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CONTENTSLesson Title Page Lesson Title Page1-1 Relations and Functions..................1 5-8 Area of Triangles ...........................351-2 Composition and Inverses of

Functions ....................................21-3 Linear Functions and 6-1 Graphs of the Trigonometric

Inequalities ..................................3 Functions ..................................361-4 Distance and Slope .........................4 6-2 Amplitude, Period, and1-5 Forms of Linear Equations ..............5 Phase Shift ...............................371-6 Parallel and Perpendicular 6-3 Graphing'Trigonometric

Lines ...........................................6 Functions ..................................386-4 Inverse "l'rigonometric

Functions ..................................392-1 Solving Systems of Equations .........7 6-5 Principal Values of the2-2 Introduction to Matrices ...................8 Inverse Trigonometric2-3 Determinants and Multiplicative Functions ..................................40

Inverses of Matrices ....................9 6-6 Graphing Inverses of2-4 Solving Systems of Equations Trigonometric Functions ...........41

by Using Matrices .....................10 6-7 Simple Harmonic Motion ...............422-5 Solving Systems of Inequalities ....112-6 Linear Programming......................12

7-1 Basic TrigonometricIdentities ...................................43

3-1 Symmetry ......................................13 7-2 Verifying Trigonometric3-2 Families of Graphs ........................14 Identities ...................................443-3 Inverse Functions and 7-3 Sum and Difference

Relations ....................................15 Identities ...................................453-4 Rational Functions and 7-4 Double-Angle and Half-Angle

Asymptotes ...............................16 Identities ...................................463-5 Graphs of Inequalities ...................17 7-5 Solving Trigonometric3-6 Tangent to a Curve .......................18 Equations .........................".........473-7 Graphs and Critical Points of 7-6 Normal Form of a Linear

Polynomial Functions................19 Equation....................................483-8 Continuity and End Behavior..........20 7-7 Distancefrom a Point to a

Line ...........................................49

4-1 Polynomial Functions ....................21

4-2 Quadratic Equations and 8-1 Geometric Vectors ...................._...50Inequalities................................22

4-3 The Remainderand Factor 8-2 Algebraic Vectors ..........................51Theorems ..................................23 8-3 Vectors in Three-Dimensional

4-4 The Rational Root Theorem ..........24 Space ........................................528-4 PerpendicularVectors ...................534-5 Locating the Zeros of a

Function ....................................25 8-5 Applications with Vectors ..............544-6 Rational Equations and 8-6 Vectors and Parametric

Partial Fractions ........................26 Equations ..................................554-7 Radical Equations and 8-7 Using Parametric Equations

Inequalities................................27 to Model Motion ........................56

5-1 Angles and Their Measure ............28 9-1 Polar Coordinates .........................575-2 Central Angles and Arcs ...............29 9-2 Graphs of Polar Equations ............585-3 Circular Functions ...'......................30 9-3 Polar and Rectangular5-4 Trigonometric Functions of Coordinates ..............................59

Special Angles ..........................31 9-4 Polar Form of a Linear5-5 Right Triangles ..............................32 Function ....................................605-6 The Law of Sines ..........................33 9-5 Simplifying Complex5-7 The Law of Cosines ......................34 Numbers ...................................61

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IIIGlencoeDivision,Macmillan/McGraw-Hill

CONTENTS

Lesson Title Page Lesson Title Page9-6 Polar Form of Complex 13-6 The Mandelbrot Set.......................93

Numbers ...................................62

9-7 Products and Quotients of 14-1 Permutations.................................94Complex Numbers in

14-2 Permutationswith RepetitionsPolar Form ................................63and Circular Permutations.........95

9-8 Powers and Roots of14-3 Combinations ................................96

Complex Numbers ....................64 14-4 Probability and Odds.-....................9714-5 Probabilities of Independent

10-1 The Circle ......................................65 and Dependent Events .............9810-2 The Parabola.................................66 14-6 Probabilities of Mutually Exclusive10-3 The Ellipse ....................................67 or Inclusive Events ....................9910-4 The Hyperbola...............................68 14-7 Conditional Probability ................10010-5 Conic Sections ..............................69 14-8 The Binomial Theorem and

10-6 Transformations of Conics ............70 Probability ...............................10110-7 Systems of Second-Degree

Equations and Inequalities .........7110-8 Tangents and Normals to 15-1 The Frequency Distribution .........102

the Conic Sections ....................72 15-2 Measures of CentralTendency .........................'.......103

11-1 Rational Exponents .......................73 .15-3 Measures of Variability ................10415-4 The Normal Distribution ..............10511-2 Exponential Functions ...................74

15-5 Sample Sets of Data ...................10611-3 The Number e ...............................7515-6 Scatter Plots ................................107

11-4 Logarithmic Functions ...................7611-5 Common Logarithms .....................7711-6 Exponential and Logarithmic 16-1 Graphs ........................................108

Equations ..................................78 " 16-2 Walks and Paths .........................10911-7 Natural Logarithms ........................79 16-3 .Euler Paths and Circuits ..............110

16-4 Shortest Paths and MinimalDistances ................................111

12-1 Arithmetic Sequences and 16-5 Trees ...........................................112Series ........................................80

16-6 Graphs and Matrices ...................11312-2 Geometric Sequences and

Series ................................._......8112-3 Infinite Sequences and 17-1 Limits ............................................114

Series ........................................82 17-2 Derivatives and12-4 Convergent and Divergent DifferentiationTechniques .......115

Series ................................i.......83 17-3 Area Under a Curve .................... 11612-5 Sigma Notation and the nth 17-4 Integration ...................................117

Term .........................................84 17-5 The Fundamental Theorem12-6 The Binomial Theorem ..................85 of Calculus ..............................118

12-7 Special Sequences andSeries ........................................86

12-8 Mathematical Induction .................87 .

13-1 Iterating Functions with RealNumbers ...................._..............88

13-2 Graphical Iteration of LinearFunctions ..................................89

13-3 Graphical Iteration of theLogistic Function .......................90

13-4 Complex Numbers andIteration .....................................91

13-5 Escape Points, PrisonerPoints, and Julia Sets ...............92

iv

GlencoeDivision,Macmillan/McGraw-Hill

NAME DATE

1-1 Practice WorksheetRelations and Functions

State the domain and range of each relation. Then state whether the relation is afunction. Write yes or no.

1. {(-1, 2), (3, 10), (-2, 20), (3, 11)} 2. {(0, 2), (13, 6), (2, 2), (3, 1)

3. {(1, 4), (2, 8), (3, 24)} 4. {(-1,--2), (3, 54), (-2,-16), (3, 81)}

Given that x is an integer, state the relation representing each of the following byfisting a set of ordered pairs. Then state whether the relation is a function. Write yesor no.

5. y = 3x 2 - 5 and 0 < x < 5 6. y2 = 3x 2 and x = -3

7. 13y+41 =xandO<x<3 8. lyi = Ixl andO<x<2

The symbol [x] means the greatest integer not greater than x. If f(x) = [2x] • 3x, find,each value.

9. f(o) 10. f(o.5) 11./(-3.5) 12. f(x- 1)

Given f(x) = 13x - 41 + 5, find each value.

13. f(l/ 14. f(0.5) 15. f(-0.5) 16. f(5d)\°]

Name all values of x that are not in the domain of the given function.1

x - 2 18. f(x) =: _ 2x + 5117. f_(x) - x + 4

19. f(x) = k/_x2 -25 20. f(x) - x - 10_16

21', f(x) x2 + 25 22. f(x) - x - 7-- x2_25 _--1

1Glencoe Division, Macmillan/McGraw-Hill

NAME DATE

1-1 Practice WorksheetRelationsand FunctionsState the domain and range of each relation. Then state whether the relation is afunction. Write yes or no.

1. {(-1, 2), (3, 10), (-2, 20), (3, 11)} 2. {(0, 2), (13, 6), (2, 2), (3, 1)

{-2, -1, 3}, {2, 10, 11, 20}; no {0, 2, 3, 13}, {1, 2, 6}; yes

3. {(1, 4), (2, 8), (3, 24)} 4. {(-1,-2), (3, 54), (-2,-16), (3, 81)}

{1, 2, 3}, {4, 8, 24}; yes {-2,-13}, {-16,-2, 54, 81};no

Given that x is an integer, state the relation representing each of the following bylisting a set of ordered pairs. Then state whether the relation is a function. Write yesor no.

5. y = 3x 2 - 5 and 0 < x < 5 6. y2 = 3x 2 and x = -3

{(1,-1), (2, 7), (3,22), (4,43)};yes {(-3, 3_/3-3,-3Vr3); no

7. 13y+41 =xandO<x<3 8. lyl = Ixl andO<x<25 2

{(1,-1),(1,- _),(2,-_),(2, =2)}; {(1, 1),(1,-1)}; nono

The symbol [x] means the greatest integer not greater than x. If f(x) = [2x] - 3x, findeach value.

9. f(O) 10. f(0.5) 11. f(-3.5) 12. f(x - 1)

0 -0.5 3.5 [2x]-3x+l

Given f(x) = 13x- 41+ 5, find each value.

13. f(1) 14. f(0.5) 15. f(-0.5) 16. f(5d)

/ \

8 7.5 10.5 115d-41+5

Name all values of x that are not in the domain of the given function.

_ - 2 18. f(x) - 117. f(x)- x+4 12x+5J

-4 _52

19. f(x) = _- 25 20. f(x) - x - lO

-5< x< 5 -4 _<x _<4

21. f(x) - x2 + 25 22. f(x) - x - 7x 2 - 25 _ : ]

___5 ___1-

T1, GlencoeDivision,Macmillan/McGraw-Hill

NAME DATE

1-2 Practice WorksheetComposition and Inverses of Functions

2 and g(x) = x2 - 2, find each function.Given f(x) - x + 4

1.(f+g)(x) 2.(f-g)(x) 3.(f'g)(x) 4. (f/(x)\_/

Find If o g](x) and [g o f](x).

5. f(x)= 1_x + 5 6. f(x)= 2x 3-3x 2+1

g(x) = x - 3 g(x) = 3x

7. f(x)=2x 2 5x+1 8. f(x)=3x 2-2x+5g(x) = 2x - 3 g(x) = 2x - 1

Determine if the given functions are inverses of each other. Write yes or no. Showyour work.

9. f(x) = 3x - 5 10. f(x) = x - 10 11. f(x) - 2x- 35

g(x) - x + 5 3x- 53 g(x) = x + 10 g(x) - 3

12. f(x) = 2x 13. f(x) = 3x - 7 14. f(x) = 4(x + 2)

1 x_22 g(x) = _ x+ 7 g(x) = 4g(x) - x

Find the inverse of each function. Then state whether the inverse is a function.

15. f(x) = 3x + 7 16. f(x) = x5 17. f(x)= x 2 + 4

Glencoe Division, Macmillan/McGraw-Hill

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1-2 Practice WorksheetComposition and Inverses of Functions

_ 2Z_and g(x) = x 2 - 2, find each function.Given f(x) - x + 4

1.(f+g)(x) 2.(f-g)(x) 3.(f'g)(x) 4. (f/(x)\s/

x3+4x2-2x-6 -x3-g'_2+2x+ 10 2x2-4 2

X+4 _ _'+_ ' X+4 _ (x+4)(X2-2) '

x _ -4 _ ¢ =4 x_¢ -4 x_i-4, _+V_

Find [f o g](x) and [g o f](x).

5. f (x) = 1-_x + 5 6. f(x) = 2x 3- 3x 2 + 1

g(x) = x - 3 g(x) = 3x

X +4, 54X3-27X2+ 1,1-_X + 2 6X3-gX2+3

7. f(x) = 2x 2 - 5x + 1 8. f(x) = 3x 2- 2x + 5g(x)= 2x- 3 g(x) = 2x- 1

8xz. 34x+ 34, 12xz- 16x+ 1O,4x2-10x-1 6x2-4x+9

Determine if the given functions are inverses of each other. Write yes or no. Showyour work.

9. f(x) = 3x - 5 10. f(x) = x - 10 11. f(x) = 2x - 35

g(x) - x + 53 g(x) = x + 10 g(x) = 3x-53

yes yes no

12. f(x) 2x 13. f(x) = 3x - 7 14. f(x) = 4(x + 2)

=2_ 1 x -2g(x) x g(x) = -_ x + 7 .g(x) = 4

no no yes

Find the inverse of each function. Then state whether the inverse is a function.

15. f(x) = 3x + 7 16. f(x) = x 5 17. f(x) = x 2 + 4

f_l(x) 1 s.=_x-_, yes f-l(x)=ffx, yes f-l(x)=_+_v/x-4;no

T2Glencoe Division, Macmillan/McGraw-Hil

Find the zero of each function.

4. f(x) = 0.2x + 10 5. f(x) = 11.5x 6. f(x) = 13x - 9

7. f(x) = -3 8. f(x) = -5x + 6 9. f(x) = 0.3x + 0.2

Graph each equation or inequality.

10. y = 3x-2 11. 1- y = 2x 12. x->-2

......-- _.... y _ _- _ _--- y ...... _'=_ ._

*- c x _ x 5

i '

13. y= 12x+41 14.-y>2x+2 15. -4-<x-2y-<6

__ __d _ ...... y .... : ...... y&--....... -.---- __ ...........

*" -- 0 __x_ _ x ..... (: .....

............. i

I i I ' ,

3Glencoe Division, Macmillan/McGraw-Hill \

NAME DATE

1-3 F'l"acticeWorkstl 2c_tLinear Functions and InequalitiesWrite an inequafity that describes each graph.

1. 2. 3.

_- _ ...............__- ___ _._ ......

.... ___ ............ ............ _ ............. __

-- -_-_-- _ ........ E,

-_-_+4y-> -2x-4 -2 -< y_< 4 _-3-< Y-< 5Find the zero of each function.

4. f(x) = 0.2x + 10 5. f(x) = 11.5x 6. f(x) = 13x - 99

-50 0 _3

7. f(x) = -3 8. f(x) = -5x + 6 9. f(x) = 0.3x + 0.26 2none5 3

Graph each equation or inequality.

10. y = 3x-2 11. 1-y = 2x 12. x -> -2

.... L_L__...... ___z L__ _ _y __

.... _- _ ........ _- __.__ _ __ _--

.... _-t .... _- _-- _ ........_ . T ,37 ..... . --

_ ____/-:--- ..... _.... ; -........ __ _ x__ _ _ x

-_ _-_--- .............

__ • x .'_:-2_ __-/ ............... _ ...... _ __

T3Glencoe Division, Macmillan/McGraw-Hill

NAME DATE

1-4 Practice WorksheetDistance and SlopeFind the distance between the points with the given coordinates. Then, find theslope of the line passing through each pair of points.

1. (-2, 2), (8,-3) 2. (2,-1), (7, 9)

3. (-7, 3), (5, 4) 4. (a, b + 4), (a, 2b - 5)

Determine whether the figure with vertices at the given points is a parallelogram.

5. (-2, 3), (-1, 7), (5, 7), (4, 3) 6. (0, 0), (3, 4), (10, 4), (0, 5)

7. (2, 3), (8, 4), (11, 9), (5, 8) 8. (-2, 3), (-8, 4), (-11, 9), (-5, 8)

Collinear points lie on the same line. Find the value of k for which points with eachset of coordinates is collinear. (Remember, the slope of each line is constant.)

9. (2, 3), (6, k), (10, 5) 10. (k, 3), (-3, 2), (-1, 1)

11. (-3, 1), (2, 8), (7, k) 12. (-3,-3), (-1, k), (1, 19)

13. Prove that the diagonals of an isoscelestrapezoid are congruent using analytic methods.

y,

B(c, b) C(a + c, b)

0 A(O, O) Dla, O) x

I


NAME DATE

1-4 Practice WorksheetDistance and SlopeFind the distance between the points with the given coordinates. Then, find theslope of the line passing through each pair of points.

1. (-2, 2), (8,-3) 2. (2,-1), (7, 9)

_1 5,23. (-7, 3), (5, 4) 4. (a, b + 4), (a, 2b - 5)

1145, 12 ib- 91,undefined

Determine whether the figure with vertices at the given points is a parallelogram.

5. (-2, 3), (-1, 7), (5, 7), (4; 3) 6. (0, 0), (3, 4), (10, 4), (0, 5)

yes no

7. (2, 3), (8, 4), (11, 9), (5, 8) 8. (-2, 3), (-8, 4), (-11, 9), (-5, 8)

yes yes

Collinear points lie on the same line. Find the value of k for which points with eachset of coordinates is collinear. (Remember, the slope of each line is constant.)

9. (2, 3), (6, k), (10, 5) 10. (k, 3), (-3, 2), (-1, 1)4 -5

• 11. (-3, 1), (2, 8), (7, k) 12. (-3, -3), (-1, k), (1, 19)15 8

13. Prove that the diagonals of an isoscelestrapezoid are congruent using analytic methods.

AC = _/((a-c) - 0) 2 + (b-O) 2 y

= N/(a-c)2+b 2 ,

BD = _/(a-c) 2 + (0_b)2 // ."_B(c'b) C(a+c,b)

= _v/(a-c)2 + b 2 / \

Hence, AC = BD o A(0, 0) O(a, O) ;

SO, the diagonals are congruent.

T4 \'


NAME DATE

1-5 Practice WorksheetForms of Linear EquationsWrite the slope-intercept form of the equation of the line through the point with thegiven coordinates and having the given slope.

1. (-3, 2), 5 2. (-1, 7),-1

43. (3, 8), -4 4. (-9, 4),

15. (-6, 6), 1 6. (11,-11),- _

7. (2, 2), 0 8. (3,-3),-1

Write the slope-intercept form of the equation of the line through the points withthe given coordinates.

9. (-2, 3), (1, 4) 10. (2, 5), (3,-1)

11. (0, 0), (3,-7) 12. (0, 0), (5, 7)

13. (3, 3), (6, 3) 14. (-2,-3), (-1,-3)

15. (-10,-10), (10, 23) . 16. (4, 7), (7, 4)

' 5Glencoe Division,Macmillan/McGraw-Hill

NAME DATE

1-5 Practice WorksheetForms of Linear EquationsWrite the slope-intercept form of the equation of the line through the point with thegiven coordinates and having the given slope.

1. (-3, 2), 5 2. (-1, 7),-1

y = 5x+ 17 y = -x+6

43. (3, 8),-4 4. (-9, 4),

4y = -4x+20 y = _-x+ 16

15. (-6, 6), 1 6. (11,-11),- _

1 11y = x+12 y = -_-x - 2

7. (2, 2), 0 8. (3,-3),-1

y= 2 y=-x

Write the slope-intercept form of the equation of the line through the points withthe given coordinates.

9. (-2, 3), (1, 4) 10. (2, 5), (3,-1)

1 11y = -_x + _- y =-6x + 17

11. (0, 0), (3,-7) 12. (0, 0), (5, 7)

7 7y=--_x y= _x

13. (3, 3), (6, 3) 14. (-2,-3), (-1,-3)

y=3 y--3

15. (-10,-10), (10, 23) 16. (4, 7), (7, 4)

33 13Y- 20 X-F 2 y=-x + 11

T5GlencoeDivision,Macmillan/McGraw-Hill

NAME DATE

1-6 Practice WorksheetParallel and Perpendicular LinesWrite the standard form of the equation of the line that is parallel to the given lineand passes through the given point.

1. y = -2x + 5; (0, 4) 2. y = 3x + 3; (-1,-2)

3. y = 3x + 8; (5, 2) 4. y = -x - 4; (6, 10)

5. 2x - 5y = 12; (15, 13) 6. 7x + 7y = 4; (1, 5)

Write the standard form of the equation of the line that is perpendicular to the givenline and passes through the given point.

7. y = 2x + 6; (0,-3) 8. y = 7x - 9; (3, 4)

9. y = -x - 1; (6, 5) 10. 2x - 5y = 6; (0, 9)

11. 3x + 4y = 13; (2, 7) 12.-7x + 7y = 6; (5, 5)

13. For what value ofk is the graph ofkx + 6y - 9 = 0 parallel to the graph of4x - 3y - 10 = 0? For what value ofk are the graphs perpendicular?

14. For what value of k is the graph of 4x + ky - 7 = 0 parallel to the graph of3x - 4y - 10 = 0? For what value ofk are the graphs perpendicular?

15. Show that the triangle with vertices R(-3, 5), S(12, 2), and T(3, -4) is a righttriangle.


NAME DATE

1-6 Practice WorksheetParallel and Perpendicular LinesWrite the standard form of the equation of the line that is parallel to the given fineand passes through the given point.

1. y = -2x + 5; (0, 4) 2. y = 3x + 3; (-1,-2)

2x+y-4 = 0 3x-y+ _ = 0

3. y = 3x + 8; (5, 2) 4. y = -x - 4; (6, 10)

3x-y-13 = 0 _+y-_6 = O

5. 2x - 5y = 12; (15, 13) 6. 7x + 7y = 4; (1, 5)

2x-5y+35 = 0 x+y-6 = 0

Write the standard form of the equation of the line that is perpendicular to the givenline and passes through the given point.

7. y = 2x (0,-3) 8. y = 7x - 9; (3, 4)

x+2y 0 x+Ty-31 = 0

9. y = -x - 1; (6, 5) 10. 2x - 5y = 6; (0, 9)

x-y-_ = 0 5x+2y-18 = 0

U. 3x + 4y = 13; (2, 7) 12. 7x + 7y = 6; (5, 5)

4x-3y+ 13 = 0 x-y = 0

13, For what value ofk is the graph ofkx + 6y - 9 = 0 parallel to the graph of4x - 3y - 10 = 0? For what value ofk are the graphs perpendicular?

-8, -g_2

14. For what value of k is the graph of 4x + ky - 7 = 0 parallel to the graph of3x - 4y - 10 = 0? For what value of k are the graphs perpendicular?

_1_fi633 _

15. Show that the triangle with vertices R(-3, 5), S(12, 2), and T(3, -4) is a righttriangle.

Slope ST: -4-2 2= 5-(-4) _ 3, 3-12 -- 3' Slope RT: _ -_°

Since the slopes are negative reciprocals, ST_LFiT=•Hence, A RSTis a right triangle.

T6Glencoe Division,Macmillan/McGraw-Hill

NAME DATE

2-1 Practice WorksheetSolving Systems of EquationsState whether each system is consistent and independent,consistent and dependent, or inconsistent.

1.3x+4y=5 2. 3x-3y=122x - 5y = 8 -x +y =-4

Solve each system of equations algebraically.

5. 3x- 2y = 7 6. 4x- 3y = 15x +y = 4 2x +y = 5

7. 3x+4y=8 8. 2x-y=6-3x-4y=10 x+y=6

9. 3x-2y=-9 10. 7x-y=94x+5y=ll 2x+3y=19


NAME DATE

2-1 Practice Worksheet

Solving Systems of EquationsState whether each system is consistent and independent,consistent and dependent, or inconsistent.

1. 3x+4y = 5 2. 3x- 3y = 122x- 5y=8 -x+Y =-4

consistent and independ_ consistent and dependen_

Solve each system by graphing.

3. 3x-y=6 4. 2x+3y=12x+y=6 x+y=6

\ \/ \

\ ; \l\ / _,, \

\ / "\ \",4 (3, 3) _ \

[ \"\\ \-,,,\ "_ [6, 0) =

/ \ x_

Solve each system of equations algebraically.

5. 3x-2y=7 6. 4x-3y=15x +y = 4 2x +y = 5

(3, 1) (3, =1

7. 3x+4y=8 8. 2x-y=6-3x - 4y = 10 x +y = 6no solutions (4, 2)

9. 3x-2y =-9 10. 7x-y = 94x + 5y = 11 2x + 3y = 19

(-1, 3) (2, 5)


NAME DATE

2-2 Practice WorksheetIntroduction to MatricesUse matrices A, B, and C to find each sum, difference, or product.

A --- -1 1 4 B 2 -7 - C - 8 10 -95 -2 3 4 4 -6 12 14

1. A+B 2. A-B

3. B-A 4. -2.4

5. AB 6. AA

7: CA 8. CB

9. (CB)A 10. C(BA)

Find the values of x and y for which each matrix equation is true.

8GlencoeDivision,Macmillan/McGraw-Hill

NAME DATE

2,-2 Practice WorksheetIntroduction to MatricesUse matrices A, B, and C to find each sum, difference, or producL

A - -1 1 4 B = 2 -7 -2 C - 8 10 295 -2 3 4 4 2 -6 12 14

1. A+B 2. A-B

[ i1 -6 2 -3 -8 -69 2 5 1 =6 -_

3. B -A 4. -2A

3-8 -6 2 -2 -8-1 6 -1 -10 _ -6

5. AB 6. JL,4

[ ] [o t19 4 0 17 ='t]0 1153 51 40 27 7 6

7. CA 8. CB

46 -34 84 86 -58 -32

9. (CB)A 10. C(BA)

70 264-242 70 264 =242]

Find the values of x and y for which each matrix equation is true.

,,[_] =[_] ,_[_]:I_x_]5'5 g'a


NAME DATE

2-3 Practice WorksheetDeterminants and Multiplicative Inverses of a MatrixFind the value of each determinant.

-3 5 7 9

2 -1 3 1 -1 0

3. 2 1 4 4. 2 1 4-3 1 -2 5 -3 5

Find the inverse of each matrix, if it exists.

o[:o:] o(

Solve each system by using matrix equations.

7. 3x+y=23 8. 2x-3y= 17

2x +y = 18 3x +y = 9

9. 2x+5y=28 10. 3x+4y=6

3x - 2y = -15 2x - 3y = 21

11. 4x-3y=-16 12. 7x-3y=4

2x+5y= 18 x+2y=-14


NAME DATE

2-3 Practice WorksheetDeterminants and Multip/icative Inverses of a MatrixFind the value of each determinant.

1 __1 _ _-3 5 7

31 62

2 -i 3 1 -1 0_. 2 i 4 4. 2 I 4

-3 1 -2 5 -3 5

11 7

Find the inverse of each matrix, if it exists.

°[1::] ,does not exist !I5-3 s]23 1 °

Solve each system by using matrix equations.

7. 3x+y=23 8. 2x-3y=172x +y = 18 3x +y = 9

(5, 8) (4,-3)

9. 2x+5y=28 10. 3x+4y=63x - 2y = -15 2x - 3y = 21

(-1, 6) (6,-3)

•11. 4x-3y=-16 12. 7x-3y=42x + 5y = 18 x + 2y =-14

(-1, 4) (-2,-6)

T9GlencoeDivisionMacmillan/McGraw-Hill

NAME DATE

2-4 Practice WorksheetSolving Systems of Equations by Using MatricesSolve each system of equations by using augmented matrices.

1. 4x- 3y = 1 2. 3x- 2y=93x + 2y = 5 12x - 8y = 40

3. x+y+z=3 4. x+y=53x - 2y - z =-4 3x + z = 2x+y -z =-1 4y -z = 8

5. 4x - 3y - 6z = 2 6. 3x - 5y + z 82x + 3y + 3z= 2 7x +y = 4lOx - 6y + 3z = 0 4y - z = 10

7. x - y + z = l 8. 2x - 3y - 4z = 2x - 2y + z = 2 -6x - 6y + 6z = 52x -y +z =-1 4x + 4y - 2z = 3

9. 4x-6y+2z=-9 lO.x-z=52x + 4y - 2z = 9 2x +y = 7x-y+3z=-4 y+32:=12


NAME DATE

2-4 Practice WorksheetSolving Systems of Equations by Using MatricesSolve each system of equations by using augmented matrices.

1. 4x- 3y=1 2. 3x- 2y=93x + 2y = 5 12x - 8y = 40

(1, 1) (4, 1)

3. x+y+z=3 4. x+y=53x - 2y - z =-4 3x + z = 2x +y-z=-i 4y-z=8

(0, 1, 2) (10,-5,-28)

5. 4x-3y-6z=2 6. 3x-5y+z=82x + 3y + 3z = 2 7x +y = 4lOx-6y+3z=O 4y-z=lO

(122, 3' 31) (2.2,-11.4,-55.6)

'7. x - y + z = l 8. 2x - 3y - 4z = 2x - 2y + z = 2 -6x - 6y �6z= 52x - y + z =-1 4x + 4y - 2z = 3

9. 4x-6y+2z=-9 lO.x-z=52x + 4y - 2z = 9 2x + y = 7x -y + 3z =-4 y + 3z= 12

(13 -1) (20,-33, 15)2' 2_ • '

T10Glencoe Division,Macmillan�McGraw-Hill

NAME DATE

2-5 Practice WorksheetSolving Systems of InequalitiesSolve each system of inequafities by graphing and name thecoordinates of the vertices of each polygonal convex set. Then,find the maximum and minimum values for each function on thatseL

1. x-O x+2y-6 2. x->O 2x+3y-12

y >- 0 f(x, y) = 2x -:y y >- 0 f(x, y) = 2x + 3yy ! y

5. y-<-x+ 8 4x - 3y ->-3 6. 3x- 2y-> 0 y-O

x + 8y >- 8 f(x, y) = 4x - 5y 3x + 2y <- 24 f(x, y) = 7y - 3xY Y !

