NAME:_________________________

St. Francis High School Advanced Algebra Mastery Skills Workbook Use this workbook to help prepare for the Mastery Skills test that will be given in the first week of school to all students enrolled in Pre-Calculus or Honors Pre-Calculus. The format of the test is multiple choice. Do the first Practice Test – it contains samples of the types of problems on the test. If you are having trouble with any section more problems can be found in the following pages of the workbook or by searching the internet or in workbooks available in bookstores. The test will be taken WITHOUT calculators. Answers are provided at the back of the workbook. Work through the skill sections during the summer. A sample multiple choice test is included at the end of the workbook. Take this sample test the week before schools starts and brush up on any sections that you found difficult. You will be asked to do extra work on the skills you do not successfully master. Good luck.

Advanced Algebra Mastery Skills Workbook

Page 1

’15-‘16

ADVANCED ALGEBRA MASTERY SKILLS PRACTICE

TEST

The test you will take on the first day of school has question similar to this test. You are not expected to get 100%, but you should get most problems in each skill correct. Practice for this and you will start your year off right. NO CALCULATOR!!! SKILL AA1: Writing the equations of lines, parallel and perpendicular lines 1. Write the equation of the line in 2. Write the equation of the line slope-intercept form with points (3,7) going through the points (-2, -6) and and (4, 10) in slope intercept form (0,-5)

4. Write an equation that is parallel to through the point (-5, 8)

y = -4x + 7

SKILL AA2: Vertical and horizontal lines 6. What is the equation of this line?

SKILL AA3: Systems of equations 8. Solve using systems of equations

4x + 3y = 8 -6x + y = -12

Advanced Algebra Mastery Skills Workbook

3. Are these equations parallel, perpendicular, or neither?

2x  4 y  2 4 x  2y  3

5. The slope of the line that is perpendicular to the line with equation -2x + 9y = 20 is:

7. Write the equation of the horizontal line that goes through the point (11,-3)

9. Solve using systems of equations

-x + y = 8 3x + y = -4

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’15-‘16

10. Solve using systems of equations 8x  4y  20 2x  y  5

SKILL AA4: Properties of exponents (7 x)(2 x)3 12. Simplify.

 4 p4  15. Simplify. (3p r )    r 

11. In a basketball game between the Chicago Bulls and the Cleveland Cavaliers, the total number of points scored was 185. The Bulls scored 53 more than half that of Cleveland. Let x represent Chicago’s points and y represent Cleveland’s points. How many points did each team score?

13. Simplify.

y5 y 3

2

2 2 4

16. Simplify.

14. Simplify.

35x 12 y 3 z 4 15x 4 y 13 z 4

15 x 3  10 x 5x 2

SKILL AA5: Factoring 17. Factor Completely.

8a4 b3  36a4 b2

18. Factor Completely.

6 p2  11p  10

19. Factor Completely.

3y 2  18y  15

20. Factor Completely.

16d 2  49

21. Factor Completely.

25n2  30n  9

Advanced Algebra Mastery Skills Workbook

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’15-‘16

SKILL AA6: Quadratics Given y   x 2  4 x  5 22. Solve for the zeros of the quadratic.

23. Find the vertex of the parabola.

26. Graph y   x 2  4 x  5

24. Find the y-intercept



25. Find the x-intercept(s)

y

                  

x 















 

       

SKILL AA7: Radicals 27. Simplify completely:

3

270

28. Simplify completely:

3

8a 7b12

29. Simplify and perform the indicated operation.

48 y  4 27 y

30. Simplify and rationalize the

8 denominator of: 12

31. Simplify and rationalize the denominator of:

Advanced Algebra Mastery Skills Workbook

2y

6

32

Page 4

32. Simplify completely

9 12  12 50 3 2

’15-‘16

SKILL AA8: The six trig functions Use the triangle below to answer #33-35

To answer #33-35, use the triangle at the left to state the ratio that satisfies the given trig function. 33. cotθ =

34. cscθ =

5 θ

35. cosθ =

6 36. In a right triangle, θ is an acute angle and

sin  

37. In a right triangle, θ is an acute angle and

7 What is tan  ? 25

sec  

38. A 7 m ladder is leaning against the side of a building to reach a 2nd floor window. If the window is 5 meters off the ground, find the measure of the acute angle the ladder makes with the building (set up the equation).

Advanced Algebra Mastery Skills Workbook

4 7 What is cot  ? 10

39. You are standing at a bus stop and spot a stained glass window near the top of a cathedral that is 350 feet tall. The angle of elevation from the ground to the window is 72°. What is the equation you would use to find how far the bus stop is from the bottom of the cathedral?

