7 5n n

5) 15 2 8x 6)

3 15

4

r 7) 4 1 8 5p p 8) 3 6 15y

III. Lines, Graphs and Applications a) Linear forms and graphs Slope-Intercept form: bmxy , where m = slope and b y-intercept Point-Slope form: 1 1( )y y m x x , where m = slope and 1 1( , )x y is a point

 

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01202403604806007208409601080

0 5 10 15

Value in

 $

Months (since purchase)

Computer ValueBertha's Height

0

4

8

12

16

20

24

28

32

0 2 4 6 8

Age in Years

Hei

gh

t in

In

ches

General form: CByAx , where ,, BA and C are integers

Slope, given two points 1 1( , )x y and 2 2( , )x y : 12

12

xx

yym

, this is x

y

or run

rise

i) Write each line in CByAx form.

1) 345 xy 2) 2

9 1y x 3) 1.2 0.51y x

4) 0.4 1.2y x 5) 763 ( 2)y x 6) 6

54 ( 10)y x ii) a) Write the given line in slope-intercept form, bmxy . b) Find coordinates of x- and y-intercepts. c) Graph the line.

1) 3 4 8x y 2) 2 4x y 3) 0.5 0.3 3x y

4) 4 3 3x y 5) 27

41

65 yx 6) 1.2 3 0x y

iii) Find the slope of the line passing through each pair of points; simplify.

1) (3, 5) and (6, 2) 2) (-9, 7) and (-8, 5) 3) (-3, 4) and (-3, -4) 4) (-6, 0) and (4, 3) 5) (-12, 3) and (8, 3) 6) 1 2

2 3( , ) and 3 48 5( , )

7) 34( ,1) and 1 4

2 3( , ) 8) (1.8, 5) and (-3, 2) 9) (-2, 5.3) and (-6, 2.1)

iv) a) Graph the line. b) Find the slope of the line. c) Find coordinates of x- and y-intercepts (when possible).

1) 4 3y x 2) 54 4y x 3) 2y x 4) 3y x

5) 4x 6) 1y 7) 3y 8) 2x b) Applications: Interpreting slope and y-intercept i) a) Find two points; calculate the slope of the line. b) Find the linear equation in bmxy form. c) Verbally interpret slope and y-intercept. 1) 2)

 

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ii) Applications in y mx b form

1) The cost, in dollars, for x minutes of use during a one-month period for a certain cell phone plan is ( ) 0.05 27.50C x x .

a) Verbally interpret slope and y-int. b) Find (300)C ; verbally interpret. c) If ( ) 35C x , find .x Interpret! 2) The average age at death, in years, of Russian males t years after 1985 is ( ) 0.2 61.3A t t . a) Verbally interpret slope and y-int. b) Find (10);A verbally interpret. c) When will that age fall to 55 years? 3) Ralph has been on a diet for t months. His weight, in pounds, is modeled by ( ) 4.5 230W t t . a) Verbally interpret slope and y-int. b) When will Ralph reach a weight of 200 lbs.? 4) The price of admission, in dollars, to visit Disneyland t years after 1960 is modeled by the formula

ttP 81.025.5)( . a) Verbally interpret slope and y-int. b) Find );20(P verbally interpret. c) Equations of lines and Linear Applications i) Find the equation of each line. When possible, write solution in bmxy form.

1) through (-4, 1) with slope 56 2) through (6, -2) with slope 5

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3) through (-3, 3) and (3, 1) 4) through (-12, 5) and (8, 5) 5) through (1, 5), perpendicular to the y-axis 6) through (-2, 4), perpendicular to the x-axis 7) through (1, 5), parallel to the y-axis 8) through (-2, 4), parallel to the x-axis 9) through the origin and (3, -6) 10) through (5, -2), parallel to 64

3 xy

11) through (1, -2), perpendicular to 3 10y x 12) through (-1, -3) with undefined slope 13) through (-1, -3) with slope = 0 14) with x-int. of (5, 0) and y-int. (0, -3) 15) through (-4, 0), perpendicular to 652 yx 16) through (2, -5), parallel to 4 3 6x y ii) Linear Equations - Applications 1) A one-day car rental cost Monica $28, plus 20 cents per mile driven. a) Find a linear equation for the cost of a one-day car rental. What does x represent? Explain. b) How much would it cost to drive the rental car 81 miles? c) If Julia paid $62.40 for a one-day rental, how many miles did she drive? 2) Each month a real estate agent is paid $500, plus 3% of sales. a) Find a linear equation that represents the agent’s monthly salary. b) If the agent sells $400,000 in properties during March, determine the agent’s earnings. 3) Water boils at 100 C , or 212 F, and it freezes at C0 , or 32 F. a) Write a linear equation that expresses the Fahrenheit, F, as a function of C, Celsius. Find: b) Fahrenheit temperature if 30C

c) Celsius temperature if 77F

 