__ __ -L. ......................

....... ....11


NAME DATE '

2'6 PracticeWorksheetLinear ProgrammingSolve each problem, if possible, ff not possible, state whether theproblem is infeasible, has alternate optional solutions, or isunbounded. V /

1. Toys-A-Go makes toys at Plant A and Plant B. Plant A has \materials to make up to 1000 toy dump trucks and fire engines.Plant B has materials to make up to 800 toy dump trucks andfire engines. Plant A can make 10 toy dump trucks and 5 toyfire engines per hour. Plant B can produce 5 toy dump trucksand 15 toy fire engines per hour. It costs $300 per hour tooperate Plant A and $350 to operate Plant B per hour. Howmany hours should each plant be run in order to minimize cost?

2. Roadster Skateboard makes skateboards at Plants X and Y.Plant X must make at least 720 standard and deluxe boards.

Plant Y has materials to make up to 520 standard and deluxeboards. Plant X can make 12 standard and 5 deluxe boards perhour. Plant Y can make 10 standard and 8 deluxe boards perhour. It costs $250 to operate Plant X per hour and $225 tooperate Plant Y per hour. How many hours should each plant berun in order to minimize cost?


NAME _ DATEi /

2-6 Practice WorkSheetLinear ProgrammingSolve each problem, if possible, ff not possible, state whether the

_roblem is infeasible, has alternate optional solutions, or is_ unbounded _.

/ '\_

_';" 1. Toys-A'Go makes toys at Plant A and Plant BI Plant A hasmaterials to make up to 1000 toy dump trucks and fire engines.Plant B has materials to make up to 800 toy dump trucks and/

/ • fire engines. Plant A can make 10 toy dump trucks and 5 toyfire engines per hour. Plant B can produce 5 toy dump trucksand 15 toy fire engines per hour. It costs $300 per hour tooperate Plant A and $350 to operate Plant B per hour. Howmany hours should each plant be run in order to minimize cost?

Plan__: _8 hours2oobl Plant I8:24 hours

,50_ " minimum cost = $34,800

100 L(0, 50)_• '_

5O

(o,o)20 40 60 80/100120140 a

(100, 0)

2. Roadster Skateboard makes skateboards at Plants X and Y.Plant X must make at least 720 standard and deluxe boards.

Plant Y has materials to make up to 520 standard and deluxeboards. Plant X can make 12 standard and 5 deluxe boards perhour. Plant Y can make 10 standard and 8 deluxe boards perhour. It costs $250 to operate Plant X per hour and $225 tooperate Plant Y per hour. How many hours should each plant berun in order to minimize cost?

infeasible,5o_ii..........................

,°°i \,,,,75i

25

T12Glencde Division, Macmillan�McGraw-Hill

NAME DATE

3-1. Practice Wor :shc,,,,.tSymmetryFind the coordinates of P" if P and P" are symmetric with respectto point M.

1. P(-7, 4), M(O, O) 2. P(5, 4), M(-3, 2) 3. P(-5, 5), M(IO, 12)

The graphs below are portions of complete graphs. Sketch acomplete graph for each of the following symmetries: withrespect to (a) the x-axis, (b) the y-axis, (c) the line y - x, and(d) the line y - -x.

4. 5. 6...... Y___'__......... Y Y

_ .. _

-_ I I I

Determine if each function is an even function, an odd function, orneither.

7. y =x 2 + x - 2 8. y = _ - 25 9. y = -2x 7 + x 5 - x 3

Determine whether the graph of each equation is symmetric withrespect to the origin, the x-axis, the y-axis, the line y = x, or theline y =-x.

10. xy = -12 11. y = _ �Vxx+ 6 12. y2 - x29 16

f,


NAME DATE

3-1 Practice Wol'l_ ilh_,,_t

SymmetryFind the coordinates of P" if P and P" are symmetric with respectto point M. ,

1. P(-7, 4), M(O, O) 2. P(5, 4), M(-3, 2) 3. P(-5, 5), M(IO, 12)

(7,-4) (-11, 0_ {25, le)

The graphs below are portions of complete graphs. Sketch acomplete graph for each of the following symmetries: withrespect to (a) the x-axis, (b) th e y-axis, (c) the line y = x, and(d) the line y = -x.

4. 5. 6.

ii- -.... I• --7-__ ______;_ __ "_------i____y--Determine if each function is an even function, an odd function, orneither.

7. y = x 2 + x - 2 8. y = V_- 25 9. y =-2x 7 + x 5 - x3neither even odd

Determine whether the graph of each equation is symmetric withrespect to the origin, the x-axis, the y-axis, the line y = x, or theline y = -x.

10. xy = -12 11. y =_+ _x + 6 12. y2 - x2__ 16

y = x, origin x-axis x-axis, y=axis,origin

T13GlencoeDivision,Macmillan�McGraw-Hill

NAME DATE

3-2 Practice WorksheetFamilies of GraphsThe graph of f(x) is shown. Sketch a graph ofeach function basedon the graph of f(x).

1. a. g(x) = -2f(x) b. h(x) = 2f(X)

---_!'--_, -F- - -- g"_-' ...... .,,--I----h_'___ .... _

2. a. g(x) = -2[(x) + 1 b. h(x) = 2[ (-x)__ _f(c)__. ................

....j..............• "'- -L: - _:..... _ "- .... _ ..... _

For each parent graph, describe the transformation(s) that havetaken place in the related graph of each function.

3. f (x) = x 2 4. f (x) = [x]a. y = 2x 2 a° y = 3Ix] - 1

b. y = -0.5(x - 2) 2 b. y = 0.5[x] - 1

c.y=3x 2+ 1 c.y= [-x]

Sketch the graph of each function.

5. f(x) = - (x - 1)2 + 1 6. f(x) = 2 Ix + 2 - 3

li!i ...........

----t-- --J --- --

_ -4 ....I _ _


NAME DATE


Families of GraphsThe graph of f(x) is shown. Sketch a graph of each function basedon the graph of f(x).

1. a. g(x) = -2f(x) b. h(x) = 2f(x)

I I - / _ . I_ I io ;, - I o I;_ f v'-_ _ - _ i - - I -_ I A _ _ _ / _ _ /\ I__ /I \ _ - _ _ L _

- J / I_ _ / _ -

i

_ _ '! I __ I _ _ "* a _ -

2. a, g(x).= -2f(x) + 1 b. h(x) = 2f(-x)

- t_ / - - / \ - - __ / \ _II / _ _ _

I_ / -. \- I I - _ / o I_ ;_ _ Io 2__- I - _ / _ _ - / l- I \ / - _ l _ _ I _ _

I _o r - _ I I _ _ ...._ t \_ ;_ ....

For each parent graph, describe the transformation(s) that havetaken place in the related graph of each function.

3. f (x) = x 2 4. f (x) = [x]a. y = 2x 2 a.y = 3[x] - 1 vertical stretch,vertical stretch translation 1 unit downb.y ---0o5(x- 2)2 vertical shrink, b,y = 0.5Ix]- 1vertical shrink,reflect over x-axis, move 2 units translation 1 unit downrightc.y-- 3x2 vertical stretch, e.y -- [-x] reflection overtranslation 1 unit up the y-axis

Sketch the graph of each function.

,5. f(x)=-(x- 1)9+1 6. f(x)=21x+21-3

_ I i _ __ / __ I I r_ I" I _ __ /I '

I I \ r._ 1 I rN . ., \ /_ I I/o _ x__ _ \ / o x__

I I/ \ x /I I \I /I

_ i /I _ _ _

: T14Glencoe Division, Macmillan/McGraw-Hill

NAME : DATEr

3-3 ]_racticeWorksheetInverse Functions and RelationsFind the inverse of each function.

1. y = -5x+ 7 2. y = -4x 2 + 1

Find the inverse of each function. Sketch the function and itsinverse. Is the inverse a function? Write yes or no.

3. y =(x- 1)a + 1 4. y = 4x 2- 1

.... ...........

........... 0 x ............ u x

I

5. y=4(x+1) 2-2 6. y=_/_-I

.... _y____L_..... y:.....

"_" 0 x

For the parent graph f(x) = x 2 describe the transformation(s) thathave taken place in the related graph of each function.

7. y= ____v/-xx- 1 8. y=_+_x+ 1

9. y=--2_x 10. y = _+2 V_-x- 1


NAME DATE

3-3 Practice WorksheetInverse Functions and RelationsFind the inverse of each function.

1. y = -5x + 7 2. y = --4X2 + 1

y_ -x+7 y=+ !___5 -- 2

Find the inverse of each function. Sketch the function and itsinverse. Is the inverse a function? Write yes or no.

3. y:(x_l)3+1 y:_/-x_l+l;yeS4, y:4x2_l y:+!_/__x;nO--2

"i ,L_-____ -- ____j____4_-_ ,-

-----ti _ =_" ............. _;_ _

_._ __-___

" - ¼_/x_: []_=_ 1 _+1o5.y : 4(x + 1)2 2 H:---- 92--_ - ,_' :

-....oo__;___....-_....---=r- _ _-_ ...........

For the parent graph f(x) = x 2 describe the transformation(s) thathave taken place in the related graph of each function.

7. y= _+N/-x-x- 1 8. y:-+V_x+ 1translatesthe inverse translates _heinverse1 unit to the right 1 uni,_u_

9.y: _+gv_ lO.y: _+2g_- 1vertically stretches translates the inversethe inverse 1 unit to _he Hght; then

vertically stretches thegraph


NAME DATE

3-4 Practice WorksheetRational Functions and AsymptotesDetermine any horizontal, vertical, or slant asymptotes in the graph of eachfunction.

x- 1 5X 2- lOx+ 1x _ 2. y- x5 4x3 3. y=1. y - x2+2x+ 1 _ x-2

For each parent graph, describe the transformation(s) that have taken place in therelated graph of each function. Then sketch the graph.

4.f(x) = !

a.y= 4x

" "6 "_

b.y - 1 1x-1

Create a function of the form y = f(x) that satisfies each set ofconditions.

5. vertical asymptote at x = 1, hole at x = 0

6. holes at x = -1 and x = O,resembles y = x 2

Graph each rational function.

7. y - x+3 8. y - x_l+2x-3x-2 x-1

Y Y


NAME : DATE

3-4 Practice WorksheetRational Functions and AsymptotesDetermine any horizontal, vertical, or slant asymptotes in the graph of eachfunction.

x - 1 3. y = 5x2 - 10x + 1x 2. y - x5_ 4x3 x - 21. y-- x2+ 2x + 1

Horizontal: y = 0 ho_iz_>H_i:y= 0 slant: y = 5×

For each parent graph, describe the transformation(s) that have taken place in therelated graph of each function. Then sketch the graph.

4.f(x) = 1_ The graph of y = _4xdoes not come ..... Y:__H+=---- -- _ k'--P--P---

a.y = 4 aS C_OSe _:0 the origin as the graph .........x - ..... _-__¢>fy = !o Tlhevertical asymptote is stiDl -- ..... _

X -- ---_-'____

x = 0 and the [_odzontal asymptote is 7*= _,--:, _:_-_ -

b.y- x-1

± i_ _ans_ated l] uni_The graph of Y = =to the right and _ uni_ dOWnoThe newverticamasv_p_ot_ i_ _ = _ _d th_horizontal asymptote i_ y = _oCreate a function of the form y = f(x) that satisfies each set ofconditions.

5. vertical asymptote at x = 1, hole at x = 0

Y= .__6. holes at x = -1 and x = O, resembles y = x 2

y= _+_Px2+_

Graph each rational function.X2 + 2x -- 3

7. y- x+3 8. y-x-2 x-1

I

........ .........--4--- -- -7

......... _- ....... -_- -_

.....' *i i:,--;'_ .___-, -_z _-, I ...... - '_


NAME DATE

3-5 WorksheetGraphs of InequalitiesGraph each inequafity.

1. y>-2x+ 1 2. y<21x- 11

I...... Y y,, _. _---- --

3. y > (x - 1)2 / 4. y > -+ V_x + 1

..... y _ ...... yJ

.............. ._

..... _ ....... -_ i 5

Solve each inequality.

5. 12x+51 >3 6. 3x- 101 -<5

7. 14x-lOI-<6 8. 2x-71 >8

9. I-x+ll >-1 10. 2x+51 <0


Solve each inequality.

5. 1_+51 >3 6. 13x-101-<5

5 _<x_<5}{xlx < -4 or x > -1} {x I_

7. 14x-101 -<6 8. 12x-71 >8

{xl 1 -<x<_4} {x Ix<--_ orx> I

9. I-x+ll >-1 10. 12x+51 <0

all real numbers •Q


NAME DATE

3-6 Practice WorksheetTangentto a CurveFind the derivative of each function.

1. f(x) = 1._ 4 -3.5x 3 + 2._ 2. f(x) = 3x -3 - 5x -2 -

2 5 4. f(x)= 1 3 13. f(x)=-Z-- x--_ -_X + X2+3

Find the slope of the line tangent to the graph of each function atthe given point.

4x 3 -- 71 x 2 _ 4, (2,-2) 7. y - , (1, 0)5. y = 3x 2, (1, 3) 6. y = _ 2

Find the equation of the line tangent to the graph of each functionat the given point. Write the equation in slope-intercept form.Graph the function and the tangent.

8. y = 2x 2 - 1, (-1, 1) 9. y = x 2 - 5x + 1, (0, 1)

--i y y •.......... m ................

_--= ........................../ i

_____ ............ ,.............

I u x

______....... -; -- ...............I

_______..........................

10. Find the coordinates of the point(s) at which the line tangent tothe graph off(x) = X2 - 1 has slope -2.


NAME DATE

3-6 Practice W0rksheetTangent to a CurveFind the derivative of each function.

1. f(x) = 1.2x 4 - 3.5x 3 + 2.4x 2. f(x) = 3x-3 - 5x-2 - 4X

4.8 x 3 - 10.5x 2 + 2,4 -9x -4 + 10x -3 - 4

2 5 4. f(x)= 1 3 13. f(x) = xS x2 _X + X2+3

-10x -6 --I- 10x -'3 x 2 + x

Find the slope of the line tangent to the graph of each function atthe given point.

1 X 2 _ 4, (2,-2) 7. y " 4x3 - 7 , (1, 0)5. y = 3x 2, (1, 3) 6. y = _ 2

6 2 6

Find the equation of the line tangent to the graph of each functionat the given point. Write the equation in slope-intercept form.Graph the function and the tangent.

8. y = 2x 2 - 1, (-1, 1) 9. y = x 2 - 5X (0, 1)

y = -4x- 3 y = -5x + 1

_- .... ....t--- _ ..... _. ......

m - -_

10. Find the coordinates of the point(s) at which the line tangent tothe graph off(x) = x 2 - 1 has slope -2.

(-1,0)


I

I NAME IL DATE3-7 Practice Worksheet

Graphs and Critical Points of Polynomial FunctionsFind the critical points for each function. Then determine whethereach point is a minimum, a maximum, or a point of inflection.

1. f(x) = x 2 - 6x + 1 2. f(x) = 2x 2 - 10x d- 3

3. f(x) = -3x 2 + 9x 4. f(x) =-x 2 + 5x- 1 i

5. f(x) = x3 + x 2 - x 6. f(x) =:x4- 10x 2+9

Find the x- and y-intercepts of the graph of each function.

7. f(x) = 2x 2 + 5x - 3 8. f(x) = x 2 - 10x + 21

9. f(x) = (x + 1)(x - 1)(x + 2) 10. f(x) = x 3 + 5x 2 + 6x

U.f(x)=(x-2) 3 12. f(x)=x 4+2x 2+1

13. Graph f(x) = 2x 3 + 3x.

_._____ ......

-- --'F"- .........

I __r._ ........ m

__ ....._ .........

-T --1- -- ....


NAME DATE

3,.7 Practice Worksheet

Graphs and Critical Points of Polynomial FunctionsFind the critical points for each function. Then determine whethereach point is a minimum, a maximum, or a point of inflection.

1. f(x) = x 2 - 6x + 1 . 2. f(x) = 2x 2 - lOx + 3

(3,=8), rain (52_ 19)2, mill3. f(x) = -3x 2 + 9x 4. f(x) = -x2 + 5x - 1

5. f(x) = x a + x 2 - x 6. [(x) = x4 - lOx2 + 9

1-1'1)' max' 27 (-_/5, = 16), ITi,n5 ), rain (0, 9), max(N/-5, - 16}, rain

Find the x- and y-intercepts of the graph of each function.

7. f(x) = 2x 2 + 5x - 3 • 8. f(x) = x 2 - lOx + 21

1x-intercepts:-3, _ x-nntercepts: 3,7y-intercept:-3 y-intercept: 21

9. f(x) = (x + 1)(x - 1)(x + 2) 10. f(x) = x 3 + 5x2 + 6x

x-intercepts: -2, - 1, 1 x-intercepts: -3, -2, 0y-intercept:-2 y-intercept: 0

11. f(x) = (x - 2) 3 12. f(x) = x 4 + 2x 2 + 1

x-intercept: 2 x-intercepts: noney-intercept:-8 y-intercept: 1

13. Graphf(x) = 2x 3 + 3x.

____ __ _______

....l_i

-----,-- ----t6- --i


NAME DATE


Continuity and End BehaviorDetermine whether each graph has infinite discontinuity, jumpdiscontinuity, or point discontinuity, or is continuous. Then grapheach function.

x2_l2 2. y--1. y- 3x 2 x+l

__ .....+.._ ...... __ _...+ .........

-- -----I_ ..... --- -- ---_ ........

_._ - -- _ -- -- ,-_ _-............... _x _ 1 -;

: : :: :3. y - Ixl2 4. y = 2x 2 - 4x + 1

x

r I ly

............. I I

........ .. __ _ --- -_

t 2x+1, ifx<O5. y=x3-2x+2 6. y= 2x+2, ifx>O

..............I -I---I--_ () X

V

Without graphing, describe the end behavior of each function.

7. y = 2x 5- 4x 8. y = -x 2 + 13 9. y = 6- 6x- 6x 3

10. y = x 3 + 2x- 1 11. y = x 2 - 49 12. y = x 4 + x 3 + x 2 _- X


I

NAME DATE

3-8 Practice WorksheetContinuity and End BehaviorDetermine whether each graph has infinite discontinuity, jumpdiscontinuity, or point discontinuity, or is continuous. Then grapheach function.

_"Y- 3x_26H_initediscontinuity 2.y - X_x+-_ poin_ discontinuity

_ ---_ ....... _---

........*= ;" =1 -'' ;'x" _ 2z; _;_ -

---_ .... :I--

3.y - fX_x point discontinuity 4. y = 2x2--4_+ 1 continuous

....... _.... _ ....... Y_ t ....

.....* .... _ _-:_-_ -- _

I 2x + 1, ifx < 0 jump5. y = x3 - 2x + 2 continuous 6. y = 2x + 2, ifx > 0L discontinuity

.... _,_-_ ............ _,, _ _

....... )- .........

*-- _ - ; "-.... Z_;--; .............. _ t _ __-._=_<o--- L -_ _ _ _: ....... '-'2__.:d:_o..... t ......... -t--_ .............

Without graphing, describe the end behavior of each function.

7. y = 2x 5- 4x 8. y = -x 2 + 13 9, y = 6- 6x- 6x 3

as x _ _, y--> _ as x--> o_, y _ -o<, asx_oo,y_-oo

10. y = x 3 + 2x - 1 11. y = x 2 - 49 12. y = x 4 + x 3 -I- x 2 _ x

as x_ =_, y_-_ as x_- oo, y_oo asx_-oo,y-_ooas x_ _, y_ oo as x_ oo,y__ _o as x-_,y-->_

T20Glencoe Division, Macmillan�McGraw-Hill

NAME DATE

4-1 Practice WorksheetPolynomial FunctionsState the number of complex roots of each equation. Then find the roots and graphthe related function.

1. 3x- 5=0 2. x2+4=0f_ V

.... __ ____ .........

........ -'- +---5--- _

3. C2 + 2c + 1 = 0 4. x3 + 2x 2 - 15x = 0! ____ Y

i ..........

i__ ..........

+--- -_ .... -_

Write the polynomial equation of least degree for each set of roots given.

5. 4,0.5 6. 3,-0.5, 1 7. 3,3, 1, 1,-2

8. 1_+2i, 3 9. --2i, 3,-3 10.-1,3--i,2-3i

Solve each equation and graph the related function.

U. x3 + 6x + 20 = 0 12. X4 + 5X 3 + 9x 2 + 45x = 0

- z-' ...... -_ ......i

......... +.___ - _ ---_

C x

• -=.... i....21


NAME DATE

4-1 Practice WorksheetPolynomial FunctionsState the number of complex roots of each equation. Then find the roots and graphthe related function.

1. 3x-5=0 2. x2+4=0

/ L,, I_ / _ -- _'_ I _

. I* 0 / x - _,/ -

- i_ __- _-, .,.--' -_, O Ex__

5one; _ two; ± 2i

3. C2 -I- 2c + 1 = 0 4. X 3 + 2x 2 - 15x = 0

i _./ \,,. I_- _ /I - _ / _\ i - _ I\ .,.

_I _.J - - -: o\ _t xo ;_ _ L. --r _

two;-1 and -1 three;-5, 0, 3Write the polynomial equation of least degree for each set of roots given.

5. 4, 0.5 6. 3,-0.5, 1 7. 3, 3, 1, 1,-22x2-9x+4 = 0 2x3-7x2+2x+3 = 0 xS-6x4+6x3+

20x2-39x+18=0

8. 1 _+2i, 3 9. _ 2i, 3,-3 10. -1, 3 -+-i, 2 +_3ix3-5x2+11x-15 x4-5x2-36 = 0 xS-9x4+37x3-

71x2+12x+130=0

Solve each equation and graph the related function.

11. x 3 + 6x + 20 = 0 12. X 4 + 5x 3 -F9x 2 + 45x = 0

- / - - I ,, --

: - i

2-/:.o '; : l_ i, -'l_

:/,L^ '

= ,1 ± 3i -5, 0, ± 3i


NAME , DATE

4-2 Practice Wo]'l sk.eetQuadratic Equations and InequalitiesSolve each equation by completing the square. Then graph the related function.

1. X 2 -- 4x + 7 = 0 2. -4x _ - 8x = 7V J,y

.......... X

..-- --_ .... -; ..........

Find the discriminant of each equation and describe the nature of the roots of theequation. Then solve each equation by using the quadratic formula and graph therelated function.

3. x2 + x - 6 = O 4. -2x2 + x- 5 = O

.... Y_......... _'y_....

_-- --_ .... -; ..........

ii ..........

Determine the critical point(s) of the graph of each function to the nearest tenth.State if the point is a relative maximum, a relative minimum, or a point of inflection.

8. f.(x) = 2x 3 - 3x 2 - 36x + 6 9. f(x) =:x 3 - 3x + 3


Determine the critical point(s) of the graph of each function to the nearest tenth.State if the point is a relative maximum, a relative minimum, or a point of inflection.

8. f(x) " 2x 3-3x 2-36x+6 9. f(x) = x3--3x+3

(-2, 50); relative maximum (-1, 5); relative maximum(3,-75); relative minimum (1,1); relative minimum


NAME DATE

4-3 Practice WorksheetThe Remainder and Factor Theorems

Divide using synthetic division.

1. (X 2 -- 5X -- 12) + (x + 3) 2. (3x 2 + 4x - 12) + (x - 5)

3. (2x 3+3x 2-8x+3)+(x+3) 4. (x 4-3x 2+l)+(x-!)

Find the remainder for each division. Is the divisor a factor of the polynomial?

5. (2x 3 - 3x 2 - 10x + 3) + (x 3) 6. (2x 4 �4x3 -x 2 + 9) - (x + 2)

7. (10x3-x2 + 8x + 29) + (x + 2) 8. (2x4 x2-14x)+ (x + 7)

Use the remainder theorern to find the remainder for each division. State whetherthe binomial is a factor of the polynomial.

9. (3x 3 - 2x 2 + x - 4) + (x - 2) 10. (X 4 -- X 3 -- 10x 2 + 4x + 24) + (x + 2)

U. (x 4 + 5X 3 -- 14x 2) + (x + 7) 12. (X 3 �X2 -- 10) + (x + 3)

Find the value of k so that each remainder is zero.

13. (2x 3 + kx 2 �7x- 3) + (x - 3)) 14. (X 3 �9x2 - 12) + (x + 4)

15. Determine how many times 2 is a root ofx 3 - 7x 2 + 16x - 12 = 0.


NAME DATE

4-3 Practice WorksheetThe Remainder and Factor Theorems

Divide using synthetic division.

1. (X2 -- 5X -- 12) + (x + 3) 2. (3x 2 + 4x - 12) + (x - 5)

x- 8, R 12 3x + 19, R 83

3. (2x 3+3x 2-8x+3) •�� œ(x4 3x 2+1)+(x-1)2X 2 -- 3x + 1 x 3_- X 2 -- 2x- 2, R-1

Find the remainder for each division. Is the divisor a factor of the polynomial?

5. (2x 3-3x 2-10x+3)+(x-3) 6. (2x 4+4x 3-x 2+9)+(x+2)

O, yes 5, no

(2)7. (10x 3 -x 2 + 8x + 29) + x + _ 8. (2x 4 + 14x 3 - 2x 2 - 14x) + (x + 7)/i

25, no O, yes

Use the remainder theorem to find the remainder for each division. State whetherthe binomial is a factor of the polynomial

9. (3x 3-2x 2+x-4)+(x-2) 10. (x 4-x 3-10x 2+4x+24)

14, no O, yes

11. (X 4 -_- 5X 3 -- 14x 2) �(x+ 7) 12. (x 3 + x2 - 10) + (x + 3)

O,yes -28, no

Find the value of k so that each remainder is zero.

13. (2x 3 + kx 2 + 7x - 3) �(x- 3)) 14. (x 3 + 9x 2 + kx - 12) + (x + 4)-8 17

15. Determine how many times 2 is a root ofx 3 - 7x 2 + 16x - 12 = 0.twice

T 23Glencoe Division, Macmillan/McGraw-Hilt

NAME DATE

Practice WorksheetThe Rational Root Theorem

List all possible rational zeros of each function. Then determine the rational zeros.

l.f(x)=x 3+3x 2-6x-8 2. f(x)=36x 4-13x 2+1

3. f(x) = x 3 - 9x 2 + 27x - 27 4. f(x) = x 3 - x2 - 8x + 12

5. f(x) = x 4-3x 3-11x 2+3x+10 6. f(x) = Sx 4-2x-4

7. f(x) = 3x 5 - 7x 2 + x + 6 8. f(x) = x 3 + 4X 2 -- 22: + 15

9. f(x)=2X 3-3x 2-2x+3 IO.f(x):=4X 3-8x 2+x+3

\

Find the number of possible positive real zeros and the number of possiblenegative real zeros. Determine all of the rational zeros.

11. f(x) = 3x3 + 7x 2 + 2x + 4 12. f(x) :=5x4 - 3x 2 + x - 7

13. f(x) = x4 - 2x 3 - 4x 2 + 11x - 6 14. f(x) := x 4 -4x 3 - 7x 2 + 34x - 24

15. f(x) = 3x 3 - 4x 2 - 17x + 6 16. f(x) 2x 3 + 3x 2 + 5x + 2


NAME DATE

.4-4 Practice WorksheetThe Rational Root TheoremList all possible rational zeros of each function. Then determine the rational zeros.