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MASTERY SKILLS PRACTICE SECTIONS SKILL AA1: WRITING THE EQUATIONS OF LINES, PARALLEL AND PERPENDICULAR LINES  Slope-intercept form of a line: y  mx  b , where m is the slope and b is the y-intercept.



y2  y1 , where ( x1 , y1 ) and ( x2 , y2 ) are points on a line. x2  x1 Point-slope form of a line: y  y1  m( x  x1 ) , where m is the slope and ( x1 , y1 ) is a point on

  

the line. o Use when given:  The slope and a point which is not the y-intercept  2 points o Start in point-slope form and convert to slope-intercept form Parallel Lines: have the same slope Perpendicular Lines: the slopes are opposite reciprocals Standard form: Ax  By  C , where A > 0 and A, B, and C are integers



Slope: m 

For #1-6, write the equation of the line in slope-intercept form. 1. Slope is 2 and y intercept is-4 2. Slope is ½ and point (0,1)

3. (-2, 2) and (0, -1)

4. (-4, 3) and (0, -5)

5. (0, 3) and (2,-5)

6. (-5, -7) and (0,-7)

For #7-10, write the equation of the line going through the given points in slope intercept form and standard form. 7. (-6, -1) and (3, 2) 8. (5,-2) and (3,5)

9. (-3,1) and (3,4)

Advanced Algebra Mastery Skills Workbook

10. (-1, -4) and (3, 5)

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11. Are these slopes parallel, perpendicular, or neither?

m

3 4

m

12. Parallel slopes are the ____________ while perpendicular slopes are the __________ ____________ of each other

4 3

13. Tell whether the lines are parallel perpendicular or neither

14. Tell whether the lines are parallel, perpendicular, or neither

Line 1 through (-1, 4) and (2,5) Line 2 through (2, 6) and (4,0)

Line 1 through (-3,2) and (5,0) Line 2 through (-1,-3) and (3, -4)

15. Are these equations parallel, perpendicular, or neither?

16. Are these equations parallel, perpendicular, or neither?

x  3 y  6

x  4y  8

17. Write an equation that is parallel to the line y = -4x + 1 and goes through the point (-2,3)

18. Write an equation that is perpendicular to the line y = -4x + 1 and goes through the point (-2,3)

19. Write an equation that is parallel to y = -6x + 2 through the point (1, -2)

20. Write an equation that is perpendicular to y = -6x +2 through (-3, 4)

x  3 y  24

Advanced Algebra Mastery Skills Workbook

Page 7

4 x  y  5

’15-‘16

SKILL AA2: VERTICAL AND HORIZONTAL LINES  Vertical lines: o Slope: m = undefined or no slope o Equation: x  a , where a is a real number  Horizontal lines: o Slope: m = 0 o Equation: y  b , where b is a real number 1. Graph y = -3 2. Graph x = 5

3. Graph y = 5

4. Graph x = -4

5. What is the equation of the line?

6. What is the equation of the line?

7. What is the equation of the line through the points (-5, 7) and (-5, 23)?

8. What is the equation of the line through the points (3, -8) and (-5, -8)?

9. Write the equation of the horizontal line that goes through the point (8,4)

10. Write the equation of the vertical line that goes through the point (9,5)

Advanced Algebra Mastery Skills Workbook

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SKILL AA3: SYSTEMS OF EQUATIONS - 2 equations with 2 variables  Two methods of solving: o Substitution:  Solve one equation for one variable and then plug that expression in for the variable in the other equation and solve for the other variable. Use that solution to find the first variable.  Used when one of the variables has a coefficient of 1 or -1. o Linear Combinations (Elimination):  Multiply one or both equations by a real number, then add the equations together. One variable should cancel out, allowing you to solve for the other variable. Once you have a solution for one of the variables, plug it into either of the original equations to find the other variable.  Used when none of the variable have a coefficient of 1 or -1. o The solution is the intersection point of the two lines: (x, y) o If while using linear combinations, all of the variables cancel out and you get an UNTRUE statement such as 0 = 42, then the system has no solution (the lines do not intersect – they are parallel) o If while using linear combinations, all of the variables cancel out and you get a TRUE statement such as 0 = 0, then the system has infinite solution (the lines are actually the same line) Solve using Substitution. 3x  2y  4 2x  y  9 3x  y  16 1. 2. 3. x  3y  17 3x  y  11 2x  3y  4

Solve using Linear Combinations (Elimination) 4x  2y  16 9x  8y  3 4. 5. 3x  4y  12 3x  13y  1

Solve using any method. 8x  3y  3 7. 3x  2y  5

8.

3x  y  2

9.

6x  3y  14

Advanced Algebra Mastery Skills Workbook

6.

Page 9

3x  6y  0 2x  2y  2

5x  3y  4  0 2x  7y  10  0

’15-‘16

10.

13.

7x  8  3y 21x  9y  42

x  4y  4 3x  2 y  19

11.

2x  7y  5 10x  35y  25

14.

4x  3 y  8 8 x  6 y  16

Set up a system of equations to solve each word problem. 16. A collection of quarters and dimes contains 44 coins and has a total value of $6.50. How many coins of each kind are in the collection?

Advanced Algebra Mastery Skills Workbook

12.

15.

6y  8x  6 2x  3y  0

4 x  2 y  14 2 x  y  7

Use any method to solve the system. 17. Tickets for a band concert cost $8.00 for the main floor and $6.00 for the balcony. If 1125 tickets were sold and the ticket sales totaled $7700, how many tickets of each kind were sold?

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’15-‘16

18. A math class has 29 students. The number of girls enrolled is one less than one-half the number of boys. How many boys and how many girls are there in the class?