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4) Two months after Wanda was born she measured 19 inches in length. By the age of six months, she had grown to 22 inches. Let x = Wanda’s age in months. a) Find the linear equation that relates Wanda’s length/height to her age, x . b) Verbally interpret the slope. c) Verbally interpret the y-intercept. d) Find Anita’s height at one year of age. e) When will her height reach 28 inches? 5) Rising at a constant rate, a plane reached an altitude of 5,000 feet above sea level four minutes after take-off. At nine minutes, it attained an altitude of 9,000 feet. a) Find the linear equation that relates the plane’s altitude to time since take-off. b) When did the plane reach an altitude of 12,000 feet? c) Verbally interpret the slope and y-intercept. 6) At birth in 1900, the life expectancy of a male in the USA averaged 46.3 years; by 1950, that figure had increased to 65.6 years. Let x be the number of years since 1900. a) Find the linear equation that relates male life expectancy to year of birth. b) Use that result to estimate the life expectancy of an American male born in the year 2000. 7) The domestic share of the US automobile market has been decreasing for decades. In 1970, US automakers commanded approximately 84% of the market; two decades later, that share had slipped to around 65%. Let x = the number of years after 1970. a) Find the linear equation that relates domestic auto market share x years after 1970. b) When did the market share fall to 50%? c) Find the slope; verbally interpret. d) Graphing Inequalities in the Plane Graph and shade the linear inequalities.

1) 3 2x y 2) 2 4 0x y

 

14 

3) 2 1x and y 4) 3 2x and y 5) 2 1x or y 6) 3 2x or y

7) 32

2y x

y x

8) 23 2

2 1

y x

y x

 

15 

e) Solving Systems of Equations i) Solve by graphing: use a straight-edge.

1) 12

2 4

1

y x

y x

2) 3 5

3

y x

y x

ii) Solve by elimination or substitution.

1) 34

12

2

1

y x

y x

2) 0.6 2.4

1.2 4.2

y x

y x

3) 5 9

3 2 8

x y

x y

4) 6 16

3 2 8

x y

x y

5) 5 3 7

4 2 12

x y

x y

6) 7 2 21

4 5 12

a b

a b

7) 3 4

4 9 5

y x

x y

8) 34 1

8 3 16

x y

x y

IV. Factoring, Expressions, Equations, Variation a) Factoring techniques i) Factor out the GCF. 1) 4 12 418 27a b ab 2) 6 4 2 435 20x y x y 3) 8 4 212 15 3y y y

4) 3 221 28 7ab a b ab 5) 2( ) ( )a b c a b 6) 2 2( 1) ( 1)x y w y

ii) Factor out the GCF and -1. 1) 3 4 2 54 12x y x y 2) 216( ) 8( )a b a b 3) 3 22 4 10n n n

4) 23 12 30a b ab b 5) ( 2) (2 )a b c b 6) 3 (5 1) (1 5 )x w y w

iii) Factor by grouping. 1) 5 5ac a bc b 2) 2 33 6 4 8n m n mn 3) 3 26 3 10 5x x x 4) 2 2 2 33 6 4 8a a b b b 5) 15 2 3 10xy x y 6) 14 12 21 8ab a b

 

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iv) Factor the Trinomials of the form 2x bx c . 1) 2 6 5b b 2) 2 8 12c c 3) 2 212 45a ab b 4) 2 217 30r rs s 5) 2 5 24x x 6) 218 7 y y

7) 3 24 20 24p p p 8) 4 3 22 22 48y y y 9) 3 25 20 105n n n

v) Factor the Trinomials of the form 2ax bx c . 1) 22 3 5w w 2) 26 11 3a a 3) 26 7 10h h 4) 28 5 3r r 5) 26 17 7v v 6) 210 18 4p p

7) 2 29 15 4x xy y 8) 2 212 4 5a ab b 9) 228 34 12r r

vi) Factor the difference of squares. 1) 2100 b 2) 24 49k 3) 2 264 81a b 4) 2 1

25x 5) 2136 25y 6) 481 p

7) 4 16n 8) 4 23 48a b a b 9) 2 2( 3 ) (2 )a b a b simplify!

vii) Factor the sum/difference of cubes. 3 3 2 2( )( )a b a b a ab b or 3 3 2 2( )( )a b a b a ab b

1) 3 64n 2) 38 1x 3) 327 125r 4) 364 125w 5) 6 327 8x y 6) 121000 27m b) Factor to Solve, to Simplify i) Solve the equations by factoring.