1. f(x) = x 3 + 3x 2 - 6x - 8 2. f(x) = 36x 4 - 13x 2 + 1

{±1 +2,±4,±8};=4,'1,2 I+! +J_ +! +! +!' -- --36 _ --18 _ --12' --9 _ --6'Lr

\ +1 +I +I_ ___11+! +l-_'-_'-2' ;-2'-3J

3. f(x) = x 3 - 9x 2 + 27x - 27 4. f(x) = x 3 - x 2 - 8x -I- 12

{±_1, ±3, ---9, ±27}; 3 {±1,_+2, _+3,___4,.±6, ___12};=3, 2

5. f(x) : x4 - 3x 3 - llx 2 + 3x + 10 6. f(x) 5x 4 - 2x - 4

{---1, ___2,+5, +10}; ___1 -2, 5 [±,+2,+4, +I+2+4 ]_ _ , ,_ -_,-_,-_;

_, none

7. f(x) : 3x5 - 7x2 +x + 6 8. f(x) : x3 + 4x2 - 2x + 15

{ ,_ +1 _+_2] {___ _+,___1<2, _+3,___6,_-_, _ ; _,-3, ___5, 5};-5

none

9. f(x) = 2x 3 - 3x 2 - 2x -t- 3 10. f(x) = 4x 3 - 8x 2 -t- x -t- 3

±1, -+3, +1 +3 -----1 3 +1 +3,_4,+-- +-- []--2'--2 ; '2 -- '- --2'--4'--2]'

1 31, 2' 2

Find the number of possible positive real zeros and the number of possiblenegative real zeros. Determine all of the rational zeros.

U. f(x) = 3x3 + 7x 2 + 2x -t-4 12. f(x) = 5x 4 - 3x2 + x - 7

O, 3 or 1; none 3 or 1; 1; none

13. f(x) : x 4 - 2x 3_- ,4x2 + llx - 6 14. f(x) = x 4 - 4x 3 - 7x 2 + 34x - 24

3 or 1; 1; {1, 2,-1 ____-3}2 3or 1; 1; {-3,1,2,4}15. f(x) = 3x 3 - 4x 2 - 17x + 6 16. f(x) = 2x 3 + 3x 2 + 5x + 2


NAME DATE

4-5 Practice WorksheetLocating the Zeros of a FunctionDetermine between which consecutive integers the real zeros of each function arelocated.

1. f(x) = 2x 3 - 7x 2 + _ + 3 2. f(x) = 2x 3 - 13x 2 + 14x - 4

3. f(x) = 2x 3 - 13x 2 + 43x - 14 4. f(x) = 2x 3 - 7x 2 + 12x - 9

5. f(x) = 4x 4 - 16x 3 -- 25x 2 + 196x - 146 6. f(x) = x 3 - 3 •

Approximate the real zeros of each function to the nearest tenth.

7. f(x)=4x 4+2x 2- 1 8. f(x)=x 3-2x+2

9. f(x)=x 4-5x 2+1 10. f(x)=2x 3+x 2-1

U.f(x)=x 3-2x 2-2x 12. f(x)=x 3-5x 2+4

Use the upper bound theorem to find the least integral upper bound and thegreatest integral lower bound of the zeros of each function.

13. f(x) = 2x 3 - x 2 + x - 6 14. f(x) = x 4 .-k x 2 - 3

15. f(x)=x 3+3x+7 16. f(x)=2x 4-x 3-6x 2-2x-30

17. For f(x) X3 -- 3x 2, determine the number and __ __ .....type of possible complex zeros. Use the location ..........principle to determine the zeros to the nearest ..........tenth. Determine the relative maxima and _-- -_-- -_-- _--relative minima. Then, sketch the graph. - .........

.... _ .... -_

25Glencoe Division,Macmillan/McGraw-Hill

NAME DATE

4-5 Practice WorksheetLocating the Zeros of a Function

Determine between which consecutive integers the real zeros of each function arelocated.

1. f(x) = 2x 3 - 7x 2 + 4x + 3 2. f(x) = 2x3 - 13x 2 + 14x - 4-1 and 0; 1 and 2; 2 and 3 5 and 6

3. f(x) = 2x3 - 13x 2 + 43x - 14 4. f(x) = 2x 3 - 7x2 + 12x - 90 and 1 1 and 2

5. f(x) = 4x4 - 16x3 - 25x2 + 196x - 146 6. f(x) = x3 - 3-4 and 3; 0 and 1 1 and 2

Approximate the real zeros of each function to the nearest tenth.

7. f(x) = 4x4 + 2x 2 - 1 8. f(x) = x3 - 2x + 2__+0.6 -1.8

9. f(x) = x 4 - 5x 2 + 1 10. f(x) = 2x 3 + x 2 - 1_+2.2, _+0.5 0.7

U. f(x) = x3 - 2x 2 - 2X 12. f(x) = X3 -- 5X2 + 4

-0.7, 0, 2.7 -0.8, 1, 4.8

Use the upper bound theorem to find the least integral upper bound and thegreatest integral lower bound of the zeros of each function.

13. f(x) = 2x 3 - x 2 + x - 6 14. f(x) = x 4 + x 2 - 3

2, -1 2, -2

15. f(x)=x 3+3x+7 16. f(x)=2x 4-x 3-6x 2-2x-30

1, -2 3, -3

17. For f(x) = x 3 -3x 2, determine the number andtype of possible complex zeros. Use the locationprinciple to determine the zeros to the nearesttenth. Determine the relative maxima and

- irelative minima. Then, sketch the graph. - !

three real roots: 3 and 0 twice; -(0, 0)relative maximum _ i \(2,-4) relative minimum i _j


NAME DATE

4-6 Practice WorksheetRational Equations and Partial FractionsSolve each equation. Check your solution.

1 6n-9 _ 2I.C 4--3c 2" 2_n + 3n n

3.4+3 -2xx-3 x x 3 4. i__55-s+8= 10-- 8

5. 1 2m _ 1 6. 3+ 2 -1m-3 m+3 d+5 d-1

Solve each inequafity.

6 2 8=, 2n+1 < n-17. 7+3> t 3n+l 3n+l

9. 1+ 3v >2 10. 2x 5x+l >3y-1 4 3

Decomposeeach expression into partial fractions.U. -3x - 29 12. 7x2 -- 12x + 11

x2 -- 4x-- 21 2x3 -- 5x2 + x + 2

13. Solve x - 5 > O.x2 - 5x + 6


NAME DATE

4-6 Practice WorksheetRationalEquationsand PartialFractions 'Solve each equation. Check your solution. '

• _" 6n - 9 _ 21. c 4 -3 2. + 3n .nc

-1, 4 94

3. 4 + 3 _ '2x 4. 15--S+8=10x-3 x x-3 s

_9 1 =5, 32'

3+ 2 --15. 1 2m _ 1 6. d+5 d 1• m-3 m+3

-6,-1 -3, 4

Solve each inequality.2n+l < n-1

6 2 8. 3n + 1 3n + 17. 7 +3> t

4 1t < -_ or t > 0 -2 < n < -_

9. 1+ 3v >2 10. 2x 5x+1 >3y-1 4 3

1Y < -2 or y > 1 x < --207

I

Decompose each expression into partial fractions.7x 2"- 12x + 11

-3x-29 12. 2X3 5x 2+x+211. x2_ 4x_ 21

5 I- 2 3 2 F 5x-7 x+3 x- 2 x- 1 2x + 1

x-5 >0.13. Solve x2 _ 5x + 6

2 < x < 3 orx> 5


NAME DATE

4-7 Practice WorksheetRadical Equations and InequalitiesSolve each equation. Check your solution.

1. N/-4x +9- 3 = 5 2. V1-7y=3

3. _/_-9 =2 4. _/_v + 5 =-3

5. V'ff+x- 5= _v/6-x 6. V'-5 +x- V'xx- 3 = 2

7. _x+1=3-N/4--x 8. %/_-2x+1=2X+5

Solve each inequality. Check your solution.

9. __z- 3 <5 10. V_x+5>l

11. V/_x + 5 <9 12. V_x +3>5

13. Solve 2 _x = V_ - 3 + 1. Check your solution.


NAME DATE

4-7 Practice WorksheetRadical Equations and InequalitiesSolve each equation. Check your solution.

1. V_-_+ 9- 3 = 5 2. N/1-7y = 355 84 7

3. _T;x- 9 = 2 4. _ + 5 =-s25 32

•2 7

5. V_+x 5 _ x 6. x/_+x _x 3 ¢,no real solutions 4

7. _x + l= 3- N/4-x 8. N_x2-2x+l=2x+5

O, 3 _43

Solve each inequafity. Check your solution.

9. V_z-3<5 10. V_x+5>l

3 _<z_<14 x> -42 3

11. V'-3-xx+ 5 < 9 12. V_x+3>5

_5 <. X < 76 X--> 113 3

13. Solve 2 Vxx = N/_ - 3 + 1. Check your solution.1


NAME DATE

5-1 Practice WorksheetAngles and Their MeasureIf each angle has the given measure and isin standard position,determine the quadrant in which its terminal side lies.

1. 7_- 2.- 2__K 3. 371 ° 4. i4_-12 3 5

5. - 156 ° 6. 1000 ° 7. 332 ° 8. - 240 °

Change each degree measure to radian measure in terms of _r.

9. 36 ° 10. -250 ° 11.- 145 ° 12. 6 °

13. 870 ° 14. 18° 15. -820 ° 16. 345 °

Change each radianmeasuretodegreemeasure.

17. -1 18. 4_ 19. -2.56 20. 12.85

21. 3v 22. - 7_ 23. 13v 24. - 17_____E_16 9 30 3

J

Find one positive angle and one negative angle that are coterminalwith each angle.

25. 70 ° 26. -- 2_" 27. --300 ° 28. 3_"5 4

Find the reference ang_foreach ang_withthe given measure.

29. -20 ° 30. 160 ° 31. -545 ° 32. 300 °

33. 10_ 34. - 5v 35. - _ 36. 7_3 8 4 -- 3


NAME DATE

5,1 Practice WorksheetAngles and Their MeasureIf each angle has the given measure and is in standard position,determine the quadrant in which its terminal side lies.

1. 7_- 2. 2_ 3. 371° 4. 14_r12 -- 3 5

II III I II

5. - 156° 6. 1000 ° 7. 332 ° 8. - 240 °III IV IV II

Change each degree measure to radian measure in terms of _.9. 36° 10. -250 ° 11. - 145° 12. 6 °

25_r 29_r5 18 36 30

13. 870 ° 14. 18° 15. -820 ° 16. 345 °87_ _ 41_ 23_18 10 9 12

Change each radian measure to degree measure.

17. -1 18. 4_ 19. -2.56 20. 12.85

- 57° 720 ° - 147° 736 °

21. 3_ 22. 7_ 23. 137r 24.- 17_-16 - 9 30 3

33.75 ° - 140° 78° - 1020°

25-28 Answers may vary. Sample answers are given.Find one positive angle and one negative angle that are coterminalwith each angle.

25. 70° 26. 2_" 27.-300 ° 28. 3_"-- 5 4

430o, _ 290 ° 8_ 12_ 60o,_ 660 ° 11_ 5_5' 5 4 ' 4

Find the reference angle for each angle with the given measure.

29. -20 ° 30. 160° 31. -545 ° 32. 300 °20° 20° 5° 60°

33. 107r 34. 57r 35. --_ 36. -- 7_r-3 -8 -4 3

".- 3vr "." vr3 8 4 3

: T28GlencoeDivision,Macmillan/McGraw-Hill

NAME DATE

5'2 Practice WorksheetCentral Angles and ArcsGiven the radian measure of a central angle, find the measure ofits intercepted arc in terms of _ in a circle of radius 10 cm.

7T _-- 4. 7T2. 3 51. 6 -- 3. 2

Ir 8. 7r5. 3_ 6. 4_- 7. 125 7 24

Given the measurement of a central angle, find the measure of itsintercepted arc in terms of _ in a circle of diameter 60 in.

9. i0° I0. 60° 11.42° 12. 50°

13. 72 ° 14. 110 ° 15. 35 ° 16. 65 °

Given the measure of an arc, find the degree measure to thenearest tenth of the central angle it subtends in a circle of radius16 cm.

17. 87 18. 5.6 19. 12 20. 25

21. 10.24 22. 7.9 23. 11 24. 6

Find the area of each sector to the nearest tenth, given its centralangle, O,and the radius of the circle.

=Z25. 0 6 'r=14cm 26. 0 = 6'r=12ft +


NAME DATE

5-2 Practice WorksheetCentral Angles and ArcsGiven the radian measure of a central angle, find the measure ofits intercepted arc in terms of rr in a circle of radius 10 cm.

7r _ 4. z•r 2. 3 51. 6 -- 3. 2

5_- cm lO_-cm 5_ cm 2_- cm3 3

_" 8. "rr5. 3_r5 6. 4v7 7. 12 24

67Tcm 407Tcm 5_ cm 5_ cm7 6 12

Given the measurement of a central angle, find the measure of itsintercepted arc in terms of _r in a circle of diameter 60 in.

9. 10 ° 10. 60 ° 11. 42 ° 12. 50 °

5_ in. 10_Tin. ?Tr in. 25__in.3 3

13. 720 14. 110 ° 15. 35 ° 16. 65 ° -i

127Tin. 55_ in. 35_____in. 6577-in.3 6 6

Given the measure of an arc, find the degree measure to thenearest tenth of the central angle it subtends in a circle of radius16 cm.

17. 87 18. 5.6 19. 12 20. 25

311.5 ° 20.1 ° 43.0 ° 89.5 °

21. 10.24 22. 7.9 23. 11 24. 6

36.7 ° 28.3 ° 39.4 ° 21.5 °

Find the area of each sector to the nearest tenth, given its centralangle, 0, and the radius of the circle.

9"[

25. 0 6 'r=14cm 26. 0 = _= -- -_,r= 12ft

51.3 cm 2 37.7 ft2


NAME DATE

5-3 Practice WorksheetCircular FunctionsFind the values of the six trigonometric functions of an angle instandardposition if the given point lies on its terminal side.

1. (-1, 5) 2. (6,-8) 3. (3, 2) 4. (-3,-4)

5.(0,-4) 6.(7,0) 7"(V2'-N/2) 8" (V_-2)2'

Suppose 0 is an angle in standard position whose terminal side liesin the given quadrant. For each function, find the values of theremaining five trigonometric functions of 0.

3 2 . quadrant IV9. cos 0 = _ ; quadrant I 10. sin 0 = - _,

30.Glencoe Division,Macmillan/McGraw-Hill

NAME DATE

5-3 Practice WorksheetCircular Functions

Find the values of the six trigonometric functions of an angle instandard position if the given point lies on its terminal side.

1.(-i,5) 2.(6,-8) 3.(3,2) 4.(-3,-4)

sin 0- s v_ sin O= _4 sin 0- 2 N/_ sin 0 =-_426 5 13 5

COS 0- _--6 COS 0 3 COS 0 3 _ COS 0---326 5 13 5

tan 0 =- 5 tan 0 =- 4 tan 0- 2 tan O= 43 3 3

csc 0- _/_ csc 0 =- 5 csc 0- _/_ csc 0=-55 4 2 4

sec 0 =- _--6 sec 0 = 5 sec 0- _/_ sec 0-- 53 3 3

,F

cot 0 =- ! cot O=_ 3 cot 0-- 3 cot 0- 35 4 2 4

5. (0,-4) 6. (7, O) 7. (V2,- V2)__ 8. (_/-52,- ½)

sin 0 = -1 sin 0 = 0 sin 0- --_/2 sin 0=- !2 2

cos 0 = 0 cos 0 = 1 cos 0- _/_ COS 0-- _v/32 2

tan O: undefinedtan 0 = 0 tan 0 = -1 tan 0=-_/a2

cSc 0 =-1 csc 0 : undefinedcscO =-_/2 csc 0 =-22v_sec 0 : undefinedsecO= 1 sec 0 = _/2 sec 0-

3

cot 0 = 0 cot O: undefinedcot 0 =-1 cot 0 =-_/3Suppose 0is an angle M standard position whose terminal side liesM the given quadrant. For each function, find the values of theremaMmg five trigonometric functions of O.

3 2 . quadrant IV9. cosO = _ ;_quadrant I 10. sin 0 = - _,

sin 0 = 4 4 _/5 2_/-5.__;tan 0 = _ cos 0 = -_-; tan 0- 5

5. 5 CSC 0 -- 3 3_/5CSC 0-- _, sec 0- 3 2; sec 0- 5

cot 0 = 3. cot 0 = __/54 2

T 30Glencoe Division,Macmillan/McGraw-Hill

NAME DATE

5-4 Practice Worksheet •Trigonometric Functions of Special AnglesFind each exact value. Do not use a calculator.

--_ 3. tan 7r7r 2. cos 4 41. sin 4

4. cos 210 ° 5. sin 300 ° 6. tan 330 °

7. sin 3_r 3__K_ 9. tan 3_r4 8. COS 4 4

10. sin 90 ° 11. csc 270 ° 12. tan 45 °

13. cos 3_ 14. tan 3_r 15. sin 3___2 2 2

Use a calculator to approximate each value to four decimal places.

16. cot (-75 °) 17. sin 634 ° 18. cos 235 °

19. sin 2 20. sec 4.28 21. cot 0.23


m

NAME DATE

5-4 Practice WorksheetTrigonometric Functions of Special AnglesFind each exact value. Do not use a calculator.

IT 7r1. sin 4 2. cos 4 3. tan 4

2 _- 1

4. cos 210 ° 5. sin 300 ° 6. tan 330 °

2 -_- --_-

7. sin 31r4 8. cos 3_E_4 9. tan 3__K_4

v_ v_2 -_ -1

10. sin 90 ° 11. csc 270 ° 12. tan 45 °

1 -1 1

13. cos 2 14. tan 15. sin 3_K2

0 undefined -1

Use a calculator to approximate each value to four decimal places.

16. cot (-75 °) 17. sin 634 ° 18. cos 235 °

-0.2679 -0.9976 -0.5736

19. sin 2 20. sec 4.28 21. cot 0.23

0.9093 -2.3864 4.2709


NAME DATE

5-5 Practice WorksheetRight TrianglesSolve each triangle described, given thetrianglo below. Roundangle measures to the nearest degree and side measures to thenearest tenth.

1. A = 39°12 ', b = 2.1 B

Ca

2. a=9, B=49 °

A C 'b

3. B = 64 °, b = 19.2 4. B = 56048 ', C = 63.1

5. A=16 °,c=14 6. a=0.4, c=0.5

7. c=21.3,A=26°20 ' 8. a=2, b=7

9. A = 55055 ', c = 16 10. a = x/-i-5, B = 18 °


NAME DATE

5-5 Practice WorksheetRight TrianglesSolve each triangle described, given the triangle below. Roundangle measures to the nearest degree and side measures to thenearest tenth.

1. A = 39°12 ', b = 2.1 B

B _ 5I]° _ _ 1.7, ©_ 2.7

a

2. a=9, B=49 °

, A C

A _ 41° b _ :10.4,c _ 13.7 b

3. B=64 ° ,b=19.2 4. B=56°48 ' ,c=63.1

A_26 ° , a_9.4, C_21]°4 A_33 ° , b_52.8, C_ 34.6

5. A=16 °,c=14 6. a=0.4, c=0.5

a _ 3.9, b _ 13.5, B _ 74° A _ 53°, B _ 37°, b _ 0.3

7. c = 21.3, A = 26°20 ' 8. a = 2,b = 7'

B _ 64°, a _ 9.4, b _ 19.1 A _ 16°, B _ 74°; c _ 7.3

9. A = 55o55 ', c = 16 10. a = k/-i--5,B = 18 °

B _- 34°, a _ 13.3, b-_ 9.0 A _ 72°, c _ 4.1, b _ 1.3


NAME DATE

5-6 Practice WorksheetThe Law of Sines

Determine the number of possible solutions, ff a solution exists,solve the triangle. Round angle measures to the nearest minuteand side measures to the nearest tenth.

1. a=13.7, A=25°26 ',B=78 ° 2. b=50, a=33, A=132 °

3. A=38 °,B= 63 ° ,c=15 4. a=125,A=25 °,b=150

5. b = 15.2, A = 12o30 ' , C = 57o30 ' 6. a = 32, c = 20, A = 112 °

7. b = 795.1, c = 775.6, B = 51°51 ' 8. b = 15, c = 13, C = 50 °

9. a = 12, b = 15,A = 55 ° 10. b = 41, A = 33 °,B = 29 °


NAME DATE

5-6 Practice WorksheetThe Law of Sines

Determine the number of possible solutions. If a solution exists,solve the triangle. Round angle measures to the nearest minuteand side measures to the nearest tenth.

1. a=13.7,A=25°26 ',B=78 ° 2. b=50, a=33,A=132 °

b = 31o2, c = 31_o0,0=76°34 ' _@_e

3. A=38 °,B= 63 °,c=15 4. a=125,A=25 °,b=150

a = 9.4, b = 13.61 C= 79° c = 243.7, B = 30°28 ',C=124o32 '

c = 28.2, B = 149°32 ',C=5O28'

5. b = 15.2,A = 12°30 ', C = 57o30 ' 6. a = 32, c = 20, A = 112 °

a = 3.5, c = 13o6, B= 110° b = 18o6, B = 32°35 ',C=35o25 n

7. b = 795.1, c = 775.6, B = 51051 ' 8. b = 15, c = 13, C = 50 °

a = 989.2, A = 78°3 ', C=50°6 ' a = 15.7, A = 67°54 ',B=62o6 '

a = 3.6, A = 12°6 ',B= 117°54 '

9. a=12, b=15, A=55 ° 10. b=41,A=33 °,B=29 °

none a = 46.1, c = 74.7,C=118 o

Answers may vary due to procedure, rounding, and method ofcomputation.

T 33Glencoe Division, Macmillan/McGraw-Hill

NAME DATE

5-7 Practice WorksheetThe Law of CosinesSolve each triangle. Round angle measures to the nearest minuteand side measures to the nearest tenth.

1. a=1.5, b=2.3, c=l.9 2. b=40, c=45,A=51 °

3. A = 52°,b = 120, c = 160 4. a = 15, b = 18, c = 17

5. A = 42°,b = 120, c = 120 6. a = 15, b = 18, c = 20

7. b = 12, a = 20, c = 28 8. a = 12.5, b = 15.1, c = 10.3

9. c = 49, b = 40, A = 53 ° 10. a = lO, c = 8, B = 100 °


NAME DATE

5-7 Practice WorksheetThe Law of Cosines

Solve each triangle. Round angle measures to the nearest minuteand side measures to the nearest tenth.

1. a=1.5, b=2.3, c=1.9 2. b=40, c=45,A=51 °

A = 40o28', B = 84o16!, a = 36.9, B = 57°24 ',/ C=55°16 ' C=71°36 '

3. A=52 °,b=120, c=160 4. a=15, b=18, c=17

a = 127.9, B = 47°41 ', A = 50°39 ', B = 68o8',C=80o19 ' C=61o13 '

5. A=42 °,b=120, c=120 6. a=15, b=18, c=20

a = 86.0, B = C=69°0 ' A = 4608 _,B = 59°53 ',C = 73°59 '

7. b=122a=20,c 28 8. a=12.5, b=15.1, c=10.3

B = 21°47 ', A = 38°13 ', A = 55°8 ', B = 82°20 ',C = 120°0 ' C = 42°32 '

9. c=49, b=40,A=53 ° 10. a=10, c=8, B=100 °

a = 40.5, C = 74o58', b = 13.8, A = 45°20 ',B = 52o2' C = 34040'

Answers may vary due to procedure, rounding, and method ofcomputation,


NAME DATE

5-8 Practice WorksheetArea of TrianglesFind the area of each triangle to the nearest tenth.

1. c=3.58,A=37°40 ',B=69°20 ' 2. a=5, b=12, c=13

3. a=11, b=13, c=16 4. C=85°,a=2, B=19 °

5. A=50 °,b=12, c=14 6. b=14, C=110 °,B=25 o

O 7. b = 15, c = 20,A= 115 ° 8. a = 68, c = 110, C = 100 °

Find the area of each circular segment to the nearest tenth, givenits central angle, 0, and the radius of the circle.

9. 8=8, r=7 10. 0=108 °,r=1.4

Tr

U. _=23 °,r=4.2 12. 0=_,r=25.25

35Glencoe,Division, Macmillan/McGraw-Hill

NAME DATE

5-8 Practice WorksheetArea of TrianglesFind the area of each triangle to the nearest tenth.

1. c=3.58,A=37°40 ',B=69°20 ' 2. a=5, b=12, c=13

3.8 square units 30 square units

3. a=11, b=13, c=16 4. C=85 °,a 2, B=19 °

71,0 square units 0°7 square units

5. A=50 °,b=12, c=14 6. b=14, C=110 °,B=25 °

64.3 square UnitS _]54.1 square units

7. b=15, c=20, A=115 ° 8. a=68, c=110, C=100 °

135.9 square units 2526.8 square units

Find the area of each circular segment to the nearest tenth, givenits central angle,/9, and the radius of the circle.

9. 0= 8'r=7 10. /9= 108 °,r=1.4


7"/"

11. /9= 23 °, r = 4.2 12. /9= _, r = 25.25



I NAME DATE

6,-1 Practice WorksheetGraphs of the Trigonometric FunctionsFind each value by referring to the graphs of the trigonometric functions.

1. sin (-720 °) 2. tan (-180 °) 3. cos (540 °)

4. tan (180 °) 5. csc (720 °) 6. sec (180 °)

Find the values of _ for which each equation is true.

7. sin 0 =-1 8. sec _ =-1 9. tan O= 0

Graph each function on the given interval

10. y = sin x; -90 ° -< x -< 90° 11. y = tan x; -90 ° -< x -< 270 °

....... y _ ...... ____y _ ..........

__....... _ ...... _ _.___ _ .......................... x X

ZZ- -- - -- - -_-_-_-_-- Z -__............../

12. y= cos x; -360 ° -< x -< 360 ° 13. y = sec x; -360 ° -< x -< 360 °

....... Y ........ _ .... Y_'......

............... iI


NAME DATE

6-1 Practice WorksheetGraphs of the Trigonometric FunctionsFind each value by referring to the graphs of the trigonometric functions.

1. sin (-720 °) 2. tan (-180 °) 3. cos (540 °)0 0 =1

4. tan (180 °) 5. csc (720 °) 6. sec (180 °)0 unde_i_®d =1]

Find the values of _ for which each equation is true.

7. sinO=-I 8. secO=-I 9. tanO=O270 ° + 360k _ 450° + 360_ 180/_where k is any whe_® _ is any where k is anyinteger integer integer

Graph each function on the given interval.

10. y = sin x; -90 ° <-x -< 90° 11. y = tan x; -90 ° -<x -< 270 °-- -----I-._ .......... -- ..........

...... -1-- -----_=.-_ -2

_ 7..... 7-_--_j_ sjn¢_ _j__ _L ........... 2__/ ...... ,

______o__2_zL - -_o-- 7..... Z-- 2-_.... / -=2 -- _ ,-1-..........----_ -7' ..... -_ .... -

"_"--_ _--- -_--1-- ---_ .... - .2 -- _

12. y = cos x; -360 ° -< x -< 360 ° 13. y = sec x; -360 ° -< x -< 360 °

----_........ -Y-- ---- --LY- :` "=cxl_- --,---L .... -k--

.... ......"- 2 -_2__ _ _ _ _o__i _ 2;0_-I,0'_r o__. _o, z _x

____ X J_ =.__ __ k ZL-__ ---- Z X_= t----_ X---.......

-------4-'-- ..........

- _--_--;-- ;-- _--


NAME DATE

6-2 Practice WorksheetAmplitude, Period, and Phase ShiftState the ampfitude, period, and phase shift for each function.

1. y =-2 sin 0 2. y = 10sec 0 3. y =-3 sin 40

/ (4)4. y=O.5sin _-_ 5. y = 2.5 cos (O +180 °) 6. y=-l.5sin 40-

Write an equation of the sine function with each amplitude, period, and phase shift.

7. amplitude = 0.75, period = 360 °, phase shift = 30 °

8. amplitude = 4, period = 3 °, phase shift = -30 °

Write an equation of the cosine function with each ampfitude, period, and phaseshift.

9. amplitude = 3.75, period = 90 °, phase shift = 4 °

10. amplitude = 12, period = 45 °, phase shift = 180 °

NAME DATE

6-'2 Practice WorksheetAmplitude, Period, and Phase Shift _l}State the ampfitude, period, and phase shift for each function.

1. y =-2 sin _ 2. y = 10 sec 0 3. y =-3 sin 40

2, 360 °, 0° none, 360°, 0° 3, 90°, 0°

= i_-- _ 5. y = 9..5 cos (0 + 180°) 6. y = -1.5 sin 40-t

0.5, 360 °, 60° 2.5, 360 °, =1]g0° 1.5, 90°, 11.25°

Write an equation of the sine function with each amplitude, period, and phase shift.

7. amplitude = 0.75, period = 360 °, phase shift = 30°

y = 0.75 sin (0 - 30 °) or y = =0.75 sin {_ - 30°)

8. amplitude = 4, period = 3°, phase shift = -30 °

y = _ 4 sin (1200 - 3600°])

Write an equation of the cosine function with each amplitude, period, and phaseshift.