19. In one week, a music store sold 9 guitars for a total of $3611. Electric guitars sold for $479 each and acoustic guitars sold for $339 each. How many of each type of guitar were sold?

20. An adult pass for a county fair costs $2 more than a children’s pass. When 378 adult and 214 children’s passes were sold, the total revenue was $2532. Find the cost of an adult pass.

21. At a pizza restaurant it costs $4 to make a small pizza that sells for $12, and it costs $6 to make a large pizza that sells for $15. In one week, the restaurant spent a total of $1100 making pizzas and sold all of them for $2910. How many small pizzas were sold?

22. A dozen eggs and five loaves of bread cost $12.40. Four dozen eggs and two loaves of bread cost $10.90. Find the price of a dozen eggs plus one loaf of bread.

23. A total of $15000 is invested in two bonds. One pays 5% simple annual interest and the other pays 7% simple annual interest. The investor wants to earn $880 in interest per year from the bonds. How much should be invested in each bond?

Advanced Algebra Mastery Skills Workbook

Page 11

’15-‘16

SKILL AA4: PROPERTIES OF EXPONENTS  Product of two powers with equal bases:

x a x b  x a b 



Quotient of two powers with equal bases: xa  x a b xb Power of a power:

x 

a b



 x ab

Power of a product:

( xy ) a  x a y a 

Power of a quotient: a

 x  xa    a  y y 



Negative exponents: 1 xn  n x Exponent of 0:

x 0  1 , provided Simplify the expression 1. x7 (x4)

x0

2.

x8 (x6)

3.

(3x9)4

6.

(17z) (3z)3

4.

(5x7)3

5.

(8y) (7y)2

7.

(x4)5

8.

(3y3)4

10.

y8 y2

11.

Advanced Algebra Mastery Skills Workbook

9.

(5x2 )3 100 x5

12.

Page 12

x9 x4

(3x2 )3 6 x5

’15-‘16

Simplify the expression. Write the answers without any negative exponents. 13. 6x5 (3x-2) 14. 7x-5 (8x9)

16.

9x-2 (6x-5)

17.

3a-11 (17a5)

19.

(-2x2)3 (3x-1y2)4

20.

(4x3)2 (-2x-3y-4)3

15.

5x-4 (2x-3)

18.

11p7 (4p-12)

21.

22.

y 9 y 5

23.

7 x 3

24.

25.

x 1 y 4 z x 2 yz 3

26.

r 5st 3 r 1s 5t

28.

z0 z3

29.

13x5 y 2 z 0 39 xy 3 z 2

31.

5  a3 a

32.

Advanced Algebra Mastery Skills Workbook

3x 2  2 x x3

Page 13

x 1 x2

5 a 5

27.

(x2y-3)0

30.

(3x4y2)12 (3x4y2)-12

33.

4  3x4 x3

’15-‘16

SKILL AA5: FACTORING  Common Monomial – factor out what each term has in common  Trinomial with Lead Coefficient other than one - factor into two binomials: o If the 3rd term (“c”) is positive the signs in BOTH binomials will be the same o If the 3rd term (“c”) is negative the signs in the binomials will be different.  Difference of Squares and Perfect Square Trinomials o

a 2  b 2  (a  b)(a  b)

o

a 2  2ab  b 2  (a  b) 2

FACTOR EACH POLYNOMIAL COMPLETELY: 1.

5x 4  45x 3

2.

4.

y 2 5y  24

5. 2v 4  6v 3  56v 2

7.

4a5b2  32a4 b3  64a3b4

8.

12 x 2 y 3  24 x 4 y

3p2  5p  8

3. 75n  30n 6

3

6. 2 x 5  24 x 4  70 x 3

9. 2m2  m  15

10. 6a2  5a  6

11. 3c 2  16c  16

12. 12 x  29 x  15

13. 15r  7r  30

2 14. 15x  14 x  8

2 15. 24b  46b  21

2

Advanced Algebra Mastery Skills Workbook

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2

’15-‘16

16. 7y 2  19y  10

17. 4 x 2  4 xy  35y 2

18. 32 x 4  8 x 2

19. 4 x 2  16

20. 64w 5  49w 3

21. 100c 2  169d 2

22. 27 x 2  48

23. 9g 2  6g  1

24. 16 z 2  40 z  25

25. 48b4  72b3  27b2

26. k 4  n4

27. 32m5  162m

Advanced Algebra Mastery Skills Workbook

Page 15

’15-‘16

SKILL AA6: QUADRATICS  

b  b2  4ac 2a b Finding the vertex (x,y) of a quadratic using x  and y=f(x) 2a Finding intercepts : x-intercept (x,0) y-intercept (0,y) Graphing quadratics

Using Quadratic Formula to solve equations x 

  Given y  x 2  x  2 1. Solve for the zeros of the quadratic.

4. Graph the quadratic

2. Find the vertex of the quadratic

Table of values

3. Find the y-intercept



y



x-intercepts: X Vertex



Y

  

y-intercept

x

       















    

Given y  x 2  8 x  15 5. Solve for the zeros of the quadratic.