1) 2 11 60 0x x 2) 2 4 32 0a a 3) 26 25 4 0y y

4) 24 4 3 0n n 5) (2 3)( 6) 13x x 6) (3 1)( 2) 20y y

7) 26 12n n 8) 2100 4 0a 9) 2194 0n

ii) Factor (when necessary) to simplify the expressions.

1) 212

243

25

15

cab

cba

2) 3 15 0

5 6

48

36

x y z

xy z 3)

22

2

xy

xyx

4) 2

6 2

2

20 6

w

w w 5)

2 2

2 2

3 6

6 16

x y xy

x xy y

6) 2

2

25 20 4

6 11 10

a a

a a

7) 2 8

2 2

3 25 8 30

24 5 15

x x x

x x x

8)

2 2 2

4 5 3

4 2

3 6 9

a b a ab

ab b b

9)

3

2

64

2 8 4

y

xy x y y

 

17 

iii) Solve by the “square-root” method. Recall: 1 i . Simplify solutions.

1) 2 36r 2) 24 81p 3) 29 16x 4) 225 36y

5) 2(3 1) 100x 6) 2(4 1) 49h 7) 26 1 24z 8) 28 3 4c

9) 2( 3) 5 50x 10) 2( 1) 3 21y 11) 2( 2) 7 68b 12) 2( 3) 9 16d

iv) Solve by completing the square. Simplify solutions.

1) 2 6 27 0y y 2) 2 8 7 0t t 3) 2 4 20 0a a 4) 2 10 6 0r r

5) 2 5 1 0b b 6) 2 4 0c c 7) 212 3 4 0x x 8) 21

3 4 1 0y y

9) 25 50 10p p 10) 22 10 6y y

v) Solve – use Quadratic formula. Simplify solutions.

1) 2 6 27 0y y 2) 2 8 7 0t t 3) 2 4 20 0a a 4) 2 10 6 0r r

5) 23 6 1 0y y 6) 22 8 3 0x x 7) 24 2 1 0a a 8) 26 4 3 0n n

vi) Solve. Find all solutions - simplify.

1) 3 8 0r 2) 3 27 0y 3) 264 81 0m 4) 3125 1 0c

5) 4 81 0a 6) 4 16 0w 7) 4 25 36 0b b 8) 4 24 7 2 0r r c) Expressions and Equations i) Simplify the expressions; solve the equations. Find “restrictions” for each equation.

1) 5 1 2

9 6 3

r r r

5 1 2

9 6 3

r r r

LCD = ____________ LCD = ____________

2) 2

5 3 11

3 4 12x x x

2

5 3 11

3 4 12x x x

LCD = ____________ LCD = ____________

3) 2

35 5

5 6 6n n n

2

35 51

5 6 6n n n

LCD = ____________ LCD = ____________

4) 6 5 1

15 5 18 6 15a a

6 5 1

15 5 18 6 15a a

LCD = ____________ LCD = ____________

 

18 

5) 2

3 4

2 2

y

y y y y

2

3 4

2 2

y

y y y y

LCD = ____________ LCD = ____________ ii) Determine the LCD, then add/subtract and simplify. 1) LCD = _____________________ 2) LCD = _______________________

65

3

65

1322

xx

x

xx

x

4 2

3 52

8 6x y xy

3) LCD = _____________________ 4) LCD = __________________

2 1 63 4a b a b 2 2

4 6

4 6 8

w w

w w w

iii) Determine the LCD, any restriction(s), then solve. 1) LCD = ___________________ 2) LCD = _____________________ restriction(s): restriction(s):

x

x 2

6

11

3 r

rr

2

3

6

51

9

72

3) LCD = ___________________ 4) LCD = _____________________ restriction(s): restriction(s):

12

3

592

22

5

22

wwww

2 2 2

2 5 6

1x x x x x

d) Literal Equations and Variation i) Solve for the indicated variable.