9. amplitude = 3.75, period = 90°, phase shift = 4°

y = _ 3.75 cos (40 - 16°)

10. amplitude = 12, period = 45°, phase shift = 180°

y = _ 12 cos (80 - 1440°)

Graph each function.

11. y = 0.5 sin x 12. y = 2 cos (3x)....... vz< _ ............ v,, v-_:_L_ i

- ,-_ _ 30

....... .-1....... _ - -_............... __ _ _t- __-

13. y = 2 cos (2x - 45 °) 14. y = tan (x + 60°)

.... -] _ _ l rl- -- L _ L- _L

- _ --_ _r__---_ __ _o ,1--_ ,_ _-

.... " i- i- _..... __--F ..... -_- I-5- _-


NAME DATE

6-3 Practice WorksheetGraphing Trigonometric FunctionsGraph each function.

1. y = 2 sin (x - 45 °) , 2. y = -2 cos (30)

....... _........ _, ......

................. i ........._ ...... _....... I "- -15 --_i I

I

) (x)3. y=-cos("- _ 4. y=sin -_ +90 °' ['_ 2

....... Y, __..... _ ....... _Y'_.......

38Glencoe Division, Macmillan/McGraw-Hill "

NAME DATE

6.13 Practice WorksheetGraphing Trigonometric FunctionsGraph each function.

1. y = 2 sin (x - 45 °) 2. y = -2 cos (30)

..... Y _-i_ V-_4_ __ _ _ _-_= ,-____-- __ _-__.____________ __ . _

__: + _

_) ( _0o)3. y= , cos I: - _ 4. y=sin x2 +

....... v_, --_x-_T ....... v,___ _ ��Œ�_,__ __ . F_ _T__5_

.... -_--_ ........ _--_

....... '1" .............. _ --- -t .....1

5. y = sinx + cosx 6. y = cos 2x- cosx

lY_"' _ _...... __ _,_:____ ---: :-_ :i=-___- o_-_

-- - ,- -- -_r_

............. "_-'2"-- --'_ - - '-- 1-, -- -- --


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6-4 Practice WorksheetInverse Trigonometric FunctionsWrite each equation in the form of an inverse relation.

1. 0.75 = sinx 2. -1 = cosx 3. 0.1 = tan 0 •

/

3 12"_ 6. cos _ = --4. _ = cos x 5. sin x - 2 13

Find the values of x in the interval 0° <_x < 360 ° that satisfy each equation.

7. x = arccos 1 8. arccos V2 12 -x 9. arcsin_=x

10. sin -1 (-1)= x 11. sin -1 V_2 -x 12. cot -ll=x

Evaluate each expression. Assume that all angles are in Quadrant L

(13. cos(cos-1½) 14. sin cos -11)

15. cos sin-1 _ 16. tan sin -1 V22 - c°s-1

17. Verify that sin -1 vo + sin_ 1 __1= 90 o. Assume that all angles are in Quadrant I.2 2


NAME DATE

6-4 Practice WorksheetInverse Trigonometric Functions

• Write each equation in the form of an inverse relation.

1. 0.75 = sin x 2. -1 = cos x 3. 0.1 = tan 0

X = arcsin 0.75 x = _vvQ$ _=1]} _ = _c_ 0=I]

3 v_ =124. _ = cos x 5. sin x - 2 6. COSc_ 13

q

x = arccos 5 _:= arcsir_v_ 122 _ -- _rCCO$ 13

Find the values of x in the interval 0° <_x <_360° that satisfy each equation.7. x = arccos 1 8. arccos V_ 1 "

2 -x 9. arcsin_=x

0°, 360° 45°, 31]5° 3_ 1]50 °

10. sin -1 (-1) = x 11. sin -1 V_2 -- x 12. cot-1 1 = x

270° 45°, 1]35° 45°, 225°

Evaluate each expression. Assume that all angles are in Quadrant I.

cos(cos . s,n(coso.5

2

15. cos (sin-Z ½) 16. tan (sin-1 _-eos-1 --_-_)

2 0

17. Verify that sin -1 -_-_ + sin -1 _1= 90o. Assume that all angles are in Quadrant I.

sin-1 _ + sin-1 ! = 60o + 300 = 90o2 2


4,0 ....nlNicGra_-_4"l_

Glencoe ONes'Ion,_acmm_

NAME DATE

6-5 Practice WorksheetPrincipal Values of the Inverse Trigonometric FunctionsFind each value.

1. Arcsin (-1) 2. Arccos 1 3. Arctan (-1)

-90 ° 0o =45o

0. os o, 0.4. Cos -1 _

60° 90° =30 °

7. cos (Cos-1 (- ½)) 8. sin (Sin-l-_ -_) 9. tan (Tan-1 --_-_)

V5 v_2 2 3.

10. Cos-1 (Cos 2) 11. Sin-1 (sin 4) 12. Tan-l(tan 3)7T 33" 77"

2 4 3

13. cos (Arcsin {) 14. sin (Arccos 3) 15. tan (Arcsin --_-_)

2 2 2

16. tan (½ Arccos 5) 17. cos (1Arcsin 6) 18. sin (2 Arccos 3)2 3_1-0 24

3 10 25

19. sin [Cos-1 (-_--_) - 4] 20. cos [Sin-1 (_-_) + f] 21. Tan[_ +Sin -1 -_-_]

0 0 0

i

T40Glencoe Division, Macmillan/McGraw-Hiil

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6-6 Practice WorksheetGraphing Inverses of Trigonometric FunctionsState the domain and range of each relation.

1. y=Sinx ”y=sinx+l 3. y=cosx-1

4. y = Cos -1 x _. y = arcsin x 6. y = Tan -1 x

Write the equation for the inverse of each function. Then graph the function and itsinverse.

7. y = Cos -1 x 8. y = Tan -1 (3x)..... Y_" _ ............. Y__ __ .....

_- - 0 x _ x

(

"n- .n')9. y = _ + Cos-lx 10. y = Sin : - 2

......... .....

___=__-;_: --_- _-_-___

"_ ..... -- ......

• Determine if each of the following is true or false, ff false, give a counterexample.

11. Cos-1 x = Cos-1 (-x)

12. Sin -1 x = - Sin -1 x .


NAME DATE

Practice WorksheetGraphing Inverses of Trigonometric FunctionsState the domain and range of each relation.

1. y = Sinx + 1 2. y = sinx + 1 3. y = cosx- 1domain: ,_maDH: domain:-90 ° _<x _<90° all teat numbers a_ireal numbersrange: 0 _<y _<2 range: 0 <- y-< 2 range:-2_ y _<0

4. y = Cos -1 x 5. y = arcsin x 6. y = Tan -1 xdomain: domain: domain:-1 _<x_< 1 -1 _<x_ 1 amlreal numbersrange: 0° _<y_< 180° range: range:

all real numbers =90° -< y-< 90°Write the equation for the inverse of each function. Then graph the function and itsinverse.

7. y = Cos -1 x 8. y = Tan -1 (3x)

..... Y._" _ ...... Y ; 1

,_ .... -T- ....._ G_ ___

.... -7-- - _5'_2 .._ _-_t_)_. _ ____ 2, _ :_ ...._- _ _ _- _--_--- ;.

; ._. - __....---- -- -_ ==1_ _ .........

1y- Cos x Y- -3 Tan x

/

"n" rr9. y=_+Cos -zx 10. y=Sinl_- > )

k..... ......... ....." -1--I -- - 71/ -

..... _ _ .... _- _ _ _, _- _--_-- -_ -__ -_. - _-__ _ / _,

....

Determine if each of the following is true or false. If false, give a counterexample.

11.(;os-zx-- Cos-Z(-_) false; Let ,_ = 1. Cos-:_- 0° butCos-_ (-1) - 180 °. So, Cos-_x ¢ (;__=;_ (-x) for all x.

12. Sin-z x = - Sin-_ x false; Let x = 1. Sin -_1 = 90° but-Sin -ix = -90 °. So, Sin-_x :/: -Sin -_x for all Xo


NAME DATE

6-7 Practice WorksheetSimple Harmonic MotionFind the amplitude, period, frequency, and phase shift.

1.3,=3sin 0-90 ° 2. Y=-2 cosO

5. y=l.5cos x+ 6. y=12cos 2_-

7w=220cosx2t 8z=O:3sin(344)

Write an equation with phase shift 0 to represent simple harmonic motion undereach set of circumstances.

9. initial position 12, amplitude 12, period 8

10. initial position O, amplitude 2, period 8_r

11. initial position -24, amplitude 24, period 6


NAME DATE

6-7 Practice WorksheetSimple Harmonic MotionFind the ampfitude, period, frequency, and phase shift.

1. y=3sin _0-90 ° 2. y=--_cosO

1 0o3, 720 ° 1 180° 1,360 o, 360'' 720 '

1 3_ 2, _, 11, 2rr, 2_' 2 _' 12

o. 1.0cos(x+1

1 2 12, 1, 1, 121,5, 12, 12'

• ' 4

1 0 o 0.3, 8 3 1220, 30°, 30' 3' 8' 3

Write an equation with phase shift 0 to represent simple harmonic motion undereach set of circumstances.

9. initial position 12, amplitude 12, period 8

1rty = 12 cos 4

10. initial position O, amplitude 2, period 87r

ty= +_2 sin _.

U. initial position -24, amplitude 24, period 6

_rty = -24 cos 3


NAME DATE

7-1 Practice WorksheetBasiC Trigonometric IdentitiesSolve for values of 0 between 0° and 90°.

1. If tan 0 = 2, find cot 0 2. If sin 0 = 2 find cos 8.• 3'

13. If cos 0 = _, find tan 0 4. If tan 0 = 3, find sec 0.

5. If sin 0 = 7 find cot 0 6. If tan 0 = 7 find sin 0' " 2 ' •

Express each value as a function of an angle in Quadrant I.

7. sin 458 ° 8. cos 892 °

9. tan (- 876 °) 10. csc 495 °

Simpfify.

11. cotA 12.tan A cos

13. sin 2 0 cos 2 0 -cos 2 0 14. cos x + sin x tan x

43Giencoe Division,Macmillan/McGraw-Hill

NAME DATE

7-1 Practice WorksheetBasic Trigonometric IdentitiesSolve for values of 0 between 0° and 90°.

1. If tan 0 = 2, find cot 0 2. If sin 0 = 2 find cos 0.• 3'

1 x/g2 3

13. If cos 0 = _, find tan 0 4. If tan 0 = 3, find sec 0.

5. If sin 0 = 7 find cot 0 6. If tan 0 = 7 find sin 0' " 2 ' "

,51 7_7 53

Express each value as a function of an angle in Quadrant I.

7. sin 458 ° 8. cos 892 °sin 82° - cos 8°

9. tan (- 876 °) 10. csc 495 °tan 24° csc 45°

Simpfify.

11. cotA 12.tan A cos

cot2A sin fi

13. sin 2 0 COS2 0 --COS 2 0 14. COSx + sin x tan x-- COS 4 0 sec x


NAME DATE

7-2 Practice WorksheetVerifying Trigonometric IdentitiesVerify that each of the following is an identity.

1. csc x = cos xcot x + tan x

2. sin3x - cos3x = (1+ sinx cosx) (sinx - cosx )

3. 1 1 _ .2sec2ysin y- 1 sin y + 1

4. 1-2sin 2r+sin 4r=cos 4r

5. tanu+ cosu -secu1 + sin u

6. tan x + sec x = CSCxsec x - cos x + tan x

Find a numerical value of one trigonometric function of each x.

7. sin x = 3 cos x 8. cos x = cot x


NAME DATE

7-2 Practice WorksheetVerifying Trigonometric IdentitiesVerify that each of the following is an identity.

1. cscx -- cos xcot x + tan x

1CSCX sin x sin x cos x cos x

. cot x + tan x cos x I- sin x sin x cos X COS2 X sin2 xsin x cos x

COS X-- -- COS X

1

2. sin 3x - COS3 X ----(1+ sin x cos x) (sin x - cos x )

sin3x-cos 3 x = (sin x-cos x) (sin2x+ sin x cos x + cos2x)= (sin x-cos x) (1 + sin x cos x)

3. 1 1 _ 2sec2ysin y- 1 sin y + 1

1 1 _ sin y + 1 - sin y_+ 1 _ 2 -- 2 sec 2 ysin y-1 sin y + 1 sin2 y- 1 -cos 2 y

4. 1 - 2 sin 2 r + sin 4 r = COS4 r

cos4 r = (1 - sin2 r) 2 - 1 - 2 sin 2 r + sin4 r

5. tanu+ cosu -secu1 + sin u

tan u + cos u _ sin u t- cos u _ sin u + sin2 u + COS 2 U1 + sin u cos u 1 + sin u (cos u)(1 + sin u)

1 + sin u- - sec u(cos u)(1 + sin u)

6. tan x + sec x = CSCXsea x - cos x + tan x

sin x+ 1 sin x+ 1 sin x + 1= = = CSC X1 - cos2 x + sin x sin2 x + sin x (sin x)(sin x + 1)

Find a numerical value of one trigonometric function of each x.

7. sinx=3cosx 8. cosx=cotxtan x = 3 csc x = 1 or sin x = 1


NAME DATE

7-3 Practice WorksheetSum and Difference IdentitiesUse the sum and difference identities to find the exact value ofeach function.

1. cos 75 ° 2. cos 375 °

3. sin(-165°) 4. sin(-105°)

5. sin95°cos55° + cos95°sin55° 6. cos160°cos40° + sin160°sin40°

7. tan (135° + 120 °) 8. tan 345 °

ff _ and fi are the measures of two first quadrant angles, find theexact value of each function.

12 3 find cos (a - fi).9. If sin a = ]_ and cos fl = -_,

12 1210. If cos a = _ and cos fl = -_, find tan (a - fl ).

11. If cos a = 8 and tan fl = 1-_' find cos (a + fi).

13 512. If csc a = ]-_ and sec fi = _, find sin (a - fl).

Verify that each of the following is an identity.

13. cos (180 ° - O)= - cos 0

14. sin (360 ° + 0) = sin 0


NAME DATE

7-3 Practice WorksheetSum and Difference IdentitiesUse the sum and difference identities to find the exact value ofeach function.

1. cos 75° 2. cos 375 °

4 4

3. sin (-165 °) 4. sin (-105 °)

4 4

5. sin 95° cos 55° + cos 95 ° sin 55° 6. cos 160 ° cos 40° + sin 160° sin 40 °

1 12 2

7. tan (135 ° + 120 °) 8. tan 345 °

2

If a and fl are the measures of two first quadrant angles, find theexact value of each function.

12 39. If sin a = ]_ and cos fl = -5' find cos (_ - fl). 6365

z2 12 36010. Ifcosa=_andcosfl= _-,findtan(a-fl). -- 319

, 21U. If cos _ = 8 and tan fl = find cos (a + fl). 221

• 13 5 1612. If csc a _-_ and sec fl = _, find sin (a- fl). 65


13. cos(180° - 0) = - cos 0

cos (180 ° - 0)- cos 180 ° cos 0 + sin 180° sin 0- (-1) cos 0 + 0 • sin 0= - cos 0

14. sin (360 ° + O)= sin 0

sin (360 ° + O)= sin 360 ° cos 0 + cos 360 ° sin 0= O. cos 0 + 1 • sin 0= sin 0

T 45GlencoeDivision,Macmillan/McGraw-Hill

NAME DATE


Double-Angle and Half-Angle Identities12 and A is in the first quadrant, find each value.If sin A =

1. cos 2A 2. sin 2A

3. tan 2A 4. cos A2

5. sin A 6. tan A_2 2

Use a half-angle identity to find each value.

7. tan -_ 8. cos _5_v_8 8

9. sin 19_r12 10. COS671°


U. tan A _ sinA 12. tan A _ 1-cosA2 1+ cosA 2 sinA


NAME DATE


Double-Angle and Half-Angle Identities12

If sin A = _ and A is in the first quadrant, find each value.

1. cos 2A 2. sin 2A

119 120169 169

3. tan 2A 4. cos A2

120 3119 13

5. sin A 6. tan A2 2

2x/_ 213 3

Use a half-angle identity to find each value.5_r

_r 8. cos 87. tan 8

._2 - "k//2" 2 + %/-2 2

9. sin 19_r 10. cos 671°12

_/2 + _ '_/2- "k/22 2

Verify that each of the following is an identity.11. tan A _ sinA 12. tan A _ 1- cosA

2 1+ CosA 2 sin A

A__l-cosA._l+cosA tan A _ sinA . 1-cosAtan _ - 1 + cos A 1+ cos A 2- -- 1+ cos A 1-cos A

/ 1- cos2 A _ sin A (1-cos A)(1 +cos A)2 1 -COS2A

sinA _sinA(1-cosA)_ 1-1 + cos A sin 2 A sin A

T 46GlencoeDivision,Macmillan/McGraw-Hill

NAME DATE

7-5 Practice WorksheetSolving Trigonometric EquationsSolve each equation for all values of x.

1. 2sin2x-5sinx+2=O 2. sin2x-2sinx-3=O

3. 3cos2x-5cosx=l 4. 2tanxcosx+2cosx=tanx+l

Solve each equation for 0° <_x < 180°.

5. 2sin2x-l=O 6. cosx=3cosx-2

7. tan x = sin x 8. cos x sin 2x = 0

9. secx=l+tanx 10, 4sin2x-4sinx+l=O

11. sin 2x = 2 cos x 12; tan 2x + tan x = 0

13. 2sin2x 1 14. cos2x+sinx=l

47Glenc0e Division, Macmillan/McGraw-Hill

NAME DATE

7-5 Practice WorksheetSolving Trigonometric EquationsSolve each equation for all values of x.

1. 2sin2x-5sinx+2=O 2. sin2x-2sinx-3=O

30° + 360_ 1150° + 360_ 27'0° + 360E_

3. 3cos2x-5cosx=l 4. 2tanxcosx+2cosx=tanx+l

120 ° + 360k'°, 24G° + 3_G£ =60 +350£, = 45° + 1180E_60° + 36G_

Solve each equation for 0 ° <- x <_180 °.

5. 2sin2x-l=O 6. cosx=3cosx-2

45°, 135° 0°

7. tan x = sin x 8. cos x sin 2x = 0

0°, 180° O°, 90% _80 °

9. sec x = 1 + tan x 10, 4 sin 2 x - 4 sin x + 1 = 0

0° 30% _50°

11. sin 2x = 2 cos x 12. tan 2 x + tan x = 0

90° 0% "i]35° 1]8Q°

13. 2sin2x=l 14. cos2x+sinx=l

15°, 75° 0°, 30°, _50°, _80°

"T47Glencoe Division, Macmillan/McGraw-Hill

NAME DATE

7-6 Practice WorksheetNormal Form of a Linear EquationWrite the standard form of the equation of each line given p, themeasure of its normal, and ¢, the angle the normal makes with thepositive x-axis.

1. p =4, _b=30 ° 2. p = 2,_b= 45 °

3. p=3, ¢=60 ° 4. p=12,_b=120 °

5. p = 8, ¢ = 150° 6. p = 15, _b= 225 °

Write each equation in normal form. Then find the measure of thenormal, p, and 4),the angle that the normal makes with the positivex-axis.

7. 3x-2y- 1 =0 8. 5x+y- 12=0

9. y=x+5 lO.y=x 2

11. x+y-5=O 12. 2x+y-l=O


NAME DATE

7-6 Practice WorksheetNormal Form of a Linear EquationWrite the standard form of the equation of each line given p, themeasure of its normal, and 4), the angle the normal makes with thepositive x-axis.

1. p=4, ¢=30 ° 2. p=2,_b=45 °

3x= _/3 y- 8_-3= O x+ y- 2 X/-2= O

a. p =3, _ =60° 4.p = 12,_= _2oox+ _/3y- 6 = 0 x-_/-3y+ 24=0

5. p = 8, _b= 150 ° 6. p = 15, _b= 225 °/...._

16 N/_ = 0 15 N/_ = 03x-V3y+ x+ y+

Write each equation M normal form. Then find the measure of thenormal, p, and 4), the angle that the normal makes with the positivex-axis.

7. 3x-2y- 1=0 8. 5x+y- 12=0

3x 2y 1 5x y 12_ %/13---- O; _ -F _ _ -- O;

12

326o; 1 - 0.28 units 11°" _ = 2.35 units_/_9. y=x+5 10. y=x-2

x y 5 x y 2

+ - - O; -- - O;

2 _ 1.41 units¢h 135o, 5 _ 3.54 units _b= 315°; v_

#

ll.x+y-5=0 12. 2x+y-l=0

+ --v3 - O; _ _ - 0.,

1 _ 0.45 unit5 _ 3.54 units _b= 27°;_b= 45°; v_


NAME DATE

7-7 Practice WorksheetDistance from a Point to a LineFind the distance between the point with the given coordinates and the line withthe given equation. Round to the nearest tenth.

1. (-1,5),3x-4y-l=O 2. (2,5),5x-12y+1=O

3. (1, - 4), 12x + 5y - 3 = 0 4. (-1, -3), 6x + 8y - 3 = 0

Name the coordinates of one point that satisfy the first equation. Then find thedistance from the point to the graph of the second equation.

5. 2x-3y +4=0 6. 4x-y+ 1=02

y=-_x ¼�4x-y-8=O

7. x+3y-4=O 8. 3x-2y=6x + 3y + 20 = 0 3x-2y + 30 = 0

Find an equation of the line that bisects the acute angle formed by the graphs ofeach pair of equations.

9. x+2y-3 =0 10. x+y-6=Ox-y+4=O x-2y-2=O

11. x+y-5=O 12. 2x+y-3=O2x -y + 7 = 0 x -y + 5 = 0


NAME DATE

7-7 Practice WorksheetDistance from a Point to a LineFind the distance between the point with the given coordinates and the line withthe given equation. Round to the nearest tenth.

1. (-1, 5), 3x- 4y- 1=0 2. (2, 5), 5x- 12y + 1=0

49 _ 3.774.8 13

3. (1,- 4), 12x + 5y - 3 = 0 4. (-1,-3), 6x + 8y - 3 = 0

11 _ 0.85 3.313

Name the coordinates of one point that satisfy the first equation. Then find thedistance from the point to the graph of the second equation.

5. 2x-3y+4=0 6. 4x-y + 1=02

y=-_x+5 4x y 8=0

( ) °.o=0,-g4., _ _3x/ig_ 3.05 (0, 1); _7

7. x+3y-4=0 8. 3x-2y=6x + 3y + 20 = 0 3x-2y + 30 = 0

(0, I )., 12 _1-05 _7.59 (0,-.3); 36_/_13 _9.98

Find an equation of the line that bisects the acute angle formed by the graphs ofeach pair of equations.

9. x +2y- 3 = 0 10. x+y-6=Ox-y + 4=O x-2y-2=O

(_/2-_/-5)x + 12_/2+_/5)y (_/2-_/5)x-12%/2 +_/5)Y-(3_v/2-- + 4%/5)_ = 0 - 2_/-2 + 6N/5 = 0

11.x+y-5=O 12. 2x+y-3=O2x-y+7=O x-y+5=O

1_/5 + 2_/2)x + 1_/5 - _/2)y 12_/2-_/-5)x+1_/2+ _/5)Y-5_/-5 + 7_/2 = 0 -(3_ _- 5_/-5) = 0


NAME DATE

8-i Practice WorksheetGeometric Vectors

Use a metric ruler and a protractor to find each sum or difference. Then, find themagnitude and ampfitude of each resultant.

6O° _o._ 310o%- -

1._+b 2. b-_

3. 2a+b 4. b+3_

Find the magnitude of the vertical and horizontal components of each vectorshown for Exercises 1-4.

5._ 6. b 7._

5OGlencoe Division, Macmillan/McGraw-Hill

NAME DATE

8-1 Practice WorksheetGeometric VectorsUse a metric ruler and a protractor tO find each sum or difference. Then, find themagnitude and ampfitude of each resultant.

SO _o._ 310 o%- -

1. _ +_) 2. /)- _

39 ram; 110° 40 ram; 139°

_ _ _ _C r

3. 2_+_) 4. l) + 3_

55 ram; 93° 6 ram; 217°

/ ,../ \

"\ ",,",,_ 140 o/ \\/

// b/ , - "_

/ 2a +/

+/3c'" \ 3x_,X _

/

• . \_

\

Find the magnitude of the vertical and horizontal components of each vectorshown for Exercises 1-4.

5._ 6._, _._x = 10 mm x _ -23.0 mm x _ 6.4 mmy -_ 17.3 mm y = 19.3 mm y_=-7.7 mm


NAME DATE

8=2 Practice WorksheetAlgebraic Vectors

Find the ordered pair that represents the vector from A to B. Then find the

magnitude of AB.

1. A(2, 4), B(.1, 3) 2. A(4, -2), B(5, -5) 3. A(-3, -6), B(8, -1)

Find the sum or difference of the given vectors algebraically. Express the result asan ordered pair.

4. 2i+6j 5. 4i- 5j

Find an ordered pair to represent EJin each equation if _,= (2, -1) and _v= (-3, 5).

6. fi =_ +,;v 7. _=_ -_¢

8. _ = 3_- 9. fi =@ -2_'

10. fit = 2_ + 3fV 11. fl = 5f,V - 29


NAME DATE

8-9, Practice Worksheet

Algebraic Vectors

Find the ordered pair that represents the vector from A to B. Then find the

magnitude of AB.

1. A(2, 4), B(-1, 3) 2. A(4, -2), B(5, -5) 3. A(-3, -6), B(8, -1)

(-3,-1);_/10 (1,-3_;_/_o 01,5);

Find the sum or difference of the given vectors algebraically. Express the result asan ordered pair.

4. 2i+ 6j 5. 41-5j(2, 6) (4,-5)

Find an ordered pair to represent iJ in each equation if _ = (2, -1) and _v= (-3, 5).

6. fa =_" +_ 7. fi =fr -gv

(-1, 4) (5,-6)

8._=3_ 9. fl =W- 2_,

(6,-3) (-7, 7)

10. fit = 2_" + 3-& U. 6 = 5-,7v- 2¢'

{-5, 13) (-19, 27)


NAME DATE

8-3 Practice WorksheetVectors in Three-Dimensional SpaceLocate points with the given coordinates. Then find the magnitude of a vector fromthe origin to each point.

1. (4, 7, 5) 2. (3,-2, 6)

For each pair of points A and B, find an ordered triple that represents AB. Then findthe magnitude of AB.

3. A(2, 1, 3), B(-4, 5, 7) 4. A(4, 0, 6), B(7, 1, -3)

5. A(-4, 5, 8), B(7, 2, -9) 6. A(6, 8, -5), B(7, -3, 12)

Write each vector as the sum of unit vectors.

7. (8,-9, 2) 8. (7, 6,-5)

Find an ordered triple to represent b ineach equation if _ = (2, -4, 5) andw =(6,-8,9).

9. _1=_7 +-& 10. _1=_-_

11. fl = 4_- + 3-,:v 12. fl = 5_7 - 2@


NAME DATE


Vectors in Three-Dimensional SpaceLocate points with the given coordinates. Then find the magnitude of a vector fromthe origin to each point.

1. (4, 7, 5) 2. (3,-2, 6)

Z, Z

/,---- "6

5; / i//./- _ /

/ / f _-flII ..... -( I Jl ;i

/7 YI _2 i / i/i. Z ;IZ .I/ I/-- -,L/3

x�For each pair of poi_nts A and B, find an ordered triple that represents AB. Then findthe magnitude of AB.

3. A(2, 1, 3), B(-4, 5, 7) 4. A(4, 0, 6), B(7, 1, -3)

(-6, 4, 4); 2_/17 (3, 1,-9);

5. A(-4, 5, 8), B(7, 2, -9) 6. A(6, 8, -5), B(7, -3, 12)

(11,-3,-17);_9 (1,-11,17);_11

Write each vector as the sum of unit vectors.

7. (8,-9, 2) 8. (7, 6,-5)

8f - 9f+2_ 7f +6f - 5_Find an ordered triple to represent IJ in each equation if I/ = (2, -4, 5) and# --(6,-8,9).

9. C[ =-_ +-& 10. f_ =_7 --,;v

(8, -12, 14) ('4, 4, -4)

11. fi = 4_" + 3¢v 12. fi[ = 5_" - 2-,;v

(26,-40, 47) (-2,-4, 7)


NAME DATE

8-4 Practice WorksheetPerpendicular VectorsFind each inner product and state whether the vectors are perpendicular. Write yesor no.