8. Graph the quadratic

6. Find the vertex of the quadratic

Table of values



y



x-intercepts: X Vertex

7. Find the y-intercept



Y

  

y-intercept

       

x 













    

Advanced Algebra Mastery Skills Workbook

Page 16

’15-‘16

Given y  x 2  2 x  8 9. Solve for the zeros of the quadratic.

12. Graph the quadratic

10. Find the vertex of the quadratic

11. Find the y-intercept

Table of values 

y



x-intercepts:



X Vertex



Y

   

y-intercept



x

          

















 

       

Given y   x 2  7 x  10 13. Solve for the zeros of the quadratic.

16. Graph the quadratic

14. Find the vertex of the quadratic

15. Find the y-intercept

Table of values 

x-intercepts:



X Vertex

y

 

Y

   

y-intercept

           

x 















 

       

Advanced Algebra Mastery Skills Workbook

Page 17

’15-‘16

Given y  x 2  x  3 17. Solve for the zeros of the quadratic.

20. Graph the quadratic

18. Find the vertex of the quadratic

Table of values

19. Find the y-intercept



y



x-intercepts:

 

X Vertex

Y

    

y-intercept

          

x 















 

       

Advanced Algebra Mastery Skills Workbook

Page 18

’15-‘16

SKILL AA7: RADICALS  Simplifying Square and Cube Roots with variables o Break the numbers to primes and look for pairs or triplets 

363x10 y 9  3 1111 x5 

2

y 

4 2

y  11x5 y 4 3 y

 3 16 x 4 y 5  3 2  2  2  2  x3 x  y 3 y 2  2 xy 3 2 xy 2 Simplify the following: 1.

3

4.

7.

10.

13.



3

117a 6

3.

x9

6.

48y 4

2.

40x5

5.

3

160w8

8.

3

128b3c5

11. 2a 3 8a3b5

39 y 2

14.

3

16a 4b6

3

120x 2 y 3

32a 4 b2

9.

56w3 2

48

12.

3

y5 27 x3

15.

3

81m7

Rationalizing Denominators of Square Roots – Simplify first and last. o Multiply the numerator and denominator by the same radical to make pairs in the denominator. 

18  5



8  18

2  3  3 3 2  5  3 10    5 5 5  5  8 8  2 8 2 4 2     3 2  3  3 3 2  2  3  2

Simplify and rationalize the denominator of the following: 16.

24 12

21x 17. 8x

Advanced Algebra Mastery Skills Workbook

2

18.

Page 19

6 3b3

’15-‘16

19.



1 162

22.

3 15

25.

20 x3 9 y2

20.

23.

26.

20 x 24 x

21.

3r 3 28r

24.

24 x 3 y 4

27.

32

6g 4 4g

4 11 8

3y4 48

Adding/Subtracting Square and Cube Roots. o Simplify FIRST, then only Add/Subt. similar radicals (same radicand)   28. 2 3 3 y  3 3 y

5 x 7  3 7 x 2  5 x 7  3x 7  2 x 7 5 2  3 6  4 18  5 2  3 6  12 2  7 2  3 6 29. 6 32  3 8

Advanced Algebra Mastery Skills Workbook

Page 20

30.

8a3  a 18a

’15-‘16

31.



200 y  3 8 y

32.

3

40  2 3 5

33.

20 x5  4 x 180 x3

34. 3 3 135 x  2 3 5 x

35. 5 12a3  2a 3a  75

36. 2 54  7 150  3 144

37. 10 9t  3 36t  50t

38. 5u 3 24u 2  2 3 81u 5

39. 4 3 48  3 162

Multiplying/Dividing Square Roots. o Multiply the numbers outside & keep them out then multiply the numbers inside the radical and simplify the new radical if possible.  o







2  3 3 2  1  3 2  2  1 2  9 2  3  9  10 2

Reduce the numbers outside then reduce the numbers inside the radical and simplify if possible. 

o



FOIL binomial multiplication and simplify if possible 

o



3x 2 6 2 x 3  6 x3 2  3  3  18 x3 2

15 2 5 5  3 5 3     3 3 6 3 3  3 

Each term in the numerator divided by the denominator 

9 2 3 8 9 2 3 8    3 1 4  3 2 1 3 2 3 2 3 2

Advanced Algebra Mastery Skills Workbook

Page 21

’15-‘16



40. 3 2 2 8

43.

z





4z  5



46. 5 2  3







49.



52.

9 8  6 18 3 2

3 3



41. 5 18 2 8



44.

2 3





45. 4 3 3





48.



51.

9 3  18 27 3 3

54.

6 18  8 27 2 3



5 3

50.



62 3

53.

2 2 3 3 6 6

Advanced Algebra Mastery Skills Workbook

20ab2  35ab3

42.

18 5 2  18

47.

2



3 5



2

Page 22



3

933

6 3





2

’15-‘16

SKILL AA8: The Six Trig Functions (SOH-CAH-TOA)

sin   

opp hyp

cos  

adj hyp

tan  

opp adj

csc  

1 sin 

1 cos 

sec  

1 tan 

cot  

Evaluating the six trig functions given a picture (using common right triangle “families”)

θ

θ

17

13 3

12

15

θ 4

1.

sin  

csc  

sec  

cos  

sec  

cot  

tan  

cot  

sin  

csc  

sec  

cos  

cot  

tan  

sin  

csc  

cos   tan  

2.