1) ;Ax By C y 2) ;p V

M Va

3) ;G n dn n

4) ;E

I rR r

5) ;A

P AnA b

6) 21 2

1 1 1; R

R R R

 

19 

ii) Variation Equations Step 1: Write the Variation Equation. Step 2: Find k, the “Constant of Variation”. Step 3: Find the missing value, if any.

1) T varies directly as 3 m , and m = 8 when T = 20. Find T when m = 271 .

2) y varies directly as x and inversely as the cube of w . Find the variation equation if 2y when

10x and 5w . Find y when 48x and 2w . 3) If B varies jointly as z and the square root of v , and 20B when 3z and 100v , find B when

6z and .64v 4) If W varies inversely as the square of r , and 8

3W when 4r , find W when .2r

5) The intensity I of a light varies inversely as the square of the distance d from that light source. If I = 80 2/W m (watts per square meter) when d = 5 meters, find I when d = 2 meters. 6) Elmo’s profit, P , varies directly as the square root of the number of hot dogs he sells, n . If Elmo profited $30 from the sale of 25 hot dogs, calculate his profit if he sells 36 hotdogs. 7) Sarina’s income, I , varies directly as the number, n , of automobiles she sells. If she earned $2600 from the sale of four cars during one week, how many cars did she sell during a week in which she earned $4550? 8) The distance, d , to the horizon (in miles) is directly proportional to the square root of the height, h , of an individual’s eyes above the ground (in feet). For example, Ann can see three miles since her eyes are four feet above the ground. How far can she see if her eyes were instead 9 feet above the ground? 9) Sid’s transit time, ,t to work varies inversely as his bicycle’s speed, s . He can bicycle to work in one-third of an hour if he averages 15 mph. a) How long will the trip take if his speed to 20 mph? b) How far does Sid live from work? 10) The mass, m (in grams), of a finch varies jointly as its length, L , and the square of its girth, g (both in cm). A finch 8 cm long with a girth of 4 cm has a mass of 32 g. Find a finch’s mass if its length and girth are 10 cm and 6 cm, respectively. 11) The current, I , of an electrical circuit is inversely proportional to its resistance, R . If the current is 15 amps when the resistance is 12 ohms, find the current when the resistance is 9 ohms. Also, find the resistance when the current is 25 amps.

 

20 

V. Relations and Functions a) Domain and range i) Find the domain and range for each relation; determine if a function.

1) {( 3,1),(2,4),(0, 6)}S 2) {( 2,1),(6, 5),(8,1),( 3, 3)}S 3) {(4, 2),(3,8),(4,3)}S domain: domain: domain: range: range: range: function: YES NO function: YES NO function: YES NO ii) Find the domain and range for each relation - use [Interval, Notation). Determine if a function.

1) 2) 3) 4) 5) 6)

iii) Calculate the domain of each function. Express answer in [Interval, Notation).

1) 2

4( )

3 10

xf x

x x

2)

23 10

4

x xy

x

3) ( ) 14 2g x x

4) 23

( )14 2

xf x

x

5) 3( ) 14 2f x x 6)

3

4( )

14 2h x

x

7) 3( ) 8 2f x x x 8) 4 3

5 2 6

xy

x

9) 2( )

4

xf x

x

 

21 

b) Graphing Parabolas i) Graph each parabola; label 3 or more points. Find vertex, intercepts, domain and range. 1) 2y x vertex: ( , ) 2) 2 4y x vertex: ( , )   

 

 

 

Domain: Range: Domain: Range: x-int.: ( . ) y-int.: ( . ) x-ints.: ( . ), ( . ) y-int.: ( . ) 3) 2( 3)y x vertex: ( , ) 4) 22y x vertex: ( , ) Domain: Range: Domain: Range: x-int.: ( . ) y-int.: ( . ) x-int.: ( . ) y-int.: ( . )

 

22 

5) 213 3y x vertex: ( . ) 6)

2( 2) 1y x vertex: ( , )

Domain: Range: Domain: Range:

x-ints.: ( . ), ( , ) y-int.: ( . ) x-ints.: ( . ), ( , ) y-int.: ( . ) ii) Step 1: “Complete the Square” to write each quadratic function in the form: 2( )y a x h k Step 2: Identify the vertex of the quadratic function. Step 3: Explain how the given quadratic function is obtained from: 2( )f x x 1) 2 6 2y x x 2)