1. (3, 6). (-4, 2) 2. (-1, 4). (3,-2) 3. (2, 0). (-1,-1)

4. (7,-3). (-2, 4) 5. (-3, 0). (4,-3) 6. (7,-4) • (-5;-8)

7. (-2, O, 1). (3, 2,-3) 8. (-4,-1, 1)- (1,-3, 4) 9. (0, O, 1)-(1,-2, O)

Find each cross product. Then verify that the resulting vector is perpendicular tothe given vectors.

10. (1, 3, 4) x (-1, O,-1) 11. (3, 1,-6) x (-2, 4, 3)

12. (3, 1, 2) x (2,-3, 1) 13. (4,-1, O) x (5,-3,-1)

14. (-6, 1, 3) x (-2,-2, 1) 15. (0, O, 6) x (3,-2,-4)


NAME DATE

8-4 Practice WorksheetPerpendicular VectorsFind each inner product and state whether the vectors are perpendicular. Write yesor no.

1. (3, 6). ('4, 2) 2. (-1, 4). (3,-2) 3. (2, 0)" (-1,-1)

O;yes =_; _o -2; no

4. (7,-3). (-2, 4) 5. (-3, 0)" (4,-3) 6. (7,--4). (-5,-8)

-26; no -12; n_ =3; no

7. (-2, O, 1). (3, 2,-3) 8. (-4,-1, 1). (1,-3, 4) 9. (0, O, 1). (1,-2, O)

-9; no 3; no O;yes

Find each cross product. Then verify that the resulting vector is perpendicular tothe given vectors.

10. (1, 3, 4) × (-1, O, -1) 11. (3, 1, -6) x (-2, 4, 3)

(-3,-3, 3); yes (27, 3, 14); yes

12. (3, 1, 2) x (2,-3, 1) 13. (4,-1, O) × (5,-3,-1)

(7, 1,-11); yes (1, 4,-7); yes

14. (-6, 1, 3) × (-2, -2, 1) 15. (0, O, 6) × (3, -2, -4)

(7, 0, 14); yes (12, 18, 0); yes


NAME DATE• j

8-5 Practice WorksheetApplications with VectorsSolve. Make a diagram to help you. Round all angle measures tothe nearest minute. Round all other measures to the nearest tenth.

1. A plane flies due west at 25-0 kilometers per hour while the wind blows south at70 kilometers per hour. Find the plane's resultant velocity and direction.

2. A plane flies east for 200 km, then 60 ° south of east for 80 km. Find the plane'sdistance and direction from its starting point.

3. One force of 100 units acts on an object. Another force of 80 units acts on theobject at a 40 ° angle from the first force. Find the magnitude and direction of theresultant force on the object.

/


NAME DATE


Applications with VectorsSolve. Make a diagram to help you, Round all angle measures tothe nearest minute. Round all other measures to the nearest tenth.

1. A plane flies due west at 250 kilometers per hour while the wind blows south at70 kilometers per hour. Find the plane's resultant velocity and direction.

259.6 km/hr; 15°39 , sou_h o_ west

2. A plane flies east for 200 km, then 60 ° south of east for 80 km. Find the plane'sdistance and direction from its starting point.

249.83 kin; 16°6, south of east

3. One force of 100 units acts on an object. Another force of 80 units acts on the

object at a 40 ° angle from the first force. Find the magnitude and direction of theresultant force on the object.

169.3 units; 17o41, from the first force


NAME DATE

8-5 Practice WorksheetVectorsand ParametricEquationsWritea vector equation of the line that passes through point P and is parallel to a.Then write parametric equations for the line.

1. P(-2, 1), gL= (3, -4) 2. P(3, 7), h = (4, 5)

3. P(2,-4), _ = (1, 3) 4. P(5,-8), h = (9, 2)

Write the equation of each line in parametric form.

5. y = 3x - 8 6. y = -X + 4

7. 3x- 2y=6 8. 5x + 4y= 20

Write an equation in slope-intercept form of the line with the given parametricequations.

9. x=2t+3 10. x=t+5

y = t - 4 y = -3t

11. x=t+4 12. x=7t+3

y = t- 9 y =-6t + 8

Set up a table of values and then graph each line from its parametric form.

13. x=5t+3 14. x=3t-9

y =-2t + 7 y =-2t + 5

y Y10

10:

55

-10 -5 -0 5 x-5 O- 5 x

-5-5


NAME DATE

8-6 Practice WorksheetVectors and Parametric EquationsWrite a vector equation of the line that passes through point P and is parallel to a.Then write parametric equations for the line.

1. P(-2, 1), h = (3, -4) 2. P(3, 7), a = (4, 5)

(x + 2, y- 1} - t(3,-4} (x- 3, y- 7} = _, 5}x =-2 + 35 x = 3 + 4_y = 1 - 4t y = 7 + 55

3. P(2, -4), & = (1, 3) 4. P(5, -8), &= (9, 2)

(x- 2,y + 4_- t(1,3_ {_- 5,y +8}= _, 2_x=2+t x=5+gty =-4 + 3_ y = =8 + 25

Write the equation of each line in parametric form.

5. y=3x-8 6. y=-x+4x=$ x=$y=3t-8 y==$+4

7. 3x- 2y = 6 8. 5x+ 4y = 20x=t x=_

3 5y= _t- 3 f= =_f + 5

Write an equation in slope-intercept form of the line with the given parametricequations.

9. x=2t+3 y-- _ X-- -- lO, x=t+5 _--=3M-t- _5y = t - 4 2 y = -3t

6_+ 74It. x=t+4 y= x- 13 12.x=7t+3 H-- 7 7y = t - 9 y =-6t + 8

Set up a table of values and then graph each line from its parametric form.

13. x=5t+3 14. x=3t-9'y =-2t + 7 y =_-2t + 5

y, x

10

- _,, (-9, 5)_

-5 O- 5 x

-5-5


/

NAME DATE

8-7 Practice WorksheetUsing Parametric Equations to Model MotionFor each exercise,a) write parametric equations to represent the path of the object,b) graph the path, andc) solve.Round your answers to the nearest hundredth.

1. A rock is tossed at an initial velocity of 50 m/s at an angle of 8 ° with the ground.After 0.8 seconds, how far has the rock traveled horizontally and vertically?

2. A toy rocket is launched at an initial velocity of 80 ft/s at an angle of 80 ° with thehorizontal. How long will it take for the rocket to be 10 feet horizontally from itsstarting point. What will its vertical distance be at that point?

3. A bullet is shot at a target 200 feet away. If the bullet is shot at a height of 5 feet,with an initial velocity of 150 ft/s, and at an angle of 10° with the horizontal,when will it reach the target, and what will be the height where it hits?

4. A disk is thrown from a height of 5 meters at an initial velocity of 65 m/s at anangle of 10 ° with the ground. After 0.5 second, how far has the disk traveledhorizontally and vertically?

\

56Glencoe Division .'Graw-Hill

NAME DATE

8-7 Practice WorksheetUsing Parametric Equations to Model MotionFor each exercise,a) write parametric equations to represent the path of the object,b) graph the path, andc) solve.Round your answers to the nearest hundredth.

1. A rock is tossed at an initial velocity of 50 m/s at an angle of 8 ° with the ground.After 0.8 seconds, how far has the rock traveled horizontally and vertically?

a) x = 50t cos 8°; b) yy = 50t sin 8° - 4.9t 2 30

c) x _ 39.61 m; y _ 2.43 m zo1.0-

O 10 20 30 40 50 60 70 80 x

2. A toy rocket is launched at an initial velocity of 80 ft/s at an angle of 80 ° with thehorizontal. How long will it take for the rocket to be 10 feet horizontally from itsstarting point. What will its vertical distance be at that point?

a) x = 80t cos 80°; b) ,ooY:y = 80t sin 80° - 16t 2 80 /_

c) 0.72 s; 48.43 ft _o4O20

0 2040 60 80100x

3. A bullet is shot at a target 200 feet away. If the bullet is shot at a height of 5 feet,with an initial velocity of 150 ft/s, and at an angle of 10 ° with the horizontal,when will it reach the target, and what will be the height where it hits?

a) x = 150t cos 10°; b) 1[y- 150t sin 10° - 16t 2 + 5

c) 1.35 s; 11.00 ft 12 f _6

O 40 80 120 160 200 240 280 320 x

4. A disk is thrown from a height of 5 meters at an initial velocity of 65 m/s at anangle of 10 ° with the ground. After 0.5 second, how far has the disk traveledhorizontally and vertically?

a) x = 65t cos 10°; b) 1:

y = 150t sin 10° - 16t 2 + 5 8c) 32.01 m; 9.42 m 4

2

0 10 20 30 40 50 60 70 x


NAME DATE

9-1 Practice WorksheetPolar Coordinates

Graph the point that has the given polar coordinates.

1. (2.5, 0 °) 2. (3,-135 °) 3, (-1,-30 °)90 ° 90 ° 90 °

15_ ° o 15_ ° o 15_ ° o

270 ° 270 ° 270 °

4. (-2,4) 5. (1,_) 6. (1,-_3_)2_ 2 _ 2_ 2 _ 2_ 2

3 3___ 3 3 3_ 3 3 3__ 32 2 2

Name four different pairs of polar coordinates that represent point A.

7. 90° 8.

15_ ° o 15_°' °

180_ o 180_ o

210° _o 330o 210° _o 330o270 ° 270 °

Graph each polar equation.

9. r=3 10. 0=60 °2qT -_ 90°

__ 150_30°'oO 180_0

7_6 _/ _ _x/ 11_6 210° _ 330°; 4__ 6 _240° , ,, ' 300 °

3 3___ 3 27_0o2 57Glencoe Division, Macmillan�McGraw-Hill

NAME DATE

9-1 Practice WorksheetPolar Coordinates

Graph the point that has the given polar coordinates.

1. (2.5, 0 °) 2. (3,-135 °) 3. (-1,-30 °)90 ° 90° ' 90 °

o o 156°_0 °

150_30 15__0° ° -.__]_)_I_ o

180__) 0° 180_: 0 180_ "0

270 ° 270 ° 270 °

o11 ) o(1 )2___ 2 __ 2_ 2 _ 2__= 2

5_ __ 5_ _r

=0 _ _O

1Tr _- _ 6 6

3 3_ 3 3 3_ 3 3 3_rr 32 2 2

Answers will vary. Sample answers are given.Name four different pmrs of polar coordinates that represent point A.

7. 90° (2.5, 135°) 8. ,0o (3, 240 °)__o_ 12.5, -225 °) _=-/_/_; (3, -120 °)

(-2.5, -45 °) (-3, -60 °)150_30 ° 150_30 °

,eo_" dO(-2.5, 315 °) 180_'_ 0o(--3, 300 °)210° _o 3300 210° _o 330'

270 ° 270 °


9. r=3 10. 0=60 °__ 90 o

qT___: O 180_ "_0°7 1"_ 210 ° _ 330 °

"_4_ _/'--_V _5,_ _240o, i , 300 °3 3__ 3 , 270 °

2 T57Glencoe Division, Macmillan�McGraw-Hill

NAME DATE

9-2 Practice WorksheetGraphs of Polar EquationsGraph eachpolar equation. Identify the classical curve it represents.

1. r=2+2cosO 2. r=3sin30

2_ 2 _ 2_ 2

__0 _-7 _ 7_ /

4_ / I _ 5_3 3_ 3 3 3___ 3

2 2

3. r=2+3cosO 4. r=2+2sint_

__ IT2

3: __o

5_ _

_T7_ _r) _T --

6 _ 6

4'n / I _ 5T_3 3___ 3 3 3"n 3

2 2

5. r = 0 6. r2 = 5 cos 20

2__ 2 _ 2"rr 2

3 ,3 . 3 3

_r 6

4_ '5'rr 4_ _ I _ 5_3 3,'n 3 3 3,'n 3

2 2


NAME DATE

9-2 Practice Worksheet LGraphs of Polar EquationsGraph each polar equation. Identify the classical curve it represents.

1. r=.2+2cosO 2. r=3sin30

carclioJd rose2-_ 2 _ 2

5_ w_ _ _T

_7__ 1_O _7_i/ _ O

" 4w / I _ 5_3 3_ 3 3 _ 3

-2 2

3. r=2+3cosO 4. r=2+2sinO

iima_on cardioid,rr

2"n 2 _ 2"K 2 ._

5_ 5"n X

0 _" ' _0

6 4w/ _ \5_ 6 "'n" / I _ 5_

p

3 3___ 3 3 3-n 32 2

5. r = 0 6. r2 = 5 cos 20

spiral of Archimedes memniscate

_0 _ _0

. 7__1_ 7_.-_l,rr

3 3'1T 3 3 3W 32 2

T 58 •. Glencoe Division, Macmillan/McGraw-Hill

NAME DATE

9-3 Practice WorksheetPolar and Rectangular CoordinatesFind the polar coordinates of each point with the given rectangular coordinates.

(1 1)1. (0, 4) 2. 2' 2 2 ' 2

4. (4, O) 5. (-1,- %/3) 6. (2, 2)

Find the rectangular coordinates of each point with the given polar coordinates.

7. (6, 120 °) 8. (-4, 45 °) 9. (3, 300 °)

10.(4,6) . 11. (O,V) 12. (3,,-_)

Write each rectangular equation in polar form.

13. x2+y2=9 14. y=3

15; X2 _y2 = 1 16. X2 + y2_ 2y = 0

Write each polar equation in rectangular form.

17, r=4 18. 0-3

19. rcosO=5 20. r=-3secO


m

NAME DATE _

9-3 Practice WorksheetPolar and Rectangular Coordinates

Find the polar coordinates of each point with the given rectangular coordinates.

( ) ()1 3. -1. (O,4) 2. g,-V_2 V_2' 21

4. (4, O) 5. (-1,- _v/-3) 6. (2, 2)

Find the rectangular coordinates of each point with the given polar coordinates.

7. (6, 120°) 8. (-4, 45°) 9. (3, 300 °)

(o(2_/3, 2) (0, O) ( 3_v_ 3V_)2 _ 2

Write each rectangular equation in polar form.

13. X 2 + y2 = 9 14. y = 3r = +_3 r sin 0 -- 3 or r = 3 csc 0

15. x 2 -y2 = 1 16. x 2 + y2 _ 2y = 0

r 2 = 1 - sec 20 _ = 2 sin 0cos 20

Write each polar equation in rectangular form.

17. r=4 18. O- 7r3

x2+y2- 16 y -_x

19. rcosO=5 20. r,=-3secOx= 5 _:=-3


l _ DATE9-4 IPractice WorksheetPolar Form of a Linear Function


1. l=rcos(O-30 °) 2. 3=rcos(O+60 °)I

- i r_- - -_ I i_ _ __ I

I i- -_ -I

_ I ol 1_ * o x_ I i- _ _

I

I I.--_. -- --I

I

1.53. 2.5 = r cos (O + 30 °) 4. r- cos (0+ 90°)

I i_ m --

I_" 0 ,x O, _"

I- I - _- __-

Write each equation in polar form. Round d_to the nearest degree.

5. y+x=O 6. y=2x-1

7. 2 =y-2x 8. x = 11

, Write each equation in rectangular form.

cos 0

(v) (4)11. rcos O- -2=0 12. rcos O- -3=0

60, Glencoe Division, Macmillan/McGraw-Hill

NAME DATE

9-4 Pr_,:tice WorksheetPolar Form of a Linear FunctionGraph each polar equation.

1. l=rcos(O-30 °) 2. 3=rcos(O+60 °)

__ ....... -r- -- -,4 _- ......._--mE.....

i <_ -- --

....... o l___L__x

3. 2.5 = r cos (O + 30 °) 4. r= 15cos (0 + 90°;

.... )_7 ............

*" _ "*" -i T

Write each equation in polar form. Round ¢ to the nearest degree.

5. y+x=O 6. y=2x-1

0 = r cos (0- 45°) _/5 = Fcos (0- 333 °)5

7. 2 =y-2x 8. x = 11

2x/g- r cos (0 - 153°) 11 = r cos 05

Write each equation in rectangular form.

( _ 29. rcos _'+-_ =1 10. r- cosOr---

X+|'- _ __ =/_)y-2=O x 2 0

( (11. rcos 0--_I_ -2=0 12. reos O- _r4 _-b -3=0/

_) - _) -- i_ :) - =(-\ x+_¢ ;!)y 4 = O (% x+ ¢; y 6 0


NAME DATE


•Simplifying Complex NumbersSimplify.

1. i65 + 129 2. i243(1 - 3i)

3. (3 + 2i) + (4 + 5i) 4. (-6- 2i)- (-8- 3i)

5. (2- 3i)(5 + i) 6. (V_2 + 2i)(V2-2i)

7. (3 + 4i) 2 8. (3 + X/_)(3 + X/_

9. 4- 7i 10. 4 + 3i- 3i 1 - 2i

2 12. (6 -- i)211. 6+5i

7

4 14. N/__ 3113. _v/-_+ 2i

Find values for x and y that make each sentence true.

15. 3x - 7yi = 21 - 56i 16. 7x + 8yi = 49 - 64i


NAME DATE

9-5 Practice WorksheetSimplifying Complex NumbersSimplify.

1. i65 +/29 2. i243(1- 3i)

2i -3 - i

3. (3 + 21) + (4 + 5i) 4. (-6 - 2i) - (-8- 3i)7+7i 2+i

5. (2 - 3i)(5 + i) 6. (V_ + 21)(V2 - 2i)13- 13i 6

7. (3 + 4i) 2 8. (3 + _v/5X/_)_i+ N/_)-7 + 24i 4 +

9. 4-7i 10. 4+3i- 3i 1 - 2i

7- 4 2 _1_t_11 -+ 3i

11. 2 12. (6 - i)26+5i

12 10 i 35 - 12i61 61

Find values for x and y that make each sentence true.

15. 3x - ?yi = 21 - 56i 16. ?x + 8yi = 49 - 64i

x = 7, y = 8 x = 7, y = -8


NAME DATE

9-6 Practice WorksheetPolar Form of Complex NumbersExpress each complex number in polar form.

1. 3 + 3X/-3i 2. 7i

3. -X/3 + i 4. 4

5. 1-i 6.-1+i

7. X/2- X/-2i 8.-X/-2 + X/-2i

9. 2X/3 + 2i 10. 1.5 + 1.5i

11. 5 - 5X/3i 12. 1 x/-3 i2 2

Express each complex number in rectangular form.

13. 2(cos _+i sin _) 14. _/-2(cos _+i sin _)

15. 1.5(cos _ + isin _) 16. 4(COS _ + isin _)

17.4(cos_+isin_) 18. 3(cos_+isin.J_ -_)


NAME DATE


Polar Form of Complex NumbersExpress each complex number in polar form.

1. 3 + 3%/3i 2. 7i

6(cos 60° + i sin 60°) 7(cos 90° + i sin 90°)

3.-k/3 + i 4. 4

2(cos 150° + i sin 150°) 4(cos 0° + i sin 0°)

• 6.-1 i

5. _(cos 315° +/sin 315 °) _2(cos 135° +/sin 135 °)

7. X/2- X/2i 8. -X/2 + X/2i

2(cos 315 ° + i sin 315 °) 2(cos 135° + i sin 135°)

9. 2%/3 + 2i 10. 1.5j_L5i4(cos 30° + i sin 30°) _/4.5(cos 45° + i sin 45°)

1 _/3.11. 5 - 5X/3i 12. 2 2 t

lO(cos 300° + i sin 300 °) cos 240 ° + i sin 240 °

Express each complex number in rectangular form.

13. 2(cos _+i sin _) 14. V2(cos _+i sin _)

,o.1.5(cossin ,°.,(cos-,-,sin• 3_ F 3 -2_/3 + 2i

4 _i

17. 4 cos + i sin 18• 3 cos T + i sin

-2X/3 - 2i 3%/2 F 3_ i2 2

T62_ Glencoe Division, Macmillan/McGraw-Hill

I

NAblE DATE

9-7 PracticeWorksheet

Products and Quotients of Complex Numbers in Polar FormFind each product or quotient.

1. (3 - 2i)(4 + i) 2. (4 - i) - (2 + 2i)

3. (4- N/-3i) + (2 + N/3i) 4. (N/3 - 2i)(4N/3 + i)

Find each product or quotient. Then write the result in rectangular form.

5. 3 cos_-+isin .3 cos +isin

4)6. 4.5 cos_+isin_- -3 cos_-+isin

v) 4)7. 7 cos + i sin + 14 cos _- + i sin

8. 3(cos Y_ + i sin _-_)2 • (cos -_ + i sin -_)

9. 18(cos _-_ + i sin -_-_) • 6(cos _ + i sin _)

63


NAME DATE

9-7 Practice WorksheetProducts and Quotients of Complex Numbers in Polar FormFind each product or quotient.

1. (3 - 2i)(4 + i) 2. (4 - i) - (2 + 2i)

3 5g114- 5g 4 ,_

3. (4- N/3i) + (2 + N/-3i) 4. (N/-3- 2i)(4N/-3 + i)

7 '7

Find each product or quotient. Then write the result in rectangular form.

°_(cos__in_/._Icos__io_/9(cos 2rr + i _in 2_r);

6. 4.5 cos_ +isin +3 cos_ +isin

_o1.5 cos 7_-+ g_in .o _.45 + O°3!;i

12 12

7. 7 cos +isin +14 cos_ +isin

g cos 1_--

cos _:)(cos _:)8. 3 _+isin 2. _+isin

6 COS 37-.- I- _Sin .3.7,-.-/; =5°80- 1.55]"_2 12 ]

9. 18(cos _ + i sin _) • 6(cos -_ + i sin _)


NAME DATE

9-8 Practice WorksheetPowers and Roots of Complex NumbersFind each power. Express the result in rectangular form.

1. (1- i)5 2. (2-2iV3) 5

3. (-2 -2iN/3) 3 4. (1 + i)8

Find each root. Express the result in the form a + bi, with a and b to the nearesthundredth.

1 1

7. (0 - 27i) 3 8. (8 - 8i) 3

1

9. _/-_-3i 10. (0- i)-3

11. _ 12. _/_-2iN/3


NAME DATE

9-8 Practice WorksheetPowers and Roots of Complex NumbersFind each power. Express the result in rectangular form.

1. (1 - i)5 2. (2 - 2iVY) 5=4 + 4i 542 + 512i.V3

3. (-2- 2iVy-)3 4. (1+ i)864 16

5. (V_(I+ i))6 6. (v_2 -_/22 i) 5

-64i _-22+ -_-i:_/_

Find each root. Express the result in the form a + bi, with a and b to the nearesthundredth.

1 17. (0 - 27i) _ 8. (8 - 8i) 3

3i 2,17 - 0.58i

1

9. _ 10. (0- i) -3

1.76 + 2.43/ -i

11. _ 12. _/_-2iN/-3

1.08 + 0.72i 0.71 + 1.22i


NAME DATE

10-1 Practice WorksheetThe Circle

Write the standard form of each equation. Then graph the equation

1. x 2 + y2 _ 2y - 15 = 0 2. x 2 + 4x + y2 = 0

.... Y_............. _ ......

3. X2 + y2 _ 8X -- 6y + 21 = 0 4. 4X 2 + 422 -- 16X -- 8y -- 5 = 0Y Y ___

.............. I

.... i I

-- ---- i _._ .......... ._

"*" -- _ _ - _ _---- _ x

Write the standard form of the equation of the circle that passes through the pointswith the given coordinates. Then identify the center and the radius of the circle.

5. (-3,-2), (-2,-3), (-4,-3) 6. (0,-1), (2,-3), (4,-1)

7. (1,-1), (5, 3), (-3, 3) 8. (-1, 0), (2, 3), (-1, 6)

9. Write the equation of the circle that passes through (-1, 3) and has its center at

(2,4).


NAME DATE

10-1 Practice WorksheetThe Circle

Write the standard form of each equation. Then graph the equation

1. X 2 + y2 _ 2y -- 15 = 0 2. X_ + 4X + y2 = 0

___1____: .Y_ = ............. y_........

.... _:__;........- _ _ _ _ :---(--_ ....... _

7' ....

x 2 + (y- 1)2 = 16 (x + 21t: + y2 = 4

3. x2 + y2 _ 8x - 6y + 21 = 0 4. 4x 2 + 4y 2 - 16x - 8y - 5 = 0i r y

......... Z--_ .................

...................... _,,_ =_....

........ t...._" ---- 3 x _... t

Z Z--ZZ-ZZZZZZ Z ...... __........ _-I

(x- 4):!+ (y- 3)2 = 4 (x + 21_ + (y- I) 2 = 2_Write the standard form of the equation of the circle that passes through the pointswith the given coordinates. Then identify the center and the radius of the circle.

5. (-3,-2), (-2,-3), (-4,-3) 6. (0,-1), (2,-3), (4,-1)

(x + 3)2 + (y + 3)2 = 1; (x- 2],2 + (y + 1)2 = 4;(-3,-3); 1 (2,-1); 2

7. (1,-1), (5, 3), (-3, 3) 8. (-1, 0), (2, 3), (-1, 6)

(x- 1)2 + (y- 3)2 = 16; (x + 1)2 + (y- 3)2 = 9;(1, 3); 4 (-1, 3); 3

9. Write the equation of the circle that passes through (-1, 3) and has its center at(2, 4).

(x- 2)::+ (y- 4)2 = 10


NAME DATE

10-2 Practice Worksheet "The Parabola

For each equation,a. write the standard form,b. find the coordinates of the focus and vertex, and the equation of the directrix and

axis of symmetry, andc. graph the equation.

1. X 2 -- 2X + 1 = 8y -- 16 2. y2 + 6y + 9 = 16 -- 16X

v y

Write the equation of the parabola that meets each set of conditions. Then graph theequation.

3. The parabola has its focus at (1, 3) 4., The focus is at (2, 1), and the equationand the vertex at (1, 2). of the directrix is x = -2.

..... j ...... x

4- .............. ...... _r .......


NAME DATE

10-2 Practice WorksheetThe Parabola

For each equation,

a. write the standard form,b. find the coordinates of the focus and vertex, and the equation of the directrix and

axis of symmetry, andc. graph the equation.

1. X 2 -- 2X + 1= 8y -- 16 2. y2 + 6y + 9 = 16 -- 16X

_ _ _____ _ ____-_____r - r - --

ii '---- -- ---r-- ............

- ...... xz=_-z= .........

........ l..... -_-

(x 1)2 = 8(y- 2); (y + 3)2 = -16(x- 1);focus: (1, 4); vertex: (1, 2); focus: (-3,-3); vertex:(1,-3);directrix: y- 0; directrix: x = 5;axis of symmetry: x- I axis of symmetry: y- 3

Write the equation of the parabola that meets each set of conditions. Then graph theequation.

3. The parabola has its focus at (1, 3) 4. The focus is at (2, 1), and the equationand the vertex at (1, 2). of the directrix is x = -2.

_ __.... / ......... 4_..... __ ,..-"_...... ¢.....=_---

-_ ._> ..... $ --Z'-ill--..... 4- _.......

V2-_ _ .ilt'--=-_ ---'- ....

(x - 1)2 = 4(y - 2) (y - 1)2 = 8x


NAME DATE

10-3 Practice WorksheetThe EllipseFor each equation, find the coordinates of the center, foci, and vertices of theellipse. Then graph the equation.

1. 4x 2 + y2 _ 32x + 4y + 64 = 0 2. _x 2 71-93,2 - 8x - 36y + 4 = 0

= = .......

............ _- -- ZZ -ZZ -Z- ZZ-Z- -ZZ

Write the equation of the ellipse that meets each set of conditions.

3. The foci are at (-2, 1) and (-2, -7), and a = 5.

4. The length of the semi-major axis is 6 units, and the foci are at (0,2) and (8, 2).

45. The center is at (1, 3), one vertex is at (1, 8), and c _a 5"

State whether the graph of each equation is a circle, parabola, or ellipse. Justifyyour answer.

6. X2 4- 4y 2-- 2X -- 163I + 1 = O 7. X2 + 4y -- 16 = 0

8. x2 + y2 + 6x + 2y + 7 = 0 9. 4X 2 4- 422 -- 20x - 24 = 0


t. or,te,"(1,2cen er: (4,-2}; 7foci. (4, -2_ _/::;); feci: (1 +__V 5, 2 ;vertices: (4, 0), (4,-4}, {3,-2), vertices: (-2, 2), (1, 4),(5,-2) (4, 2), (_, 0)

Write the equation of the ellipse that meets each set of conditions.

3. The foci are at (-2, 1) and (-2, -7), and a = 5.

(Y + 3)2 -t- (x + 2) 2 _25 9

4. The length of the semi-major axis is 6 units, and the loci are at (0,2) and (8, 2).

{x- 4)2 + {y- 2)2 _36 20

45. The center is at (1, 3), one vertex is at (1, 8), and c _a 5"

(Y- 3)2 '-I- {'_'- 1)2 -- "_25 9

State whether the graph of each equation is a circle, parabola, or ellipse. Justifyyour answer.