3.

θ 5 2

12

5 θ 6

4.

sin  

csc  

sec  

cos  

sec  

cot  

tan  

cot  

sin  

csc  

cos  

tan  

Hint: don’t forget to rationalize the denominators!

Advanced Algebra Mastery Skills Workbook

5.

Hint: don’t forget to rationalize the denominators!

Page 23

’15-‘16



Finding one of the trig functions given another (using Pythagorean Theorem) 6. In a right triangle, θ is an acute angle and cos   What is sin  ?

8. In a right triangle, θ is an acute angle and sin   What is cot  ?

10. In a right triangle, θ is an acute angle and cos   What is sec  ?

12. In a right triangle, θ is an acute angle and tan   What is sec  ?

Advanced Algebra Mastery Skills Workbook

7 10

7. In a right triangle, θ is an acute angle and

5 6

9. In a right triangle, θ is an acute angle and

5 8

11. In a right triangle, θ is an acute angle and

1 2

13. In a right triangle, θ is an acute angle and

sin  

tan  

cos  

Page 24

sin  

4 What is tan  ? 7

7 What is csc  ? 3

9 What is csc  ? 10

7 What is csc  ? 10

’15-‘16

 

Word problems using sin-cos-tan with angle of depression and angle of elevation. Leave your answer in calculator ready form. For example, if the equation to solve the given trig problem is

sin 40 

7 7 , then the answer must be given as x  x sin 40

14. A fireman rests his ladder against a building, making a 57° angle with the ground. The bottom of the ladder is 28 feet from the base of the building. How long is the ladder?

15. A pilot of an airplane in flight looks down at a point on the ground that is some distance away. The angle of depression is 28°, and the plane's altitude is 1200 meters. What is the distance from the pilot to the point on the ground?

16. On a bright and sunny summer day, Johnny was driving his mother crazy playing video games inside, so his mother told him to get out of the house and go fly a kite. Unfortunately, his kite got caught at the top of a very tall tree. While standing 25 ft. from the base of the tree, Johnny put the string to the ground at an angle of elevation of 31°. How tall is the tree?

17. Suppose you are flying a kite and it gets caught at the top of a tree. You’ve let out all 100 feet of string for the kite, and the angle the string makes with the ground is 75°. Instead of wondering how to get your kite back, you wonder “How tall is that tree?” and “How many feet am I away from the base of the tree?” Since you are so great at trig after having the awesome math teachers at SFHS, you can now answer your own questions! 

Advanced Algebra Mastery Skills Workbook

Page 25

’15-‘16

18. A ladder 20’ long reaches the top of a wall when its foot is 13’ from the base. How high is the wall?

What is the angle of the ladder to the ground?

19. A 10 m ladder is leaning against the side of a building. If the foot of the ladder is 3 meters from the foot of the building, find the measure of the acute angle the ladder makes with the building.

20. An elderly couple is trapped 17 m up in their burning house. If a fireman has a 26 m ladder and it makes a 43° angle with the ground to clear the greenhouse under the window, how would you determine if they could be saved?

sin 43 to 17, if it is close to 17, then the couple can be saved 26 b) compare 26sin 43 to 17, if it is close to 17, then the couple can be saved a) compare

c) compare 17sin 43 to 26, if it is close to 26, then the couple can be saved d) compare

sin 43 to 26, if it is close to 26, then the couple can be saved 17

Advanced Algebra Mastery Skills Workbook

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’15-‘16

ADVANCED ALGEBRA MASTERY SKILLS PRACTICE TEST MULTIPLE CHOICE TESTTETESTTETESTTESTTEST The test you take the first week of school is very similar to this test. You should get most problems in each skill correct. Practice for this and you will start your year off right. NO CALCULATORS SKILL AA1: Writing the equations of lines, parallel and perpendicular lines 1. Write the equation of the line in 2. Write the equation of the line slope-intercept form with points (3,6) going through the points (-5, -6) and (0,-2) and (3, 10) in slope intercept form

3 a) y   x  2 8

8 b) y   x  2 3

3 8 c) y  x  2 d) y  x  2 8 3 4. Write an equation that is parallel to through the point (9, 8)

a) y  3x  19

1 b) y  x  35 3

c) y  3x  35

d) y  3x  8

a) y  2x  4

1 5 b) y  x  2 2

c) y  2x  5

d) y  2x  2

y = -3x + 1

SKILL AA2: Vertical and horizontal lines 6. What is the equation of this line? a) x  2

b) y  2

c) y  2

d) x  2

SKILL AA3: Systems of equations 8. Solve using systems of equations

8x + 3y = 16 -6x + y = -12

3 x  y  2 3 x  y  3

a) parallel

b) perpendicular

c) neither 5. The slope of the line that is perpendicular to the line with equation 3x + 4y = 20 is:

a)

4 3

b)

4 3

c)

3 4

d)

3 4

7. Write the equation of the vertical line that goes through the point (8,-6)

a) x  6

b) y  6

c) x  8

d) y  8

9. Solve using systems of equations

-2x + y = 8 3x + y = -7 a)

a) (2, 24)

3. Are these equations parallel, perpendicular, or neither?

b) (2,0)

c) (2, 24) d) (2,24) Advanced Algebra Mastery Skills Workbook

 3,2 

7  c)  1,   2  Page 27

5  b)  3,   2   d) (8, 7) ’15-‘16

10. Solve using systems of equations 9x  6y  20 3x  2y  1

11. In a basketball game between Montini and St. Francis, the total number of points scored was 119. St. Francis’ score was 49 points less than twice that of Montini. Let x represent Montini’s points and y represent St. Francis’ points. How many points did each team score?