2 8 9y x x 3) 2 3 2y x x 4) 2 5 1y x x 5)

22 8 4y x x 6) 23 36 6y x x

7) 2 10 5y x x 8) 213 4 2y x x 9)

212 8 3y x x

c) Algebra of functions Given ( ) 5 2f x x , 2( ) 2 8g x x and 2( ) 3 2h x x x , find and simplify:

1) 74( )f 2) ( 3)g 3) (1)h 4) 4 ( 3.2)f 5) 2

3( )h

6) 52( )g 7) (0.2)h 8) (3) 2g 9) (3 2)g 10) 1

2 ( 4)h

11) ( ) ( )g t h t 12) ( ) ( )h a g a 13) ( ) ( )f r g r 14) ( ) [ ( ) ( )]f x g x h x

15) ( )

( )

g x

h x 16)

( )

( )

f a

g a 17) ( 3)f x 18) ( ) 3f x 19) ( ) 1g x

20) ( ) 1g x 21) (2 )h x 22) 3( )h x 23) ( ) ( )h a h h a 24) 2( )xh

 

23 

d) The Difference Quotient

For each function, find and simplify: ( ) ( )f a h f a

h

1) ( ) 9 4f x x 2) ( ) 7 3f x x 3)

2( ) 3 1f x x 4) 2( ) 2 5f x x

5) 2( ) 4 3f x x x 6) 2( ) 5 12f x x x 7) 2( ) 3 2f x x x 8) 2( ) 6f x x x

9) 6

( )f xx

10) 2

( )3

f xx

11) 2

1( )

4f x

x 12) 2

5( )f x

x

VI. Exponential and Logarithmic Properties, Equations and Applications a) Exponential graphs and equations

i) Graph each exponential function – include asymptote; label 3+ points. Find domain, range, equation of asymptote and x- and y-intercepts (approximate, if necessary).

1) 12 4xy Asymptote: 2) 12 1xy Asymptote:  

 

 

Domain: Range: Domain: Range: x-int.: ( , ) y-int.: ( , ) x-int.: ( , ) y-int.: ( , )

 

24 

3) 22 xy Asymptote: 4) 5 2xy Asymptote: Domain: Range: Domain: Range:

x-int.: ( , ) y-int.: ( , ) x-int.: ( , ) y-int.: ( , ) 5)

12( ) 2xy Asymptote: 6) 11

34 ( ) xy

Asymptote:

Domain: Range: Domain: Range:

x-int.: ( , ) y-int.: ( , ) x-int.: ( , ) y-int.: ( , )

 

25 

7) 2 1yx Asymptote: 8) 12 yx Asymptote:

Domain: Range: Domain: Range: x-int.: ( . ) y-int.: ( . ) x-int.: ( . ) y-int.: ( . ) ii) Solve the exponential equations. [Hint: Make bases same!]

1) 3 12 16x 2) 2 2 1162 x 3)

12 1

273 x 4) 0.410 0.01x

5) 4097 32

y 6) 3 25 25 150n 7) 3 1 1819 ( )x x 8) 34(25) 500x

9) 1 410 (1000)r 10) 2 5 148 ( )y y 11) 24 8n n 12) 91

981 ( )x x

13) 3 4 2 2( )x x xe e e 14) 5 3 1 35 5 25a a 15) 00001.0104 x 16) 542 xx ee   

b) Logarithmic properties and equations Exponential and Logarithmic properties

Laws of Exponents: )0( x Laws of Logarithms: )1,0,,( xxba

1) baba xxx 1) log log logx x xa b ab

2) bab

a

xx

x 2) log log logx x x

aa b

b

3) ( )a b abx x 3) log log rx xr a a

  Common Log: )0( x Natural Log: )0( x , 718.2e

10log logx x xx elogln

 

 

26 

i) Write in exponential form. 1) 3125log5 2) 1

4 16log ( ) 2 3) 4000,10log

4) 2389.7ln 5) ln50 2 7n 6) 2log (3 1) 4x ii) Write in logarithmic form. 1) 36 216 2) 32

152 3) 210 0.01

4) 10 e 5) OD G 6) 2(0.2) 25 iii) Simplify the expressions with Laws of Logarithms.