6. X 2 + 4y 2 -- 2X -- 16y + 1 = 0 7. X 2 -t- 4y - 16 = 0

ellipse; _ + _ = 1 parabola; x2 = 4(-1)(y- 4)_6 4

8. x2+y2+6x+2y+?=0 9. _+4y 9-20x-24=0

circle; (x + 3)2 + (y + 1)2 = 3 circle; x- + y2 = _


NAME DATE

10-4 Practice WorksheetThe hyperbolaFind the coordinates of the center, the foci, and the vertices and the equations of theasymptotes of the graph of each equation. Then graph the equation.

1. 4X 2 --y2 _ 8x + 20 = 0 2. X 2 -- 4y 2 - 4x + 24y - 36 = 0

......... _ ..... Y__ .............. --,-t .........

"" _".... o_...... -l--"

State whether the graph of each equation is a circle, ellipse, parabola, or hyperbola.

3. 2X 2 + 3y 2 -- 6 = 0 4. 25X2 -- 9y 2 + 100X -- 54y -- 206 = 0

Graph each equation.

5. (x41)2 - (y92)2 = 1 6. xy =-1

_ - ........_---- __

.... _- -- .... -I-- ---_- *---_--_....n--;

.... I .........

Write the equation of the hyperbola that meets each set of conditions.

7. The length of the conjugate axis is 6 units, and the vertices are at (-3, 0) and (5, 0).

8. The vertices are at (2, 1) and (2, 7), and focus is at (2, 8).


NAME DATE

10-4 Practice WorksheetThe hyperbolaFind the coordinates of the center, the foci, and the vertices and the equations of theasymptotes of the graph of each equation. Then graph the equation.

1. 4X 2 --y2 _ 8x + 20 = 0 2. X 2 -- 4y 2 - 4x + 24y - 36 = 0

____ __-y _ -_ ........-_- - -_ 7" .... _- ..._

• __=______:,_..... _.- ............ _____ O_ _ _

..... / --_v,, --- .......

_- _ _ _ __......_ - __ ._.......

center: (1, 0); Ven_er: (2,_;

vertices: {_, _+4); vertices: {0_3)_(4_3);

asymptotes: ] = ___2(x- _]) asy_p_o_ee: _ - 3= _ ½(x- 2)

State whether the graph of each equation is a circle, ellipse, parabola, or hyperbola.

3. 2x 2 + 3y 2 - 6 = 0 4. 25x 2 - 9y 2 + 100x - 54y - 206 = 0

ellipse hyperbo_

Graph each equation.

5. (x - 1)2 (Y + 2)2 = 1 6. xy = -14 9

-E::_ -_ .... -7<- -- -" ..... _ ,: =v--

-_:-- _.... .- 7-- :__=_____::--::::---x---c....... .......

1_ t i_ _____:__,:__-_ ...... _...........-4---,- _- /= I--- ...... [--...........

...... i---_ ...... _............... r_ ...........1" ' I _ i

Write the equation of the hyperbola that meets each set of conditions.

7. The length of the conjugate axis is 6 units, and the vertices are at (-3, 0) and (5, 0).

(x- 1)2 y_ _

8. The vertices are at (2, 1) and (2, 7), and focus is at (2, 8).

(y- 4)2 _ (_- 2)_ _7


NAME DATE

10-5 Practice WorksheetConic Sections

Determine the eccentricity of the conic section represented by each equation.

l. 4X 2 zr 9y 2 + 54y + 45 = 0 2. (y + 1)2 = 3(X + 4)

Identify the conic section represented by each equation. Write the equation instandard form and graph the equation.

3. X 2 -- 4y = -4 4. X 2 + y2 _ 6x - 6y = 18, _ ...... _ J_ ___

i

5. 4x 2-y2-8x+6y=9 6. 9x 2+5y 2+ 18x=36_ ...... _]'_ .... __ .... _ ......

Write the equation of the conic that meets each set of conditions.

7. a hyperbola with a vertical transverse axis 10 units long, center at (2, 0)i ande=2

8. an ellipse with center at (2, -2), a vertical semi-major axis 4 units long, ande = 0.5


NAME DATE

10-5 Practice WorksheetConic SectionsDetermine the eccentricity of the conic section represented by each equation.

1. 4x 2+9y 2+54y+45=0 2. (y+l) 2=3(x+4)

3

Identify the conic section represented by each equation. Write the equation instandard form and graph the equation.

3. x 2 - 4y = -4 4. X 2 Jr-y2 _ 6x - 6y = 18

.....4-

parabola; x2 - 4(y- I) circne; (x- 31,_+ (y- 3)2 -3@

5. 4X 2 --y2 _ 8x + 6y = 9 6. 9x 2 + 522 9- lSx = 36

......._-I--- _......__L.......13 _:-3]__ _ :..... _ ........- -F_-2 ..... ,L____.......

! ............ ...... _ _- ...._ -----._..... ___-__ ......

...... __( _12 v -=--1---........ --JL- - .9---r

hyperbola;(x-I)2 (Y-3)2 1 ellipse{x+1)2 y2- ; _ -11 4 5 9

Write the equation of the conic that meets each set of conditions.

7. a hyperbola with a vertical transverse axis 10 units long, center at (2, 0), ande=2

y2 (x- 2)2 _ 125 75

8. an ellipse with center at (2, -2), a vertical semi-major axis 4 units long, ande = 0.5

(Y+2)2 -t- (x-2) 2 _ 116 12


NAME DATE

10-6 Practice WorksheetTransformations of Conics

Identify the graph of each equation. Write the equation of the translated graph ingeneral form. Then draw the graph.

1. x 2 + y2 = 9 for T(_2,1) 2. 2x2 - 4x + 3 - y = 0 for T(1'

............. ....Z_ZZ- Z_-ZZ-ZZ--

_ ___ __ _ -_- --- ___*- .... _ ....... _ "--I-}-- --a .......

II

............... __ ._,+,,__p__ ..........

I

Suppose the graph of each equation is rotated about the origin for the given angle.Find an equation of the rotated graph.

3. x 2 --y2 = 16, 0 = 45 ° 4. 21x 2 - lOV'-3xy + 31y 2 = 144, 0 = 30 °

Identify the graph of each equation. Then find 0 to the nearest degree.

5. 2x 2 - 4xy + y2 + 3 = 0

6. X2 + 3xy + 322 = 3

7. 3x 2 + 4V3xy -y2 = 15

7OGlencoe Division, Macmillan/McGraw-Hill

NAME DATE

10-6 Practice WorksheetTransfOrmations of Conics

Identify the graph of each equation. Write the equation of the translated graph ingeneral form. Then draw the graph.

1. x 2 + y2 = 9 for T(_2,1) 2. 2x2 - 4x + 3 - y = 0 for T(1' -1)I I ly_.............. ..... ....

i -/ -- ....

circle; (x + 2_2 + {y- 1)2 = 9 p_rqbola; {x- 2)2 =

Suppose the graph of each equation is rotated about the origin for the given angle.Find an equation of the rotated graph.

3. X2 --y2 = 16, 0 = 45 ° 4. 21x 2 - lOV'3xy +31y 2 = 144, 0 = 30 °

y2xy=8 x2 _ -19 4

Identify the graph of each equation. Then find _ to the nearest degree.

5. 2x 2 - 4xy + y2 + 3 = 0

ellipse; 38°

6. X 2 + 3xy + 3y 2 = 3

hyperbola; 45°

7. 3x 2 + 4V/-3xy -y2 = 15

hyperbola; 30°


NAME DATE

10-" Practice WorksheetSystems of Second-Degree Equations and Inequalities.Graph each system of equations. Then solve. Round the coordinates of thesolutions to the nearest tenth.

1. 2x - y = 8 2. X 2 --y2 = 4x2 + y2 = 9 y = 1...... Y' ..... I

2- " o__- - ........

Graph each system of h_equafities.

5. 3 >- (y - 1)2 + 2x 6. (x - 1) 2 + (y - 1)2 < 4y -> -3x + 1 8y 2 + x2 _ 16

..... x__ _ __ x__


NAME DATE-

]0-8 Practice WorksheetTangents and Normals to the Conic SectionsFind the equations of the tangent and the normal to the graphs of each equation atthe point with the given coordinates. Write the equations in standard form.

1. x 2 - 5 = 2 -y2, (1, V6) 2. 2x 2 + 4x + 2y = 10, (1, 3)

3. 2(x - 1)2 + y2 = 10, (2, 2_/2) 4. 2x 2 + 2y 2 -- 10, (1,2)

5. 9(x + 1)2 + 16(y + 3) 2 = 144, (-1, 0) 6. y2 _ x 2 _ 6x = 25, (0, 5)

Find the length of the tangent segment from each point to the graph of the givencircle.

7. (6, 2), x 2 + y2 = 13 8. (5, 2), 3(x + 1)2 + 3(y + 1)2 = 27

9. Find the equations of the horizontal lines tangent to the graph ofx 2 + y2 = 81.

10. Find the equations of the normals if the lines tangent to the graph of9x 2 + 422 = 36 are vertical.


NAME DATE

10-8 Practice WorksheetTangents and Normals to the Conic SectionsFind the equations of the tangent and the normal to the graphs of each equation atthe point with the given coordinates. Write the equations m standard form.

1. x 2 - 5 =2 -y2, (1, V6) 2. 2x 2 + 4x + 2y = 10, (1, 3)

o; �o._- 4y+ 7 =0

3. _2-xl_ + y2= 10, (__2V2) 4. 2x2+ 2y 2= 10, (1,2)2y 6V2 = 0; x + 2y- 5 = 0;_//-2x- y = 0 2x- y = 0

5. 9(x + 1)2 + 16(y + 3) 2 = i44, (-1, 0) 6. y2 _ x 2 _ 6x = 25, (0, 5)

y = O;x = -1 3x- 5y + 25 = O;5x + 3y- 15 = 0

Find the length of the tangent segment from each point to the graph of the givencircle.

7. (6_ 2_rX 2 �y2= 13 8. (5, 2), 3(x + 1)2 + 3(y + 1) 2 = 273V3 6

9. Find the equations of the horizontal lines tangent to the graph ofx 2 + y2 = 81.

y=_9

10. Find the equations of the normals if the lines tangent to the graph of9x 2 + 422 = 36 are vertical.

y=0


NAME DATE

11-1 Practice WorksheetRational ExponentsEvaluate.

1. _/_ 2.2-5"2 7 3.

Express using rational exponents.

7. N/_x5y6 8. _ 9. _ylO

10. _y5 11. _3 12. _r3-6aSb5

Express using radicals.1 3 1 3 5

13. 64_ 14. y_ 15. 2_a_b_

2 3 1 2 1 3

16. xsy5 17. (s2t)3v3 18. (x6y3)2z2

Simpfify.

19. (X-2) 3 • (X3) -2 20. (3y3)(3y) 3 ' 21. X 7 " X 5 " X -7 " X -5


NAME . DATE11-1 Practme WorksheetRational ExponentsEvaluate.

z. _-_6 3. 2-5.27 8.6 4 8

16,384 t 728 !12

Express using rational exponents.

7. N_x5y 6 8. _ 9. _v/144x6y 105 7

xy3 12x3#

10. _/27xl°y 5 11. %/1024a 3 12. _/3-6aSb 51 3 1 5

27gx2y 32a_- 6_a2b_-

Express using radicals, ii1 3 1 3 5

13. 64_ 14. yY 15. 2Ya_bY

_/_ or 4 _ k/2a31_

2 3 1 2 1 3

16. xsy5 17. (s2t)_v_ 18. (x6y3)yzY

Simplify.

19. (x-2) 3. (X3) -2 20. (3y3)(.3y) 3 21. X 7" X 5" X -7" X -5 i

X-12 81y° 1 :,i,

i[i'


NAME DATE


Exponential FunctionsUse a calculator to evaluate each expression to the nearest ten thousandth.

1. 3V_2 2. 4_2 3. 5_6

6. y = -2 x + 1 7. y = 2-x -1

-- -5 - _ --7 ..... ................... _ .... ___........ __ .......... ..--__

........ _".... 5 ...... "_

Graph each inequality.

8. y > 2x 9. y-> (0.5)x...... Y_ _ .......... )"......: _

...... :" ....... .- F--I-T P-.....


NAME DATE

11-2 Practice WorksheetExponential FunctionsUse a calculator to evaluate each expression to the nearest ten thousandth.

1. 3_2 2. 4 v _2 3. 5 _6


i

NAME DATE

11-3 Practice WorksheetThe Number e

Use a calculator to evaluate each expression to the nearest ten thousandth.

1. e2"3 2. e4"6 3. _e

Use a graphing calculator to graph each equation. Sketch the graph below. Thendetermine the interval in which each function is increasing or decreasing.

7. y = 2ex + : 8. y =-3e x

..... ................ _x ........... _ ....

Given the original principal, the annual interest rate, and the amount of time foreach investment, and the type of compounded interest, find the amount at the endof the investment.

9. P = $1250, r = 8.5%, t = 3 years, quarterly

10. P = $2575, r = 6.25%, t = 5 years 3 months, continuously

75• Glencoe Division, Macmillan/McGraw-Hill

NAME DATE

11-3 Practice WorksheetThe Number e

Use a calculator to evaluate each expression to the nearest ten thousandth.

1. e2"3 2. e4.6 3. _e

9.9742 99.4843 1.6487

4. 2_e 4 5. 3N/_e3 6. e-2

7.5873 13o4451 0.1353

Use a graphing calculator to graph each equation. Sketch the graph below. Thendetermine the interval in which each function is increasing or decreasing.

7. y = 2ex + 1 8. y = --3ex

..... y__ _ y I

............... --_........_____ ..................... _ -_--_._ .... __ .............

..........___ .... 7_.......... __2 7--

....._.___ ___ _;__ __,___

increasing for all x decreasing all x

Given the original principal, the annual interest rate, and the amount of time foreach investment, and the type of compounded interest, find the amount at the endof the investmenL

9. P = $1250, r = 8.5%, t = 3 years, quarterly

$1608.77

10. P = $2575, r = 6.25%, t = 5 years 3 months, continuously

$3575.03


NAME DATE

11-4 Practice WorksheetLogarithmic FunctionsWrite each equation in logarithmic form.

1. 25 = 32 2. 5 -3 -- 1 3. 6 -3 -- 1125 216

Write each equation in exponential form.

1 _ 24. log 3 27 = 3 5. log 4 16 = 2 6. loglo loo

Evaluate each expression.

7. log7 73 8. loglo 0.001 9. 31°g36

10. log b b -4 11. log a a 12:341°g3 4

Solve each equation.

13. logx 64 = 3 14. log4 0.25 = x

15. log 4 (2x - 1) = log 4 !6 16. loglo _ = x

Graph each equation or inequafity.

17. y = log 2x 18. y < loglo (x - 1)

____ Y__ Y


NAME DATE

11-4 Practice WorksheetLogarithmic FunctionsWrite each equation in logarithmic form.1. 25 = 32 2. 5 -3 -- 1 3. 6-3 -- 1

125 216

- 3Rog232 = 5 _o_s 1 = -3 _°96 216"_25

Write each equation in exponential form.1 --24. log3 27 = 3 5. log4 16 = 2 6. logzo loo

33 = 27 42 = 16 _0_2 = 1100


7. log 7 73 8. lOglo 0.001 9. 31°g3 6

3 -3 6

10. log b b -4 U. log a a 12. 341°g3 4-4 1 256


13. logx 64 = 3 14. log4 0.25 = x4 -1

15. log4 (2x - 1) = log4 16 16. loglo _ =.x17 12 2

Graph each equation or inequafity.

17. y = log2x 18. y < loglo (x -1)

:_ ...... _ I__N_- !1__ _

-1- ......... _-_-• _ ......... ==._-__:.__._ _ _ __--,- ....... _- L .---___ ,-t- V-_ "

.... _ ..... ____.__

__ .... .._ _---__-7" -- _ _


NAME" DATE

11-5 Practice WorksheetCommon LogarithmsUse a calculator to find the common logarithm of each number to the nearest tenthousandth.

1. 726.5 2. 0.6351 3. 0.0026

4. 0.852 5. 16,256 6. 3.2 x 104

Use a calculator to find the antilogarithm of each number to the nearest hundredth.

17. 0.6259 8. 2.7356 9. -0.0251

10. -1.2619 11. 4.3251 12. 2.6359-3

Evaluate each expression by using logarithms. Check your work with a calculator.

13. 261 x 32 x 0.32 14. 181.72x 7.014.62

15. 2.43 × (8.9) 4 16. _ x 4.180

17. _/(2.69)(420) 18. 16.21 +


NAME DATE

11-5 Practice WorksheetCommon LogarithmsUse a calculator to find the common logarithm of each number to the nearest tenthousandth.

1. 726.5 2. 0.6351 3. 0.0026

2.8612 -0.197'2 =2.5850

4. 0.852 5. 16,256 6. 3.2 × 104-0.0696 4°2 __0 4.5051

Use a calculator to find the antilogarithm of each number to the nearest hundredth.

7. 0.6259 8. 2.7356 9.-0.0251423 544°00 0o94

10. -1.2619 11. 4.3251 12. 2.6359-3

0.05 21,139.76 0.43

Evaluate each expression by using logarithms. Check your work with a calculator.

13. 261 × 32 × 0.32 14. 181.72× 7.014.62

2672.64 275.73

15. 2.43 × (8.9) 4 16. _ × 4.180

15,246.36 3°35

17. "C/(2.69)(420) 18. 16.21-10.42 11.26


NAME DATE

.1i-6 Practice WorksheetExponential and Logarithmic EquationsSolve each equation or inequafity by using logarithms.

1. 8x=lO 2. 12x=18

3. 2.4 x -< 20 4. 1.8x-'5 = 19.8

5. 42x + 1 = 15.2 6. 35x = 85

7. x < log 2 15 8. x -> log 3 12.3

2

9. x 3 > 25.3 10. X 0"4 = 18.9

11. 32x - 2 _-- 2x 12. 41 - 2x = 32x


NAME DATE

11-6 Practice WorksheetExponential and Logarithmic EquationsSolve each equation or inequafity by using logarithms.

1. 8x= 10 2. 12x= 18

1.1073 1,1632

3. 2.4 x --20 4. 1.8x-5 = 19.8

x _< 3.4219 10.0795

5. 42x + 1 = 15.2 6. 35x = 85

0.4815 0.8088

7. x < log 2 15 8. x -->log 3 12.3x < 3.9069 x >- 2.2843

2

9. x T > 25.3 10. x °.4 = 18.9

x -> 127.2567 1552.9394

11. 32x-2=2x 12. 41-2x=32x

1.4608 0.2789


NAME DATE

i 1:7 Practice WorksheetNatural LogarithmsUse a Calculator to find each value to the nearest ten thousandth.

1. in 43.2 2. In 0.0217 3. In 985

4. In 0.0076 5. In 10 6. In 10.6

7. antiln (-0.256) 8. antiln 4.62 9. antiln (-1.62)


10. 1500 = 6e°.°43t 11. 1249 = 175e-°'°4t

12. In 6.7 = In (e 0"21t) 13. In 724.6 = In (e 6'3t)

14. Banking Ms. Cubbatz invested a sum of money in a certificate of deposit thatpays 8% interest compounded continuously. Recall that the formula for theamount in an account earning interest compounded continuously is A = Pe rt. IfMs. Cubbatz made the investment on January 1, 1989 and the account is worth$12,000 On January 1, 1993, what was the original amount in the account?

79GiencoeDivision,Macmillan/McGraw-Hill

NAME DATE

11-7 Practice WorksheetNatural LogarithmsUse a calculator to find each value to the nearest ten thousandth.

1. In 43.2 2. In 0.0217 3. In 985

3.7658 =3,8304 6,8926

4. In 0.0076 5. In 10 6. in 10.6

-4.8796 2°3026 0.5108

7. antiln (-0.256) 8. antiln 4.62 9. antiln (-1.62)0.7741 10I]o4940 0.1979


10. 1500 = 6e 0-043t U. 1249 = 175e -0.04t128.41 -49.13

12. In 6.7 = In (e 0"21t) 13. In 724.6 = In (e 6"3t)

9.06 1=05

14. Banking Ms. Cubbatz invested a sum of money in a certificate of deposit thatpays 8% interest compounded continuously. Recall that the formula for theamount in an account earning interest compounded continuously is A = Pe rt. IfMs. Cubbatz made the investment on January 1, 1989 and the account is worth$12,000 on January 1, 1993, what was the original amount in the account?

$8713.79

_T79GlencoeDivision,Macmillan/McGraw-Hill

NAME • DATE

12-1 Practice WorksheetArithmetic Sequences and SeriesSolve.

1. Find the 24th term in the sequence for which a = -27 and d = 3.

2. Find n for the sequence for which a n = 27, a = -12, and d = 3.

3. Find d for the sequence for which a = -12 and a23 -- 32.

4. Find the firstterm in the sequence for which d = -3 and a 6 = 5.

5. Find the first term in the sequence for which a 4 = -21 and a 7 = -3.

6. Find the sixth term in the sequence -3 + V_, 0, 3 - _/-2, • • •

7. Find the 45th term in the sequence -17, -11, -5, • • •.

8. Form asequence that has one arithmetic mean between 35 and 45.

9. Find the sum of the first 13 terms in the series

-5 + 1 + 7 + ... + 43.

8OGlencoe Division,Macmillan/McGraw-Hill

NAME DATE

12-1 Practice WorksheetArithmetic Sequences and SeriesSolve.

1. Find the 24th term in the sequence for which a = -27 and d = 3.42

2. Find n for the sequence for which a n = 27, a = -12, and d = 3.13

3. Find d for the sequence for which a = -12 and a23 = 32.2

i

4. Find the first term in the sequence for which d = -3 and a 6 = 5.20

5. Find the first term in the sequence for which a4= -21 and a 7 = -3.-39

6. Find the sixth term in the sequence -3 + V/2, 0, 3 - _/-2, • • •12-4x/

7. Find the 45th term in the sequence -17, -11, -5, • • •.

247

8. Form a sequence that has one arithmetic mean between 35 and 45.

35, 40, 45,...

9. Find the sum of the first 13 terms in the series-5 + 1 + 7 + ... + 43.

403


NAME DATE

12-2 Practice WorksheetGeometric Sequences and SeriesSolve.

31. The first term of a geometric sequence is -4, and the common ratio is _. Find the

next four terms.

32. The first term of a geometric sequence is 12, and the common ratio is - _. Find

the next four terms.

3. Find the ninth term of the geometric sequence V3, -3, 3V3, • • •

4. Find the fifth term of the geometric sequence 20, 0.2, 0.002, • • •.

5. Find the first term of the geometric sequence for which a 5 = 64V2 and r = V2.

6. Find the first three terms of the geometric sequence for which a 4 = 8.4 and r = 4.

17. Form a sequence that has one geometric mean between _ and 3.

38. Find the sum of the first eight terms of the series -_ + 9 + +.;..2U 100


NAME DATE

12-2 Practice WorksheetGeometric Sequences and SeriesSolve.

31. The first term of a geometric sequence is -4, and the common ratio is _. Find the

next four terms.9 27 81

-'_' 4' 16' 64

32. The first term of a geometric sequence is 12, and the common ratio is - _. Find

the next four terms.

-18, 27, 81 2432 _ 4

3. Find the ninth term of the geometric sequence V3, -3, 3V3, • • •.six/3

4. Find the fifth term of the geometric sequence 20, 0.2, 0.002, • • •.0.0000002

5. Find the first term of the geometric sequence for which a5 = 64V2 and r = _V/2.16x/5

6. Find the first three terms of the geometric sequence for which a 4 ----8.4 and r = 4.

0.13125, 0.525, 2.1

' 1

7. Form a sequence that has one geometric mean between -_ and 3.

1 _/3 3or 1 x/3 39' 3 _ 9' 3 '

3Find the sum of the first eight terms of the series _ + 9 + _ +...

978. I

1.84351


NAME DATE

12-3 Practice WorksheetInfinite Sequences and SeriesEvaluate each limit, or state that the limit does not exist.

n 2- 1

1. lim n2 2. lim 4n2-5n 3. lim 5n2 + 1n_¢¢ + 1 n_ 3n2 + 4 n_¢¢ 6n

4. lim (n-1)(3n + 1) 5. lim 5na + 1 6. lim 3n- (-1) nn_oo 5n 2 n_ 3n2 _- 1 n_ 4n 2

Write each repeating decimal as a fraction.

7. 0.49999... 8. 1.999... 9. 2.242424...

10. 0.127127... 11. 1.164164... 12. 0:6414141...

Find the sum of each infinite series, or state that the sum does not exist.

2 _5 18 3 _ 7513. _+ +-- 14. -_ +-- + ]-_ +125 ......


NAME DATE

12-3 Practice WorksheetInfinite Sequencesancl5eriosEvaluate each limit, or state that the limit does not exist.

n 2 1 3. lim 5n2 + 11. lira :_- 2. lira 4n2-5nn_¢¢ n + 1 n_ 3n2 + 4 n--,_ 6n

1 4 does not exiist3

4. lim (n- 1)(3n + 1) 5. lim 5n3 + 1 6. lim 3n- (-1) nn_oo 5n2 n_oo 3n2 + 1 n_oo 4n 2 ,,

3 does not exist 05

Write each repeating decimal as a fraction.

7. 0.49999... 8. 1.999... 9. 2.242424. • i

! 2 2 82 33

ilil

10. 0.127127''' 11. 1.164164"" 12. 0.6414141'''

164 127 ,_127 1 999 198999

Find the sum of each infinite series, or state that the sum does not exist.

22_ _ 18 3 _.5 7513. _+ +--125"'" 14. _+ +_+...

1 does not exist

T82 "GlencoeDivision,Macmillan/McGraw-Hill

NAME DATE

12-4 Practice WorksheetConvergent and Divergent SeriesUse the ratio test to'determine whether each series is convergent or divergent.

22 32 4 21. 1+-_+_+_+... 2. 0.006+0.06+0.6+...

3. 1 •2----_ 1-2.3.4 1.2,3.4.5

Use the comparison test to determine whether each series is convergent ordivergent.

5. 1+ + +-_+... 6. 1+_+_ +_+...

83Glencoe Division, Macmillan/McGraw=Hill

NAME DATE


Convergent and Divergent SeriesUse the ratio test to determine whether each series is convergent or divergent.

2 2 3 2 421. 1 + _- + _- + _ + ... 2. 0.006 + 0.06 + 0.6 + ...

divergent dive_H_

4 -t- 8 -t- 16 -t- 4. 1+_+ + +3° 1.2.3 1.2.3-4 1.2.3.4.5 ......

convergent convergent

Use the comparison test to determine whether each series is convergent ordivergent.

1 -_3 1 1 4 95. 1+_-+ +_-+... 6. 1+_ +_- +7+...

convergent divergent


NAME DATE

12-5 Practice WorksheetSigma Notation and the nth TermEvaluate each expression.

1. 7! 2. 3(4!) 3. 6!2!

Write each expression in expanded form and find the sum.5 4 5

2_

3 5

7.t=o_ (2t-3) 8'c=2_ (C-1) 2 9. i=1£ 10(1) i

Express each series using sigma notation.

10. 3+6+9+ 12+ 15 11. 3+9+27+...+243

1 1 1 1

12. T + 4- + 9 + "'" + 13. 24 + 19 + 14 + ... + (-1)lOO

3 3 3 _ 15._+ 35-----A4+ -_ +... + 6.714. ¥ + -_ + -5 +"" + 67 • 26

84Glencoe Division, Macmillan�McGraw-Hill

NAME DATE


Sigma Notation and the nth TermEvaluate each expression.

1. 7! 2. 3(4!) 3. 6!2!5040 72 1440

Write each expression in expanded form and find the sum.5 4 5

4. (k2- 2k) 5. _ 6. t(t - 1)3 =1 t=l

(32 _ 23)+(42 _ 24)+ 2_. I-- _- -I- _ -t--_-,2 2 2, 1(1-1)+2(2-1)+(52 25);-6 3(3-1)+4(4-1)+

25 5(5 -- 1); 406

7. ,7_, (2t-3) 8. _ (c-1) 2 9,. 10 it=O c=2 _=i

• 0' 1 2(2(0)-3)+(2(1)-3)+ (2-)2:+(3-1)2+ 10 +1( +(2(2)-3)+(2(3)-3);0 (4-_)2+(5-1)z; 30 2

10 +,.. ; 10

Express each series using sigma notation.10. 3+6+9+ 12+ 15 11. 3+9+27+...+243

5

I=1 I=1

1 I_ 1 I12. T + 4 + 9 + "'" + lOO 13. 24 + 19 + 14 + ... + (-i)

10 5

Z _ Z (24 - 51")I=1 I=0

3 3 3 3 15. 223 + 354 +-_ +--'+ 6.714. ]- + -_ + -_ +... + 6--7 -- 26

34 5

Z 3 Z (i+ 1)(i+ 2)I=1 2i-1 I=1 i2+1

"r84Glencoe Division, Macmillan/McGraw-Hill

NAME DATE

12-6 Practice WorksheetThe Binomial Theorem

Use Pascal's triangle to expand each binomial.