 17 19  a)   ,   6 4 

 19 17  b)   ,    6 12 

c) infinite solutions

d) no solution

SKILL AA4: Properties of exponents (6 x)(2 x) 2 12. Simplify.

a) 12x 3

b) 12x 2

3

2

c) 24x

d) 24x

c)

 3 p2  15. Simplify. (6 p r )    r 

1 y6

a) 2p r c) 2p4 r 3

c) (52,67)

d) (23,96)

1 y 12

a)

z3 3x 2

d) y 12

c)

3yz 5 x2

b)

c)

d) 18p12 r

b)

x6y6 z5 3

d)

yz 3 10 x 2

9 x  12 x 3 3x 2

16. Simplify.

a) 7x 2

4 b) 3r

5x2 y3 z 4 15 x 4 y 3 z

14. Simplify.

3

3 2 2

10

b) (56,63)

y 9 y3

13. Simplify.

a) y 3

a) (49,70)

b)

3  12x 3 x

3  4x 2 x

d)

21 x 4 3x 2

SKILL AA5: Factoring 17. Factor Completely.

9a 2b3  36a 4b 2

18. Factor Completely.

5 p2  6 p  8

a) 9a 2b 2 (b  4a 2 )

b) 27a6 b5

a) (5p  4)(p  2)

b) (5p  4)(p  2)

2 3 c) 3ab(3a b  12a b)

d) 9a 4b3 (a 2  4b)

c) (p  8)(5p  1)

d) (p  1)(5p  8)

Advanced Algebra Mastery Skills Workbook

Page 28

’15-‘16

2 y 2  22 y  48

19. Factor Completely.

9d 2  25

20. Factor Completely.

a) (2y  24)(y  2)

b) (2 y  6)(y  8)

a) (3d 5)(3d 5)

b) (3d 5)(3d 5)

c) 2(y  3)(y  8)

d) 2(y  12)(y  2)

c) (9d 25)(d 1)

d) (9d 1)(d 25)

16n 2  56n  49

21. Factor Completely.

a) (4n 7)(4n 7)

b) (4n 7)2

c) (4n 7)2

d) (4n 7)(7n 4)

SKILL AA6: Quadratics Given y  x 2  2 x  3 22. Solve for the zeros of the quadratic.

23. Find the vertex of the parabola.

24. Find the y-intercept

25. Find the x-intercept(s)

a) (3, 0) and (-1, 0) b) (-3, 0) and (1, 0) a) x = -3, x = 1

b) x = 3, x = -1

a) (-4, -1)

c) x = 3, x = 1

d) x = -3, x = -1 c) (-1, -4)

b) (1, 6)

a) (0, -4)

b) (0, -3)

c) (0, -3) and (0, 1)

d) (-2, -3)

c) (0, 3)

d) (-3, 0)

d) (3, 0) and (1, 0)

26. Which of the following is the graph of y  x 2  2 x  3 ? a)

b)

c)

y  x 







d) y

y





































 

y



 x

x

 









 













  

 





Advanced Algebra Mastery Skills Workbook

 

Page 29























x 









































’15-‘16





SKILL AA7: Radicals 27. Simplify completely:

3

80

28. Simplify completely:

3

27a5b9

29. Simplify and perform the indicated operation.

300 y  4 27 y

b) 2 3 10

a) 10 3 2

c) 8 3 10 d) 3 20 30. Simplify and rationalize the denominator of:

24 12

a) 3 3 a5b9

b) 3b 3 a 5

a) 2 3y

b) 2 3y

c) 3ab3 3 a 2

d) 3b3 3 a 5

c) 7 3y

d) 3 273y

31. Simplify and rationalize the denominator of:

a) a) 4

b) 4 3 c)

c) 2

d) 12 3

2y 3

y2 2 3

d)

y2 3

b) cos θ

Advanced Algebra Mastery Skills Workbook

b) 15  4 5

c) 4 5  15

d) 8 5  30

c) tan θ

33. ____________ =

2 10 3

34. ____________ =

2 10 7

35. ____________ =

7 3

θ 3

a) 4 5  3

To answer #33-35, select the trig function from below that results in the given ratio, using the triangle from the left. a) sin θ

7

8 15  10 27 2 3

18

b)

y2 3 2

SKILL AA8: The six trig functions Use the triangle below to answer #33-35

2y

32. Simplify completely

4

Page 30

d) csc θ

ab) sec θ

ac) cot θ

’15-‘16

36. In a right triangle, θ is an acute angle and

sin  

a)

37. In a right triangle, θ is an acute angle and

8 What is cot  ? 17

8 15

b)

17 15

cos  

c)

17 8

d)

15 8

a)

38. A 16 m ladder is leaning against the side of a building. If the foot of the ladder is 5 meters from the foot of the building, find the measure of the acute angle the ladder makes with the building.