1) 8log 8 2) 9log 9 3) 7log 7 4) 34

6log 6

5) 5lne 6) ln3e 7) 3log 103 8) 4log 124 9) 64log 4 10) log0.001 11)

542log 64 12) 16log16log 42

13) 2log (8 8) 14) 2log (8 4) 15) 5 3log(10 ) 16) 16 368 log ( )

17) 7

8log 8 18)

78(log 8) 19) 6 6log 18 log 2 20) 3 3log 2 log 54

21) 2 2log 5 log 80 22) 2log 2 log 25 23) 1 1200 5log log

24) 3log 2 log125 25) 12 122log 3 4log 2 26) 6 6 6log 9 log 8 log 2

iv) Solve – find the exact solution.

1) 2log8 x 2) 5log 3r 3) 3log (5 2) 3x

4) 5log (13 1) 2p 5) 4log 16 6 1x 6) 5log 125 8 1r

7) 3 3log ( 2) log 3x x 8) 24 4log log (2 8)y y 9) 2log log(6 40)a a

10) 2log(3 ) log(8 6)r r r 11) 2log(5 15) 4x 12) 3log 49 6n

13) 38 3log (2 1) 2w 14) 2 12 81 2log ( ) 3y 15) 702 x

16) 5 245x 17) 2 13 150x 18) 3 24 76x 19) 210 8 200y 20) 4 17 12n n 21)

1 2 39 5x x 22)

85 75te 23) 2 1 2.45te 24) 83 7 38te

25) 2ln(6 1) 8x 26) 3ln(7 3) 6p 27) log log( 18) 1r r

28) 2ln 4 lna a 29) log log( 9) 1x x 30) 122 log (5 1) 3x

c) Exponential and Log Applications 1) The amount of radio-active material (in grams) remaining t hours after the start of an experiment is

modeled by 1 52( ) 280( )

tA t .

a) Find (0)A ; verbally interpret that result. b) Find (10)A ; verbally interpret that result. c) When will exactly 35 grams remain? d) When will 10% of the original amount remain? e) Find the half-life of this substance. (This answer should involve logs.)

 

27 

2) The amount of radio-active material (in milligrams) remaining t minutes after the start of an experiment is modeled by 0.081

2( ) 500( ) tA t .

a) Find (0)A ; verbally interpret that result. b) Find the half-life of this substance. c) When will 4% of the original amount remain? (This answer should involve logs.) 3) If the half-life of a type of phosphorus is 18 days, and the initial amount present is 8.2 grams, find a

model of the form 10 2( ) ( )

tkA t A . Estimate how long it will take for the amount to decay to slightly less

than one gram. 4) The half-life of Cesium-137, a radio-active substance, is 30 years. At the site of a nuclear power plant, 315 grams are present 60 years after an accident.

a) Find a model of the form 10 2( ) ( )

tkA t A .

b) How many grams of Cesium-137were initially present? 5) The population of a rural town increases by 10% annually; the population was 800 in the year 2000. a) Model the town’s population using the following: 0( ) (1 )tP t P r , where 0P is the initial amount, r

is the rate of growth/decay, and t is time, in years, after the year 2000. b) Find (2)P ; verbally interpret that result. c) How long will it take the population to reach 1,400? (This answer should involve logs.) 6) The population of a rural town decreases by 8% annually; the population was 400 in the year 2010. a) Model the town’s population using the following: 0( ) (1 )tP t P r , where 0P is the initial amount, r

is the rate of growth/decay, and t is time, in years, after the year 2010. b) Find (1)P ; verbally interpret that result. c) How long will it take the population to reach 300? (This answer should involve logs.) 7) a) If inflation is 4% per year, and the price of bread is $3.00 per loaf, model the cost of that bread using the formula 0( ) (1 )tP t P r .

b) How much will the bread cost after 1 year? After 2 years? c) When will the bread cost $4.00? 8) The average speed (in kilometers per hour) of traffic in a certain city can be estimated by

2( ) 59.5 4log ( 1)S x x , where x is the city’s population in millions. a) What is the average traffic speed if the population is one million? b) What is the average traffic speed if the population is three million? c) At what population will the average traffic speed drop to 40 kph? (Exponential answer.)

9) The Richter number, R , represents the magnitude of an earthquake. In the formula, 0

logI

RI

, I is

the intensity of an observed earthquake, while 0I is the baseline intensity to which all earthquakes are

compared. a) Find R , the Richter number, and interpret the result, if: i) 0100I I ii) 010,000I I

b) If the Richter number is 5, find I in terms of 0I .

c) If the Richter number is 9, find I in terms of 0I .