1. (x- y)4 2. (r + 3) 5

3. (m + 2n) 5 4. (3a- b)4

Use the binomial theorem to expand each binomial.

5. (x - 5)4 6o (3x + 2y)4

7. (a - V_) 5 8. (2x - 3y) 6

Find the designated term of each binomial expansion.

9. 4th term of(2n 2 - 3m) 4 10. 3rd term of(a - 2N/-3)6


NAME DATE

12-6 Practice WorksheetThe Binomial TheoremUse Pascal's triangle to expand each binomial

l. (x-y)4 2. (r + 3)5

- 4x3y + 6x2y2 - 4xy3 + _ _ - 15t,4+ 90r3 + 270_2+405_ + 243

3. (m + 2n) 5 4. (3a- b)4mS+ lOm4n+ 40rrPr_+ 81_ - 108_b + 54_b2-80m2n3 + 80mn4 + 32_ 12ab3 +

Use the binomial theorem to expand each binomiaL

5. (x- 5)4 6. (3x + 2y)4

x4 - 20._ + 150._2 - 5eO._+ 81,_ + 216_y + 216._2y_ +625 96xJ# + I¢y _

7. (a - V2) 5 8. (2x - 3y) 6

- 5V_ + 20_ - 2ov_ + 64_- 576_zy+ 2160_ -20a- 4X/2 4320_ + 48_}Ox2y_

2e e p +72e

Find the designated term of each binomial expansion.

9. 4th term of(2n 2 - 3m) 4 10. 3rd term of(a - 2%/3) 6

-216n2m 3 180_


NAME DATE

12-7 Practice WorksheetSpecial Sequences and Series

. X 2 X 3 X 4

Use the first five terms of the exponential series ex = 1 + x + _. + _. + _. +... and

a calculator to approximate each value to the nearest hundredth.

1. e°'5 2. e1"2 3. e3"1

4. e2'7 5. e09 O. e2"2

Use the first five terms of the appropriate trigonometric series to approximate thevalue of each function to four decimal places. Then, compare the approximation tothe actual value.

_rr-- --_ 9. cos-•r 8. COS 4 37. sin 2

Write each expression or complex number in exponential form.

10. 5 + 5i 11. 1- k/-3i

12. 13(cos 3 + isin 3) 13. 7(cos _ +isin_)

Find each value.

14. In (-5) 15. In (-5.7) 16. In (-1000)


NAME DATE

12-7 Practice WorksheetSpecial Sequences and Series

x2 x* __.Use the first five terms of the exponential series ex = 1 + x + _. + -_. + +... and

a calculator to approximate each value to the nearest hundredth.

1. e0"5 2. e1"2 3. e3"1

1.65 3.29 ]7.71

Use the first five terms of the appropriate trigonometric series tO approximate thevalue of each function to four decitnal places. Then, compare the approximation tothe actual value.

__ 17"_" 9. COS-7. sin -_ 8. cos 4 32

1.0000; equal 0.7071; equal 0.50000; eq_al

Write each expression or complex number in exponential form.

10. 5 + 5i 11. 1- _/3ii'n- 4_'i

5_/2e T 2_ 3

12. 13 cos _- + i sin 13. 7 cos + i sin

iv 7_-i

13e3- 7_ 6

Find each value.

14. In (-5) 15. In (-5.7) 16. In (-1000)i_ + 1.8094 i_- + 1.7405 #_+ 8.111078

T86Gtencoe Division, Macmillan�McGraw-Hill

NAME DATE

12,8 Practice WorksheetMathematical InductionUse mathematical induction to prove that each formula is vafid for all positiveintegral values of n.

1 2 3 n _ n (n + 1)1. _+_ :1-_+...+ _ _

2. 4 + 12 + 36 +... + 4 • 3 n -1 = 2(3 n _ 1)


NAME DATE

12-8 Practice WorksheetMathematical Induction

Use mathematical induction to prove that each formula is valid for all positiveintegral values of n.

1 2 3 n _ n (n + 1)1. _+-_+_-+.. +_ 6

1 _ 1 (1 + 1), So the fo_muuais vatid 1for_ = 1oSl:-gAssume the formula is valid for t_ - k.

Sk: _ -I-_-1 2 +g+ ...+g3 k = _{k+ _}oNOWprove that it is valid

for n = k + 1,

_ 1+2 3 k k+l

= k(k+ 1) + k+6 3

6

= k2+3k+2 = (k+l) (k+2 }6 6

So, Sn is valid for all positive integers no

2. 4 + 12 + 36 + ... + 4.3 n- 1 = 2(3 n - 1) -

$1:4 = 2(31 - 1). So, the formula is valid for n = 1[]

Assume the formula is valid for n = k.Sk: 4 +.12 + 36 +, -_+ 4.3 k - 1it iS valid for n = k +'1. = 2(3k

1). Now prove that

Sk+ 1 =4 + 12 + 36+ooo+4.3 k 1 +4.3 k

=2(3 k- 1)+4.3 k

= 2.3 k- 2 + 4.3 k

= 3k (2 + 4) - 2=2.3.3k-2

= 2 (3k + 1 -1)

Thus, Sn is valid for all positive integers n.


NAME DATE

13-1 Practice WorksheetIterating Functions with Real NumbersFind the first three iterates of each function using the given initial value, ff necessary,round your answers to the nearest hundredth.

1. f(x) = x 2 - 0.5; x 0 -- 1.5 2. f(x) = x 2 - 0.5; x 0 -- -1.5

3. f(x) = x 3 - 0.5; x o = 0.75 4. f(x) - x(2.5 - x); x o = 2.1

Find the first ten iterates for f(x) = 2.3(1 - x) for each initial value. If necessary, round youranswers to the nearest hundredth.

5. x o=0.3 6. x o=1.5

7. x o = 0.99 8. x o =-0.5

Find the first ten iterates for f(x) = 3.2(1 - x) for each initial value, ff necessary, round youranswers to the nearest hundredth.

9. x o = 0.25 10. x o = 0.09

11. x o = 0.75 12. x o = 0


: NAME DATE

13-1 Practice WorksheetIterating Functions with Real NumbersFind the first three iterates of each function using the given initial value, ff necessary,round your answers to the nearest hundredth.

1. f(x) = x 2 - 0.5; xo = 1.5 2. f(x) = x 2 - 0.5; x 0 -- --1.51.75, 2.56, 6.05 1.75, 2.56, 6.05

3. f(x) = x 3 - 0.5; x 0 -- 0.75 4. f(x) = x(2.5 - x); x o = 2.1-0.08,-0.50,-0°63 0.84, 1o39,1.54

Find the first ten iterates for f(x) = 2.3(1 - x) for each initial value. If necessary, round youranswers to the nearest hundredth.

5. xo=0.3 6. xo--1.5

1.61,-1.40, 5.52,-10.40, 26.22, -1.15, 4.95,-9.09, 23.21,-58.01,135.72, -309.86, 714°_8, -51.08, 119.78, -273.19,-1642.15 630.64_ -1448.17, 3333.09

7. x o = 0.99 8. xo =-0.50.02, 2.25,-2.88, 8.92, =18.22, 3.45,-5.64, 15.27,-32.82,44.21, -99,38, 230.87, -528.70, 77.79, -176.62, 408.53,1218.31 -937.32, 2158.14, -496"1.42

Find the first ten iterates for f(x) = 3.2(1 - x) for each initial value, ff necessary, round youranswers to the nearest hundredth.

9. x o = 0.25 10. xo = 0.092.4, -4.48, 17.54, -52.93, 2.92, -6.11, 22.75, -69.60,172,58, -549.06, 1760.19, 225.92, -719.74, 2306.37,-5629.41, 18,017.31, -7377.18, 23,610.18,-57,652.19 -75,549.38

11. x o = 0.75 12. Xo= 00.8, 0.64, 1.15,-0.48, 4.74, 3.2,-7.04, 25.73,-79.14,

-11.97, 41.50,-129.60, 417.92, 256.45,-817.44, 2619.01,-1334.14 -8377.63, 26,811.62,

-85, 793.98

T88Glencoe Division,Macmillan/Mc(_raw-Hill

NAME DATE

13-2 Practice l_lorL:,,;la_tGraphical Iteration of Linear FunctionsCopy the graphs shown and perform graphical iteration for the first four iterates ofthe initial point shown.

1. 2.

_____-_"_x _____-- - _-z "_....... -_........ -' _- ....._- --- ..... _,---Z ..... - ....... _ _..... ,,

=.... .... !.......... 7-__, -_-____ ." r____c_ _, _.-__ _z _ _ ---_ ..... ---_ .... _

Graph each function and the function f(x) = x on the same set of axes. Thenperform graphical iteration for x = 1 State the slope of the linear function and the.0 :type of path that the graj _ili__ I utera _i_tpforms.

3. f(x) = 2x + 1 4. f(x) = -0.5x + 1 5. f(x) = 0.4x - 2

_ __0____ ...... _'.__ ..... __0:__

_ --

Find the coordinates of the fixed point for each function. Is the x-coordinate of thefixed point a repeller or an attractor?

6. f(x) = 3.5x- 4 7. f(x) = -2x + 3

8. f(x) = 0.6x- 5 9. f(x) = -0.45x + 3


NAME DATE

13-2 F'l'_I.:tice l_o_.l_:s_t_.,.,tGraphical Iteration of Linear FunctionsCopy the graphs shown and perform graphical iteration for the first four iterates ofthe initial point shown.

1. 2.

_ ,__)-._ _ _ _ _ __7_. 7---.......... ....+ ........ 2 ......... 7Z.... _,

22-- --- - .... _- _-v-

x:_ o -_, _ ___--- - -

Graph each function and the function f(x) = x on the same set of axes. Thenperform graphical iteration for xn = 1. State the slope of the linear function and t,betype of path that the graphical iteration forms.

3. f(x) = 2x + 1 4. qx) = -0.5x + 1 5. f(x) = 0.4x- 2

,,_ ,, _ --,,_,____t _ "_L Z------ .... --- __z_ -- - " _-_ '_'xl--_+_-I-2z.... -'_- Z........ _--Z---- - L- - -__--__ "_ - --E--E :_2"-T-_ - -_ 2L,__-_-_ ...... - 2 __--

.... =......... ...."____,x _ _ _._x__

--'- f - -- ____Z ....... ....---2 2 ...... _zz ........ _ r_- _-_ ..... - _ z_ --__x__...... Z _ ___/ I_, ' i , I

2, staircase out "0.5, spiral in 0.4 staircase, in

Find the coordinates of the fixed point for each function. Is the x-coordinate of thefixed point a repeller or an attractor?

6. f(x) = 3.5x- 4 7. f(x) = -2x + 3

(1.6, 1.6), repeller (1,1), repeller

8. f(x) = 0.6x - 5 9. f(x) = -0.45x + 3

(-12.5, -12.5), attractor ( 60 60 / attractor29 ' 29 /'\


NAME DATE

13-3 Practice WorksheetGraphical Iteration of the Logistic FunctionFind the coordinates of the vertex of the graph of each logistic function.

1. f(x) = 3.6x(1 - x) 2. f(x) = 0.25x(1 -- x)

3. f(x) = 2.3x(1 - x) 4. f(x) = 0.76x(1 - x)

Find the coordinates of the fixed points for each function. Use the graphicaliteration behavior program on pages 724-725 of your textbook to graph thefunction and determine if the x-coordinate of each nonzero fixed point is arepeller or an attractor.

5. f(x) = 2.5x(1 - x) 6. f(x) = 0.6x(1 - x)

7. f(x) = 0.5x(1 -x) 8. f(x) = 2.0x(1 -x)

Use the modified graphical iteration behavior program on pages 724-725 ofyour textbook to determine the long-term iterative behavior of the functionf(x) = ax(1 - x) for each value of a. Write period-n attractor, fixed point attractor,or chaos.

9. a = 3.21 10. a = 2.1 11. a -- 4.0

12. a = 2.25 13. a = 3.47 14. a = 1.9


NAME DATE

13-3 Practice WorksheetGraphical Iteration of the Logistic FunctionFind the coordinates of the vertex of the graph of each logistic function.

1. f(x) = 3.6x(1 - x) 2. f(x) = 0.25x(! - x)

3. f(x) = 2.3x(1 -x) 4. f(x) _ 0.76x(1- x)

-2-,(

Find the coordinates of the fixed points for each function. Use the graphicaliteration behavior program on pages 724-725 of your textbook to graph thefunction and determine if the x-coordinate of each nonzero fixed Point is arePeller or an attractor.

5. f(x) = 2,5x(1 - x) 6. f(Jc) = 0.6x(1 - x)

5' 5 ; attractor (0,.0); 3' 3 ; repeller

7. f(x) = 0.5x(1 - x) 8. f(x) = 2.0x(1 - x)

(0,0); (-1,-1); repeller (0,.0); _, _ ; attractor

Use the modified graphical iteration behavior program on pages 724-725 ofyour textbook to determine the long-term iterative behavior of the functionf(x) = ax(1 - x) for each value of a. Write period-n attractor, fixed point attractor,or chaos.

9. a = 3.21 10. a = 2.1 11. a = 4.0

period-2 fixed point chaosattractor attractor

12. a = 2.25 13. a = 3.47 14. a = 1.9

•fixed point period 4 _ fixed pointattractor attractor attractor

v


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13-4 Practice WorksheetComplex Numbers and IterationFind the firSt three iterates of the function f(z) = 3z - i for each initial value.

1. z o=1+2i 2. z o=-5i 3. z o=4

4. z o=3-i 5. z o=0.5-0.25i 6. z o=0.2i

Find the first three iterates of the function f(z) = 2z - (3 + i) for each initial value.

7. z o = 1 - 2i 8. z o =-2 - 5i 9. z o = 3 - i

lO.z o=0.5+i 11. z o=i 12. z o=1-i

Find the next four iterates of the function f(z) = Z2 4-C for each given value of c andeach initial value.

13. c=l-2i;z o=0 14. c=i;z o=i 15. c=2-3i;z o= l+i


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13-4 Practice WorksheetComplex Numbers and IterationFind the first three iterates of the function f(z) = 3z - i for each initial value.

;159 + 14i -49i 36 - 4i

27 + 41i -148i 108- 13i

27 - 13i 4.5 - 6o25i -2.2i81 - 40i 13.5 - "_9.75i -7.6i

Find the first three iterates of the function f(z) = 2z - (3 + i) for each initial value.

5i 9. = -

-5- 11i -17- 23i 3- 7i-13- 23i -37- 47i 3- 15i

10. Zo-- 0.5 + i 11. Zo= i 12. Zo.= 1 -- !-2 + i -3 + i --1 - 3_-7 + i -9 + i -5 - 7i-17 + i -21 + i -13- 15i

Find the next four iterates of the function f(z) = Z2 4-C for each given value of c ,andeach initial value.

13. c= 1-2i;z o=0 14. c=i;z_=i 15. c=2-3i;z o=3L+i1 -2i -1 + i 2- i-2 - 6i -i 5 - 7i-31 + 22i -1 + i -22- 73i478- 1366i -i -4843 + 3209i


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13-5 Practice WorksheetEscape Points, Prisoner Points, and Julia SetsDetermine whether the graph of each value is in the prisoner set or the escape setfor the function f(z) = z- + (-3 + Oi).

1. 0.5+0.5i 2. 0 3. l+V_2

Determine whether the graph of each value is in the prisoner set or the escape setfor the function f(z) = z 2 + (-2 + Oi).

4. 1+2i 5. 3-2i 6. 2.0+0i

Determine whether the Julia set for each function is connected or disconnected.

7. f(z) = z2 + (2 + i) 8. f(z) = z 2+ (-0.2 + 0.05i)

9. f(z) = z2+ i 10. f(z) = z2 + 1

11. f(z) z 2- 1 12. f(z) = z2 + (1 - i)


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13-5 Practice WorksheetEscape Points, Prisoner Points, and Julia SetsDetermine whether the graph of each value is in the prisoner set or the escape setfor the function f(z) - z2 + (-3 + Oi).

1. 0.5 + 0.5i 2. 0 3. 2

escape set escape set prisoner set

Determine whether the graph of each value is in the prisoner set or the escape setfor the function f(z) = z2 + (-2 + Oi).

4. 1+2i 5. 3-2i 6. 2.0 + Oi

•escape set escape set prisoner set

Determine whether the Julia set for each function is connected or disconnected.

7. f(z) = z 2 + (2 + i) 8. f(z) = z 2 + (-0.2 + 0.05i)discon nected con nected

9. f(z) = z 2+i lO.f(z)=z 2+1connected disconnected

11. f(z) = z 2 - 1 12. f(z) = z 2 + (1 -i)connected disconnected


L

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13-6 Practice WorksheetThe Mandelbrot Set

For each value of c given, complete the following.

a. Determine whether the Julia set associated with each value of c is connected ordisconnected.

b. Determine if the point is inside or outside of the Mandelbrot set.

c. Use the color scheme given on page 743 in the lesson to assign a color to thepoint.

1. c=-1 2. c=-i

3. c=-0.8+0.5i 4. c=2+i

5. c = -0.4- 0.6i 6. c = -0.3 + i

93Glencoe Division, Macmillan/McGraw-Hill _"

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13-6 Practice WorksheetThe Mandelbrot SetFor each value of c given, complete the following.

a. Determine whether the Julia set associated with each value of C is connected ordisconnected.

b. Determine if the point is inside or outside of the Mandelbrot set.

c.Use the color scheme given on page 743 in the lesson to assign a color to thepoint.

1. c=-1 2. c=-i

connected, _o_nected_inside, i_ide,bNack _k

3. c=-0.8+0.5i 4. c=2+i

disconnected, _is_o_®c_d,outside, out_B_®,


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14-1' Practice WorksheetPermutationsFind each value.

1. P(10, 7) 2. P(7 3) 3. P(6 4).P(5 2)P (4, 2) P (7, 4)

Solve.

4. A briefcase lock has 3 rotating 5. A golf club manufacturercylinders, each containing 10 makes irons with 7digits. How many numerical different shaft lengths,codes are possible? 3 different grips, and 2

different club headmaterials. How manydifferent combinations areoffered?

6. There are 5 different routes 7. In how many ways can the 4that a commuter can take call letters of a radio station be

from her home to the arranged if the first letter mustoffice. In how many ways W or K and no letters repeat?can she make a round tripif she uses different routes for

coming and going?

8. How many 7-digit telephone 9. How many 7-digit telephonenumbers can be formed if numbers can be formed ifthe first digit cannot be 0 the first digit cannot be 0 oror 1? 1 and no digit can be

repeated?


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14-1 Practice WorksheetPermutationsFind each value.

1. P(10, 7) 2. P(7,3_ 3. _)P (4,2) P (7,4)

11104,800 I}7.5 6o7Solve.

4. A briefcase lock has 3 rotating 5. A golf club manufacturercylinders, each containing 10 makes irons with 7digits. How many numerical different shaft lengths,codes are possible? 3 different grips, and 2

different club headQ00 materials. How many

different combinations areoffered? 42

6. There are 5 different routes 7. In how many ways can the 4that a commuter can take call letters of a radio station befrom her home to the arranged if the first letter mustoffice. In how many ways W or K and no letters repeat?can she make a round trip 27_600if she uses different routes forcoming and going? 20

8. How many 7-digit telephone 9. How many 7-digit telephonenumbers can be formed if numbers can be formed ifthe first digit cannot be 0 the first digit cannot be 0 oror 1? 8,000_000 1 and no digit can be

repeated? 483,840


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14-2 Practice WorksheetPermutations with Repetitions and Circular PermutationsHow many different ways can the letters of each word be arranged?

1. CANADA 2. ILLINI

3. ANNUALLY 4. MEMBERS

Solve.

5. A photographer is taking a 6. A person playing a word gamepicture of a bride and groom has the following letters in hertogether with 6 attendants, tray: QUOUNNTAGGRA. HowHow many ways can he many 12-letter arrangementsarrange the 8 people in a line could she make to check if aif the bride and groom stand single word could be formedin the middle? from all the letters?

7. How many ways can 3 8. Three different hardcoveridentical pen sets and 5 books and five differentidentical watches be given paperbacks are placed on ato 8 graduates if each person shelf. How many ways can theyreceives one item? be arranged if all the hardcover

books are together?

9. In how many ways can 6 10. In how many ways can 5people stand in a ring charms be placed on a Braceletaround the player who is "it"? with no clasp?


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14-2 Practice WorksheetPermutations with Repetitions and Circular PermutationsHow many different ways can the letters of each word be arranged?

1. CANADA 2. ILLINI

120 60

3. ANNUALLY 4. MEMBERS

5040 126G

Solve.

5. A photographer is taking a 6. A person playing a word gamepicture of a bride and groom has the following letters in her 'together with 6 attendants, tray: QUOUNNTAGGRA. HowHow many ways can he many 12-letter arrangementsarrange the 8 people in a line could she make to check if aif the bride and groom stand single word could be formedin the middle? 1440 from all the letters?

29,937,600

7. How many ways can 3 8. Three different hardcoveridentical pen sets and 5 books and five differentidentical watches be given paperbacks are placed on a

to 8 graduates if each ]_erson shelf. How many ways can theyreceives one item? St} be arranged if all the hardcover

books are together? 4320

9. In how many ways can 6 10. In how many ways can 5people stand in a ring charms be placed on a braceletaround the player who is "it"? with no clasp?120 12


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14-3 Practice WorksheetCombinationsEvaluate each expression.

1. C(7, 2) 2. C(10, 4)

3. C(8, 8) 4. C(10, 4). C(5, 3)

5. C(12, 4). C(8, 3) 6. P(4, 3). C(8, 6)

7. P(12, 3). C(2, 1). C(6, 5) 8. P(8, 3). C(9, 6). C(8, 2)

Solve.

9. Sally has 7 candles, each a 10. In how many ways can adifferent color. How many student choose 4 books fromways can she arrange the 2 geometry, 4 geography, 5candles in a candelabra that history, and 2 physics books?holds 3 candles?

11. Eight toppings for pizza are 12. From a list of 12 books, howavailable. In how many ways many groups of 5 books cancan Jim choose 3 of the be selected?toppings?

\

13. How many committees of 5 14. Leroy can afford to buy 2 ofstudents can be selected from the 6 CDs he wants. How

class of 25? many possible combinationscould he buy?

15. A box contains 12 black and 16. A box contains 12 black and

8 green marbles. In how 8 green marbles. In how manymany ways can 3 black and ways can 5 marbles be2 green marbles be chosen? chosen?


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14-3 Practice WorksheetCombinations


1. C(7, 2) 2. C(10, 4)

21 210

3. C(8, 8) 4. C(10, 4). C(5, 3)1 2100

5. C(12, 4)- C(8, 3) 6. P(4, 3)- C(8, 6)

27,720 672

7. P(12, 3). C(2, 1). C(6, 5) 8. P(8, 3)" C(9, 6)" C(8, 2)

15,840 790,272

Solve.

9. Sally has 7 candles, each a 10. In how many ways can adifferent color. How many student choose 4 books fromways can she arrange the 2 geometry, 4 geography, 5candles in a candelabra that history, and 2 physics books?holds 3 candles? 210 715

11. Eight toppings for pizza are 12. From a list of 12 books, howavailable. In how many ways many groups of 5 books cancan Jim choose 3 of the be selected? 792

toppings? 56

13. How many committees of 5 14. Leroy can afford to buy 2 ofstudents can be selected from the 6 CDs he wants. How

Class of 25? 53_130 many possible combinationscould he buy?15

15. A box contains 12 black and 16. A box contains 12 black and

8 green marbles. In how 8 green marbles. In how manymany ways can 3 black and ways can 5 marbles be2 green marbles be chosen? chosen? 15,5046160


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14-4 Practice WorksheetProbability and OddsA bag contains 7 pennies, 4 nickels, and 5 dimes. Three coins are selected atrandom. Find the probability of each selection.

1. all 3 pennies 2. all 3 nickels

3. all 3 dimes 4. 2 pennies, i dime

5. I penny, i dime, i nickel 6. I dime, 2 nickels

A bag contains 5 red, 9 blue, and 6 white marbles. Two are selected at random.Find the probability of each selection.

7. 2 red 8. 2 blue

9. 1 red, i blue 10. i red, i white

Sharon has 8 mystery books and 9 science-fiction books. Four are selected atrandom. Find the probability of each selection.

ll. 4 mystery books 12. 4 science-fiction books

13. 2 mysteries, 2 science-fiction 14. 3 mysteries, 1 science-fiction

From a standard deck of cards, 5 cards are drawn. What are the odds of eachselection ?

15. 5 aces 16. 5 face cards

17. 5 from one suit 18. 2 of one suit, 3 of another


Practice WorksheetProbability and OddsA bag contains 7 pennies, 4 nickels, and 5 dimes. Three coins are selected atrandom. Find the probability of each selection.

1. all 3 pennies 2. all 3 nickels1 1

16 140

3. all 3 dimes 4. 2 pennies, 1 dime1 3

56 10

5. I penny, 1 dime, i nickel 6. i dime, 2 nickels1 34 56

A bag contains 5 red, 9 blue, and 6 white marbles. Two are selected at random.Find the probability of each selection.

7. 2 red 8. 2 blue1 18

19 95

9. i red, I blue 10. I red, I white9 338 19

Sharon has 8 mystery books and 9 science-fiction books. Four are selected atrandom. Find the probability of each selection.

11. 4 mystery books 12. 4 science-fiction books1 9

34 170

13. 2 mysteries, 2 science-fiction 14. 3 mysteries, 1 science-fiction36 1885 85

From a standard deck of cards, 5 cards are drawn. What are the odds of eachselection?

15. 5 aces 16. 5 face cards33

0 108, 257

17. 5 from one suit 18. 2 of one suit, 3 of another33 429

16,627 7901


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14-5 Practice WorksheetProbability of Independent and Dependent EventsThere are 3 nickels, 2 dimes, and 5 quarters in a purse. Three coins are selected insuccession at random.

1. Find the probability of 2. Find the probability ofselecting 1 nickel, 1 dime, and selecting 1 nickel, 1 dime, and1 quarter in that order without 1 quarter in that order withreplacement, replacement.

3. Find the probability of 4. Find the probability of selectingselecting 1 nickel, 1 dime, and 1 nickel, 1 dime, and 1 quarter1 quarter in any order with in any order withoutreplacement, replacement.

A red, a green, and a yellow die are tossed. What is the probability that thefollowing occurs?

5. All 3 dice show a 4. 6. None of the 3 dice shows a 4.

7. The red die shows an even 8. All 3 dice show the samelnumber, and the other 2 show number. t

[

different odd numbers.

From a standard deck of 52 cards, 2 cards are selected. What is the probability thatthe following occurs?

9. 2 black cards selected 10. 2 black cards selected with

without replacement replacement

11. i red card and 1 spade in 12. I red card and 1 spadj_ in thatany order selected without order selected withoutreplacement replacement

\


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14-5 Practice WorksheetProbability of Independent and Dependent EventsThere are 3 nickels, 2 dimes, and 5 quarters in a purse. Three coins are selected insuccession at random.

1. Find the probability of 2. Find the probability ofselecting 1 nickel, 1 dime, and selecting 1 nickel_ 1 dime, and1 quarter in that order without 1 quarter in that order withreplacement, replacement.1 3

24 100

3. Find the probability of 4. Find the probability of selectingselecting 1 nickel, 1 dime, and 1 nickel, 1 dime, and 1 quarter1 quarter in any order with in any order Withoutreplacement, replacement.9 !50 4

A red, a green, and a yellow die are tossed. What is the probability that thefollowing occurs?

5. All 3 dice show a 4. 6. None of the 3 dice shows a 4.

1 125216 216

7. The red die shows an even 8. All 3 dice show the samenumber, and the other 2 show number.different odd numbers.

1 112 36

From a standard deck of 52 cards, 2 cards are selected. What is the probability thatthe following occurs?