 5   16 

1 a)   sin 

 16    5

1 c)   cos 

 5   16 

1 b)   cos 

 16    5

1 d)   sin 

Advanced Algebra Mastery Skills Workbook

2 3 What is csc  ? 10

5 22 22

b)

2 22 10

c)

66 3

d)

66 22

39. You are standing at a bus stop and spot a gargoyle on top of a building that is 250 feet tall. The angle of elevation from the ground to the gargoyle is 70°. What is the equation you would use to find how far you are standing from the bottom of the building?

 x    250 

b) tan70  

 250    x 

d) cos70  

a) cos70  

c) tan70  

Page 31

 x    250 

 250    x 

’15-‘16

ANSWERS PRACTICE TEST 1) y = 4x – 5 2) y  8 x  2

3) neither 4) y = -4x – 12 5) 9 6) x = 5 7) y = -3 8) (2, 0) 9) (-3, 5) 10) infinite  3 3 solutions 2 4 2 8 81 2 11) (97, 88) 12) 56x4 13) y8 17) 4a b (2b – 9) 18) (3p + 2)(2p – 5) 15) 16) 3x  14) 7 x10 6 16r x 3y 2 19) 3(y – 1)(y – 5) 20) (4d + 7)(4d – 7) 21) (5n – 3) 22) x = 5 & x = -1 23) (2, 9) 24) (0, 5) 25) (5, 0) & (-1, 0) 26)

27) 3 3 10

28) 2a2b4 3 a

29) 8 3y

30) 4 3 3

3 31) y 4

32) 3 6  20

33) 6

34)

5

61 5

35) 6 61

39) tan72  350

7

3

X

SKILL AA1: Writing the Equations of Lines; Parallel and Perpendicular Lines 1) y = 2x – 4 2) y= ½ x + 1 3) y = -3/2 x – 1 4) y = -2x – 5 5) y = -4x + 3 1

7

31

5

9

7

7) y = /3 x + 1

8) y = - /2 x + /2

9) y = ½ x + /2

10) y = /4 x – /4

13) neither

14) parallel

15) parallel

16) neither

19) y = -6x + 4

20) y = 1/6 x + 9/2

SKILL AA2: Vertical and Horizontal Lines 1) 2)

5) x = 4

6) y = -4

9) y = 4

10) x = 9

11) infinite solutions

12) (1/2, -1/3)

3)

4)

7) x = -5

8) y = -8

13) (6, -1/2)

16) 14 quarters, 30 dimes

17) 475 main floor, 650 balcony

20) $5 / adult pass

21) 80 small pizzas

Advanced Algebra Mastery Skills Workbook

6) y = -7

11) perpendicular 12) same, opposite reciprocals 17) y = -4x – 5 18) y = ¼ x + 7/2

SKILL AA3: Systems of Equations – 2 equations with 2 variables 1) (4, -1) 2) (2, 5) 3) (4, 4) 4) (-4, 0) 5) (-1/3, 0) 6) (2/3, 1/3) 10) no solution

7) (3, 7)

14) no solution

18) 9 girls, 20 boys

22) price is $3.80 Page 32

24

61

38) cos  5

37) 5 3

36) 7

8) (4/3, 2)

9) (-2, 2)

15) infinite solutions

19) 4 electric, 5 acoustic

23) $8500 at 5%, $6500 at 7% ’15-‘16

SKILL AA4: Properties of Exponents 1) x11 2) x14

81x36

4)

8)

81y12

9)

125x21

5)

392y3

x5

10)

y6

56x4

15)

6)

459z4

7)

11)

5x 4 54 x7 1 x3

12)

9x 2 51 17) 6 a 22) 1 y4

13) 18x3

14)

18) 44

19) -648x2y8

s6 r 4t 4 5 2 a a

27)

28)

16) 21) 26) 31)

x20

3)

p

5

7x3

23)

1

10 x7 20) 128 x 3 y 12

1 z3

5a5

24)

x4y 3z 2

29)

25)

xz 4 y5

30)

1

33) 4x3+3x7

32) 3  2 2 x x

SKILL AA5: Factoring 1) 5x3(x+9)

2) -12x2y(y2-2x2)

3) 15n3(5n3-2)

4) (y-8)(y+3)

5) 2v2(v-7)(v+4)

6) 2x3(x+7)(x+5)

7) 4a3b2(a-4b)2

8) (3p-8)(p+1)

9) (2m+5)(m-3)

10) (3a+2)(2a-3)

11) (c+4)(3c+4)

12) (4x-3)(3x-5)

13) (3r+5)(5r-6)

14) (5x-2)(3x+4)

15) (6b-7)(4b-3)

16) (7y+5)(y+2)

17) (2x-7y)(2x+5y)

18) 8x (2x-1)(2x+1)

19) 4(x-2)(x+2)

20) w3(8w-7)(8w+7)

21) (10c-13d)(10c+13d)

22) 3(3x-4)(3x+4)

23) (3g-1)2

24) (4z+5)2

25) 3b2(4b+3)2

26) (k2+n2)(k+n)(k-n)

27) 2m(4m2+9)(2m-3)(2m+3)

SKILL AA6: Quadratics 1) x = 1 or x = -2

2

 1 9  ,   2 4 

3) y-intercept: (0, -2)

2) vertex: 

4) Graph the quadratic:

Table of values



y

 

x-intercepts: (1,0) and (2,0)

 

 1 9  Vertex  ,   2 4 



x

       

















y-intercept (0,  2)

   

5) x = -3 or x = -5

6) vertex: (-4, -1)

8) Graph the quadratic

7) y-intercept: (0, 15)

Table of values



y



x-intercepts: (3,0) and (5,0)



Vertex  4, 1  y-intercept (0,15)

   













x 











    

Advanced Algebra Mastery Skills Workbook

Page 33

’15-‘16

9) x = -2 or x = 4

10) vertex: (1, -9)

12) Graph the quadratic

11) y-intercept: (0, -8)

Table of values



y

  

x-intercepts: (2,0) and (4,0)

 

Vertex  1, 9 

            

x 



















y-intercept (0,  8)

      

7 9 2 4

13) x = 5 or x = 2

15) y-intercept: (0, -10)

14) vertex:  , 

16) Graph the quadratic

y

Table of values

  

x-intercepts: (5,0) and (2,0)

 

7 9 Vertex  ,  2 4

 



 

x

 

















 

y-intercept (0, -10)

  

17) NO REAL ZEROS (using the quadratic formula, x 

1  11 ) 2

 1 11   2 4 

19) y-intercept: (0, 3)

18) vertex:  ,

Graph the quadratic

Table of values



y

 

x-intercepts: NONE  1 11  Vertex  ,  2 4  y-intercept (0,3)

            

x 













 

SKILL AA7: Radicals 1)

4 y2 3

6) 2 xy 30 y 2

3

11) 4a b b

16) 4 3

2) 3a 3 13

3) 2 3 6

7) 4 w4 10

8) 2ab2 3 2a

2

12)

17)

y3

y2 3x 3

21x 2 x 4

Advanced Algebra Mastery Skills Workbook

13) y 13

18)

2 3b b2 Page 34

4) 2 x 3 5 x 2 9)

4a 2 2 b

14) 2 w 7 w

19)

2 18

5)

x3 3

10) 4bc 2c

2

15) 3m2 3 3m

20)

5 6x 3 ’15-‘16

21)

g 6g 2

22)

5 5

23)

24)

r 21 14

22

25)

2 x 5x 3y

xy 2 3x 26) 2 31) 4 2y

y2 27) 4

28) 3 3 3y

29) 30 2

30) 5a 2a

32) 4 3 5

33) 22 x 2 5 x

34) 7 3 5x

35) 12a 3a  5 3

36) 29 6  36

37) 12 t  5 2t

38) 16u 3u 2

39) 5 3 6

40) −24

41) −120

42) 10ab 2 7b

43) 2 z  5 z

44) 12

45) 12  4 3 9

46) 13  6 6

47)

50) 18  12 2

3

15  5 5  3 3  15 48) 9  6 2

51) −15

52) 0 53)

49) 12  6 3 54) 3 6  12

3 2  9 4

SKILL AA8: The Six Trig Functions 1.

4.

sin  

3 5

csc 

5 3

cos  

4 5

sec  

tan  

3 4

cot  

15 17

csc 

17 15

5 4

cos  

8 17

sec  

17 8

4 3

15 8 3 5. sin   2 tan  

sin  

2 2

csc 

2

cos  

2 2

sec  

2

tan   1

51 10 58 9. csc   7 5 12. sec   2 1200 15. meters sin 28 6.

sin  

2.

sin  

cos  

cot   1

tan  

1 2

8 15 2 3 csc  3 cot  

sin  

5 13

csc 

13 5

cos  

12 13

sec  

13 12

tan  

5 12

cot  

12 5

sec   2

3

cot  

4 33 33 8 10. sec   5 10 13. csc   7 7.

3.

tan  

16. 25 tan 31 feet

3 3 11 5 10 19 11. csc   19 28 14. feet cos 57 8.

cot  

17. Tree height: 100sin 75 feet Distance from base: 100 cos 75 feet

Advanced Algebra Mastery Skills Workbook

Page 35

’15-‘16

18. Wall height:

231 feet

20. b

 3   10 

19. sin 1 

Angle of ladder to ground:

 13  cos 1    20 

MULTIPLE CHOICE PRACTICE TEST 1) D 2) A 3) C 11) B 12) C 13) B 21) B 22) A 23) C 31) D 32) C 33) C

4) C 14) A 24) B 34) A

Advanced Algebra Mastery Skills Workbook

5) A 15) B 25) B 35) AB

6) B 16) B 26) B 36) D

Page 36

7) C 17) A 27) B 37) A

8) B 18) A 28) C 38) A

9) A 19) C 29) A 39) C

10) D 20) A 30) B

’15-‘16