9. 2 black Cards selected 10. 2 black cards selected withwithout replacement replacement25 1102 4

11. 1 red card and 1 spade in 12. I red card and 1 spade in thatany order selected without order selected withoutreplacement replacement13 1351 102


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14-6 Practice WorksheetProbability of Mutually Exclusive or Inclusive EventsAn um contains 7 white marbles and 5 blue marbles. Four marbles are selectedwithout replacement. What is the probabilffy that each occurs ?

1. all white or all blue 2. exactly 3 white

3. at least 3 white 4. exactly 3 white or exactly 3 blue

Two cards are drawn from a standard deck of 52 cards. What is the probability thateach occurs ?

5. 2 spades 6. 2 spades or 2 red cards

7. 2 red cards or 2 jacks 8. 2 spades or 2 face cards

Three dice are tossed. What is the probability that each occurs?

9. exactly two 5s !0. at least two 5s

11. three 5s 12. no 5s


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Probability of Mutually Exclusive or Inclusive EventsAn urn contains 7 white marbles and 5 blue marbles. Four marbles are selectedwithout replacement. What is the probability that each occurs?

1. all white or all blue 2. exactly 3 white

8 3599 99

3. at least 3 white 4. exactly 3 white or exactly 3 blue14 4933 99

Two cards are drawn from a standard deck of 52 cards. What is the probability thateach occurs?

5. 2 spades 6. 2 spades or 2 red cards1 31

17 102

7. 2 red cards or 2 jacks 8. 2 spades or 2 face cards55 47

221 442

Three dice are tossed. What is the probability that each occurs ?

9. exactly two 5s 10. at least two 5s5 2

72 27

U. three 5s 12. no 5s

1 1125-216 216

/ T99' i Glencoe Division, Macmillan/McGraw-Hill

NAME DATE-

14-7 Practice WorksheetConditional Probability

, A card is drawn from a standard deck of 52 cards and is found to be red. Given thatevent, find each probability.

1. P (heart) 2. P (ace)

3. P (face card) 4. P (jack or ten)

5. P (six of spades) 6. P (six of hearts)

A survey taken at Stirers High School shows that 48% of the respondents likedsoccer, 66% liked basketball, and 38% liked hockey. Also, 30% liked soccer andbasketball, 22% liked basketball and hockey, and 28% liked soccer and hockey.Finally, 12% liked all three sports. Find each probability.

7. the probability Meg likes 8. the probability Juan likessoccer if she likes basketball if he likes soccerbasketball

9. the probability Kim likes 10. the probability Greg likeshockey if she likes hockey and basketball ifbasketball he likes soccer

A pair of dice are thrown. It is known that the sum is greater than seven. Find eachprobability.

11. P(numbers match) 12. P(one die shows a 1)


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14-7 Practice WorksheetConditional ProbabilityA card is drawn from a standard deck of 52 cards and is found to be red. Given that _event, find each probability. __

1. P (heart) 2. P (ace)

2 43

3. P (face card) 4. P (jack or ten)3 213 13

5. P (six of spades) 6. P (six of hearts)

0 26

A survey taken at Stirers High School shows that 48% of the respondents likedsoccer, 66% liked basketball, and 38% liked hockey. Also, 30% liked soccer andbasketball, 22% liked basketball and hockey, and 28% liked soccer and hockey,.Finally, 12% liked all three sports. Find each probability.

7. the probability Meg likes 8. the probability Juan likessoccer if she likes basketball if he likes soccerbasketball

5 511 8

9. the probability Kim likes 10. the probability Greg likeshockey if she likes hockey and basketball ifbasketball he likes soccer

1 13 4

A pair of dice are thrown. It is known that the sum is greater than seven. Find eachprobability.

U. P(numbers match) 12. P(one die shows a 1)

! 05

TIO0GlencoeDivision,Macmillan�McGraw-Hill

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14-8 Practice WorksheetThe Binomial Theorem and ProbabilitySix coins are tossed. What is the probability that each occurs ?

1. 3 heads and 3 tails 2. at least 4 headsr:

3. 2 heads or 3 tails 4. all heads or all tails

2The probability of Chris making a free throw is -5. If she shoots five times, what isthe probability of each ?

5. all missed 6. all made

7.. exactly 4 made 8. at least 3 made

When Mary and Dwayn e play a certain board game, the probability that Mary will3

win a game is _. If they play five games, find the probability of each event.

9. Dwayne wins only once 10. Mary wins exactly twice

11. Dwayne wins at least two 12. Mary wins at least three gamesgames


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14-8 Practice WorksheetThe Binomial Theorem and ProbabilitySix coins are tossed. What is the probability that each occurs ?

1. 3 heads and 3 tails 2. at least 4 heads

5 11 '_16 32

3. 2 heads or 3 tails 4. all heads or all tails

35 164 32

2The probability of Chris making a free throw is -_. If she shoots five times, what isthe probability of each ?

5. all missed 6. all made

1 32243 243

7. exactly 4 made 8. at least 3 made80 64

243 81

When Mary and Dwayne play a certain board game, the probability that Mary will3

win a game is _. If they play five games, find the probability of each event.

9. Dwayne wins only once 10. Mary wins exactly twice405 451024 512

11. Dwayne wins at least two 12. Mary wins at least three gamesgames

47 459128 512


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15-1 Practice WorksheetThe Frequency DistributionThe selling prices of a random sample of 30 single-family homes sold in a city duringthe past year are given below.

67,500 72,000 54,600 38,900 87,400 28,300

105,000 91,600 46,500 62,800 136,600 81,200

59,900 76,200 117,100 64,400 25,500 51,800 !

88,000 63,900 125,000 70,500 28,200 59,500

118,700 81,100 42,600 57,300 77,700 64,800

1. Make a frequency distribution of the data using six classes from $20,000 to$140,000.

2. List the class marks.

3. What are the class intervals?

4. Whatare the class limits?

5. Which class interval had the greatest frequency?

The ages of 100 people attending a concert are given below.

[ C.sUo.S roqen°,6. List the class marks.

7. What is the class interval?

8. What are the class limits?

9. Which age group had the most people in attendance?.

10. Which age group had the fewest people in attendance?

\


NAME DATE

15-1 Practice WorksheetThe Frequency DistributionThe selling prices of a random sample of 30 single-family homes sold in a city duringthe past year are given below.

67,500 72,000 54,600 38,900 87,400 28,300

105,000 91,600 46,500 62,800 136,600 81,200

59,900 76,200 117,100 64,400 25,500 51,800

88,000 63,900 125,000 70,500 28,200 59,500

118,700 81,100 42,600 57,300 77,700 64,800

1. Make a frequency distribution of the data using six classes from $20,000 to$140,000.

CLASS LIMITS TALLY FREQUENCY$20,000 -40,000 IIII 4$40,000 - 60,000 Jtff II 7$60,000 - 80,000 J41TIIII 9$80,000- 100,000 gff 5

$100,000- 120,000 III 3$120,000- 140,000 II 2

2. Listthe class marks. $30,000, $50,000, $70,000, $90,000,$110,000, $130,000

Whataretheclassintervals?$20,000-40,000, $40,000-60,000,8. $60,000-80,000, $80,000-100,000, $100,000 120,000,$120,000 i40,000

What are the class limits? $20,000, $40,000, $60,000, $80,000,4. $100,000, $120,000, $140,000

5. Which class interval had the greatest frequency? $60,000-80,000

The ages of 100 people attending a concert are given below.

[ Class Limits 10-13 t 13-16 16-19 t 19-22 22-25Frequency 4 " 19 28 • 36. 13

6. List the class marks. 11.5, 14.5, 17.5, 20.5, 23.57. What is the class interval? 38. What are the class limits? 1 0, 1 3, 1 6, 1 9, 22, 259. Which age group ha d the most people in attendance? 1 9-22

10. Which age group had the fewest people in attendance? 1 0-1 3

T 102GlencoeDivision,Macmillan�McGraw-Hill

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Measures of Central TendencyThe numbers of airline fatafities in the United States during theyears 1981 to 1990 are given below.

Year Fatalities Year Fatalities

1981 4 1986 1

1982 233 1987 231

1983 15 1988 285

1984 4 1989 278

1985 197 1990 39

1. Find the mean of the data.2. Find the median of the data.

3. Find the mode of the data.

The average number of points per game scored by members of abasketball team during a recent year are given below°

37.1 8.5 4.2

14.5 8.5 3.5

11.3 8.3 2.8

9.7 6.9 1.9

4. Find the mean of the data.

5. Find the median of the data.6. Find the mode of the data.

The numbers of students present at each monthly meeting of theSpirit Club are given below.

August 146 January 121

September 138 February 93October 120 March 118

November 132 April 129

December 146 May 136

7. Find the mean of the data.

8. Find the median of the data.



NAME ' DATE

15-2 Practice WorksheetMeasures of Central TendencyThe numbers of airline fatalities in the United States during theyears 1981 to 1990 are given below.

Year Fatalities Year Fatalities

1981 4 1986 1

1982 233 198 _] 23!

1983 15 1988 285

1984 4 1989 278

1985 197 1990 39

1. Find the mean of the data. 12_o7

2. Find the median of the data. _ _


The average number of points per game scored by members of abasketball team during a recent year are given below.

37.1 8.5 4.2

14.5 8.5 3.5

11.3 8.3 2.8

9.7 6.9 1.9

4. Find the mean of the data. 9,85. Find the median of the data. 8°4

6. Find the mode of the data. 8.5

The numbers of students present at each monthly meeting of theSpirit Club are given below.

August 146 January 121

September 138 February 93October 120 March 118

November 132 April 129December 146 May 136

7. Find the mean of the data. 1 27.9

8. Find the median of the data. 1 30.5

9. Find the mode of the data. 1 46

T 103_- Glencoe Division, Macmillan/McGraw-Hill

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15-3 Practice WorksheetMeasures of VariabilityFind the mean deviation, semi-interquartile range, and standard deviation for eachset of data. Make a box-and-whisker plot for each set of data.

1. 43, 26, 92, 11, 8, 49, 52, 126, 86, 42, 63, 78, 91, 79, 86

2. 1.6, 9.8, 4.5, 6.2, 8.7, 5.6, 3.9, 6.8, 9.7, 1.1, 4.7, 3.8, 7.5, 2.8, 0.1

3. 146,289,121,146,212,98,86,153,128,136,181,142

4. 1592, 1486,1479,1682,1720,1104,1486,1895,1890,2687,2450

5. 506,612,789,412,814,583,102,881,457,826

6. 26.8, 15.7, 98.4, 27.3, 14.1, 81.6, 19.4, 21.5, 46.5, 23.7, 16.7, 29.8


NAME DATE "

15-3 Practice WorksheetMeasures of VariabilityFind the mean deviation, semi.interquartile range, and standard deviation for eachset of data. Make a box-and.whisker plot for each set of data.

1. 43, 26, 92, 11, 8, 49, 52, 126, 86, 42, 63, 78, 91, 79, 86

27'.2; 22; 32.02$ 0

0 20 40 60 80 100 120 140

2. 1.6, 9.8, 4.5, 6.2, 8.7, 5.6, 3.9, 6.8, 9.7, 1.1, 4.7, 3.8, 7.5, 2.8, 0.1

2,46; 2.35; 2°93

0 2.0 4.0 6.0 8.0 10.0

3. 146, 289, 121, 146, 212, 98, 86, 153, 128, 136, 181, 142

37.1;21.25; 51.99

• _ o •F =

80 100 120 140 160 180 200 220 240 260 280 300

4. 1592, 1486, 1479, 1682,' 1720, 1104, 1486, 1895, 1890, 2687, 2450

334.84; 204.5; 433.25

1000 1200 1400 1600 1800 2000 2200 2400 2600 2800

5. 506, 612, 789, 412, 814, 583, 102, 881,457, 826

186.2; 178.5; 229.04

100 200 300 400 500 600 700 800 900

6. 26.8, 15.7, 98.4, 27.3, 14.1, 81.6, 19.4, 21.5, 46.5, 23.7, 16.7, 29.8

20.19; 10.05; 26.10e

0 10 20 30 40 50 60 70 80 90 100


NAME DATE

15-4 PracticeWorksheetThe Normal DistributionA set of 1000 values has a normal distribution. The mean of thedata is 120, and the standard deviation is 20.

1. How many values are within one standard deviation from the mean?

2. What percent of the data is in the range 110 to 130?

3. What percent of the data is in the range 90 to 110?

4. Find the range about the mean which includes 90% of the data.

5. Find the range about the mean which includes 77% of the data.

6. Find the probability that a value selected at random from the data will be withinthe limits 100 and 150.

7. Find the probability that a value selected at random from the data will be greaterthan 140.

8. Find the point below which 90% of the data lie.


NAME DATE

15-4 Practice WorksheetThe Normal DistributionA set of 1000 values has a normal distribution. The mean of thedata is 120, and the standard deviation is 20.

1. How many Values are within one standard deviation from the mean? 683

2. What percent of the data is in the range 110 to 130? 38.3%

3. What percent of the data is in the range 90 to 110? 24.15%

4. Find the range about the mean which includes 90% of the data. 87 to 153

5. Find the range about the mean which includes 77% of the data. 96 to 144

6. Find the probability that a value selected at random from the data will be withinthe limits 100 and 150. 0.7745

7. Find the probability that a value selected at random from the data will be greaterthan 140. 0.1585

8. Find the point below which 90% of the data lie. 146


NAME DATE

15-5 Practice WorksheetSample Sets of DataFind the standard error of the mean for each sample. Then find the range for a 1%level of confidence and the range for a 5% level of confidence.

1. _=50, N= 100, X=250 2. _r=4, N=64, X=100

B

3. _r = 2.6, N = 100, X = 50 4. _ = 43, N = 100, X = 110

The table below shows a frequency distribution of the time inminutes required for students to wash a car during the car wash.The distribution is a random sample of 250 cars.

Number of Minutes

Number of Cars

5. Find the standard deviation of the data in the frequency distribution.

6. Find the standard error of the mean.

7. Find the interval about the sample mean such that the probability is 0.90 thatthe true mean lies within the interval.


106Glencoe Division, Macmillan/McGraw.Hill

NAME DATE

15-5 Practice WorksheetSample Sets of DataFind the standard error of the mean for each sample. Then find the range for a 1%level of confidence and the range for a 5% level of confidence.

1. _r=50, N=100, X=250 2. cr=4, N 64, X=100

5;i 237.1 -262.9; 240.2-259.8 0.5; 98.71 -101.29;99.02-100.98

3. ¢r = 2.6, N = 100, X = 50 4. _ = 43, N = 100, X = 110

0.26; 49.3292-50.6708; 4.3; 98.906-121.094;49.4904-50.5096 101,572-118.428

The table below shows a frequency distribution of the time inminutes required for students to wash a car during the car wash.The distribution is a random sample of 250 cars.

•Number of Minutes 5 6 7 8 9 10

Number of Cars 2 4 5 1 8 5

5. Find the standard deviation of the data in the frequenc:_ distribution. 1.64

6. Find the standard error of the mean. 0.327


7.42045-8.49955


/

7.31908-8.60092


NAME DATE

15-6 Practice WorksheetScatter PlotsThe table below shows the amount of sales for eight salesrepresentatives and the years of sales experience for eachrepresentative.

I,m0unto,'esYears of Experieilce

1. Draw a scatter plot and find a median 2. Use a graphing calculator to plotfit line for the data. the data and draw a regression line.

....C .......

-z--- ±_ -__Years of Experience

3. What is the equation of the regression 4. What is the Pearson product-momentline? correlation value?

The table below shows thestatistics grades and the economicsgrades for a group of college students at the end of a givensemester.

f Statistics Grades ]_-t 5_--Economics Grades ___


"o

E ....... r'-- ....oe- ............

O ............oIII ............

............ /

Statistics Grade



NAME DATE

15-6 Practice Worksheet.

Scatter Plots

The table below shows the amount of sales for eight salesrepresentatives and the years of sales experience for eachrepresentative.

Amount of Sales $_ $3000 $5000 $8000 $2000

Years of Experience


9000 _- °

7000 / -/,n f "e- 5000 _ h -

3000 ,,-/" " - " *m

1000 /

1 2 3 4 5 6

.Years v.^4_ v= .='-=x'^r:^n'e ...........!.........'.........!.........'.........!.........'.........!i


y = 1191.5x + 531.9 0.871

The table below shows the statistics grades and the economicsgrades for a group of college students at the end of a givensemester.

Statistics Grades 95 51 49 27 42 52 67 48 46

Economics Grades 88 70 65 50 60 80 68 49 40


100 -_ '. /::"o /

80 " / --

._ 60 y -

_ / _," 40 f/ "00uJ 20

........ !...... L......' ...... !...... '..'....' ...... !...... '--.-..! ...... !!

. 20 40 60 80 100 "_Statistics Grade


y = 0.59x+ 32.06 0.724


NAME DATE

16-1 Practice WorksheetGraphsDraw each graph described below.

1. G(4, 7) 2. G(5, 10)

3. V= {A,B, C,D},E = HA, B}, 4. V= {M,N, O,P},E = {{M,N},{A, C}, {C, D}} {M, P}, {N, O}, {N, P}, {O, P}}

5. 5 vertices;deg(A) = 1, deg(B) = 2, 6. 6 vertices;deg(A) = 2, deg(B) = 1,deg(C) = 1, deg(D) = 3, deg(E) = 4 deg(C) = 3, deg(D) = 1, deg(E) = 1,

deg(F) = 3


NAME DATE

16-1 Practice WorksheetGraphsDraw each graph described below.

Sample answersa_o_iV®no

1. G(4, 7) 2. G(5, 10)

3. V= {A,B, C,D},E = I{A, B}, 4. V= {M,N, O,P},E = {{M,N},

{A, C}, {C, D}} {M, P}, IN, 0}, IN, P}, 10, P}}

A M N

P 0

D

5. 5 vertices; deg(A) = 1, deg(B) = 2, 6. 6 vertices; deg(A) = 2, deg(B) = 1,

deg(C) = 1, deg(D) = 3, deg(E) = 4 deg(C) = 3, deg(D) = 1, deg(E) = 1,deg(F) = 3

B A 8

_D E D


NAME DATE

16-2 Practice WorksheetWalks and Paths zUse the graph below to determine whether each walk is a circuit, cycle, path, trail,or walk. Use the most specific name.

F _ d -_.k

1. a, h, g

2. a,b,c,j,a

3. i, c, d

4. e,d,c,i

5. k

6.[,g,h,a

For each multigraph below, list one of the paths from A to B and state its length.

7. 8. 9.A D

C d B d _2)B c

109GlencoeDivision,Macmillan�McGraw-Hill

NAME DATE

16-2 Practice WorksheetWalks and Paths

Use the graph below to determine whether each walk is a circuit, cycle, path, trail,or walk. Use the most specific name.

Ba b

A C

f

k

1. a, h, g cycle

2. a,b,c,j,a walk

3. i, c, d path

4. e, d, c, i circuit

5. k cycle

6. f g,h,a trail

For each multigraph below, list one of the paths from A to B and state its length.

Sample answers are given.

7. 8. 9.

A D' A A a

d E

(S c cC (

C d B d

B

a, c; 2 b, c; 2 a, f, h; 3


NAME DATE

16-3. Practice WorksheetEuler Paths and Circuits

Determine whether each multigraph has an Euler path. Write yes or no.

1. • 2. 3.

C4. 5. 6.

___d __e_

b c '

Determine whether each multigraph has an Euler circuit. Write yes or no.

7. 8. 9.

eaQ2__?cg f

e d e d

10. 11. 12.

b

c c


NAME DATE

16-3 Practice WorksheetEuler Paths and CircuitsDetermine whether each multigraph has an Euler path. Write yes or no.

1. yes 2. no 3. no

" "__h e

b d a

d c

4. yes 5. yes 6. no

___b d __e_bc

Determine whether each multigraph has an Euler circuit. Write yes or no.

7. yes 8. no 9. no

de _ ._

g ,

e d e d

10.yes 11.yes 12.no

Tl10Glencoe Division, Macmillan�McGraw-Hill

NAME DATE

16-4 Practice WorksheetShortest Paths and Minimal DistancesDetermine the distance and shortest path from A to Z in the graphs below by usingthe breadth-first search algorithm.

1. A G 2. A B C

M Z

3. A B C 4. A B C

D E F

DI G IG H Z

J K Z

Determine the distance from A to all of the other vertices. Then find a minimal pathand the minimal distance from A to Z by using Dijkstra's algorithm.

5. A 2 G 6. A 2 B 9 C

'/D 2 E 3 ZM 9 Z

'7. A 1 B 6 C 8. A 1 B 3 C

4 5 5 41 D F

7 7 G I

2 2

G 4 H 8 ZJ K 2 Z


NAME DATE

16-4 Practice WorksheetShortest Paths and Minimal Distances

Determine the distance and shortest path from A to Z in the graphs below by usingthe breadth-first search algorithm.

1. A G 2.

M Z D E Z

2; one path is A, T, Zo 3; one path is A, B, C, Z.

3° A. B C 4. A B C

D _F D' E_F

,i G' IG H Z

J K Z

4; one path is A, B, E, H, Zo 5; A, B, C, E, hi.,Z

Determine the distance from A to all of the other vertices. Then find a minimal pathand the minimal distance from A to Z by using Dijkstra's algorithm.

5. A 2 G 6. A 2 B 9 C

M 9 Z D 2 E 3 Z

9; A, G, Z 9; A, B, E, Z

7. A 1 B 6 C 8. A 1 B 3 C

4 5 5 4c_2 -

1 E_' 2_ __ D _-._""'-""_ F

DI7 _j _4

7 G' I2 2

G 4 H 8 ZJ K 2 Z

14; A, B, E, F,Z 11; A,B, C, E, H,Z

TlllGlencoe Division,Macmillan�McGraw-Hill

NAME DATE

16-5 Practice Worl s| eetTrees

Determine whether each graph is a tree. Write yes or no. If, no explain.

1. 2. 3.

Solve.

4. How many vertices are there in a 5. How many edges are there in a tree

tree with 25 edges? with 20 vertices?

Find a spanning tree for each graph.

6. c D 7. A, 8°A I

I\'\ .\\

B _ L"_-'_H

A F B C O E O E F G

9. B 10. A B 11. A B C D E F

H C L _.. N_._._/O._O_ G

G E O K J I HJ I H G F

Find a minimal spanning tree for each weighted graph. State the weight of the tree.

12. 13. 14.

BL 3 C 3 D A 1 B 1 C_ D"-A

4 3 2 134 3

JI--Z-2r--E-sL4---Z--1E G DA E 4 5 12

_ _------_ F _6 .._I 1 H 1 G 1 FF E


NAME DATE

16-5 Practice Worksl leetTrees

Determine whether each graph is a tree. Write yes or no. If, no explain.

x. 2. yes 3. yes

I "

No; there is more than onepath between any two vertices.

Solve.

4. How many vertices are there in a 5. How many edges are there in a treetree with 25 edges? with 20 vertices?26 19

Find a spanning tree for each graph. Answers may vary.6. C D 7 A, 8. A I

_ B,_ _

I F/_ c,_ L_._HB K____ _

A F B C D E D E F G

9. B 10. A B 11. A B C O E F

H, _ I / C L G

I

G E D K J I HJ I H G F

Find a minimal spanning tree for each weighted graph. State the weight of the tree.

12. 17 13. 18 14. 25B3 C 3 D AIB 1 Cy D-

A

3 5 C4 3 24 3

j 2 K 5 L.___4 E G_3 ___D

A E 3 4 5 2

I 1 H 1 G 1 FF E

Tl12Glencoe Division,Macmillan/McGraw-Hill

NAME DATE


Graphs and MatricesFind M(G) for each graph.

1. A B 2. F

G ,C D

Find M(G) for each digraph.

3, j K 4o N 0

J

Draw the graph for which each matrix is M(G).DEF

° o.;[,o1_[o11] _.oB 1. 11

JKLM

HG/I 10 i] K 1011

10 L 020101 M 1111

J

71

i' 113GlencoeDivision,Macmillan/McGraw-Hill

NAME DATE

15-5 Practice WorksheetGraphs and MatricesFind M(G) for each graph.

1. A B 2" F

G'C D

_EFGHI_ABCD £ 011 01

i[OlO] OOl1 01 1 G 11 01 00101 H 00101111 0 I_11 01 0_

Find M(G) for each digraph.

3. 4.J K N 0

• Q P

_NOPQN_JKLM N 011 01

ooI ooooo0010 P 010002000 Q 1010101 01 R_I 0000_

Draw the graph for which each matrix is M(G). Sample answers are given.DEF

A[0B111] 20A 11

JKLM F

G[I 01] _H K 1011

H 100 L 0201I 010 M 1111

IJ

I M °

Tl13 _tGlencoe Division, Macmillan�McGraw-Hill

NAME DATE

17-1 Practice WorksheetLimitsEvaluate each limit.

1. lim (x2+3x - 8) 2. lim (2x + 7)x_3 x_O

(1) ( 1)3. lim 4n-_nn 4. lim 3"6n+2"_nnn ---_0 n--_O

5. lim x_ - 25 6. lim x2 - 5x + 6x --->5 x-5 x --->2 x-2

x 2 - 9

x - 2 8. lim _-_7. lim _-_x_2 x_3

9. lim N/_x 2 + 1 10. lim N/xx2 - 2x + 1x_2 x_4

Evaluate the limit of f(g(x)) as x approaches 2 for each f(x) and g(x).

11. f(x) = 3x - 1 12. f(x) = 2x - 6g(x) = 2x + 5 g(x) = 3x + 2

13. f(x) = x 2 + 1 14. f(x) = 3x + 1g(x) = x + 7 g(x) = 2X 2 -- 1


NAME DATE

17-1 Practice WorksheetLimitsEvaluate each limit.

1. lim (x2+3x - 8) 2. lim (2x + 7)x-o3 x-_O

10 7

3. lim (4n-1) ( 1)n--->O " _n 4. lim 3"6n+2"_n nn --->0

0 5

5. lim x2 - 25 6, lim x2 - 5x + 6x_5 x-5 x--->2 x-2

10 -1

x-2 x 2 -97. lim __Z: t 8. lim _2_ 7x_2 x-*3

1 24 9

9. lim _+ 1 10. lim V_x2-2x+ 1x-->2 x_4

3 3

Evaluate the limit of f(g(x)) as x approaches 2 for each f(x) and g(x).

11. f(x) = 3x - 1 12. f(x) = 2x - 6g(x) = 2x + 5 g(x) = 3x + 226 10

13. f(x) = x2 + 1 14. f(x) = 3x + 1g(x) = x + 7 g(x) = 2X 2 -- 182 22

Tl14Glencoe Division, Macmillan/McGraw-Hill

NAME. DATE

17-2 Practice WorksheetDerivatives and Differentiation TechniquesFind the derivative of each function.

1. f(x) = 2x 2 - 3x 2. f(x) = 6x 3 - 2x + 5

3. f(x) = 3x z + 4x 5 - 2x 2 4. f(x) = 3x 15 - 12x 1° + 7x 2 - 8

5. f(x) = (2x + 7)(3x - 8) 6. f(x) = (x2+1)(3x - 2)

7. f(x) = (x 2 + 5x) 2 8. f(x) = (x 2 -2x + 1)3

9. f(x) = x 2 (x 3 + 3x 2) 10. f(x) = (x 2 + 2x)(x 2 +7x)

U. f(x) = x 2 (x 4 + 2x) 2 12. f(x) = (x 2 + 3x)(x 2 + 2x) 2

x213. f(x) - x 1 14. f(x)- x2 - 3x + 1- 2x - 9

15. f(x) = _/2x2+7x - 8 16. f(x) =_ + 1


NAME DATE

17'2 PracticeWorksheetDerivatives and Differentiation TechniquesFind the derivative of each function.

1. f(x) = 2x2 - 3x _2. f(x) = 6x 3 - 2x + 54x- 3 18x2 - 2

3. f(x) = 3x7 + 4x 5 - 2x 2 4. f(x) = 3x15- 12x 1° + 7x2 - 8-21x 6 +20x 4 - 4x 45x 14- 120x9 + 14x

5. f(x) = (2x + 7)(3x - 8) 6. f(x) = (x2+l)(3x - 2)12x + 5 9X 2 - 4x + 3

7. f(x) = (x2 + 5x) 2 8. f(x) = (x2 -2x + 1)34x3 + 30X 2 -t- 50X 6x5 - 30x4+ 60x3- 60x2 +

30x- 6

9. f(x) = x 2 (x3 + 3x 2) 10. f(x) = (x2 + 2x)(x2 +7x)5x4 + 12x3 4x3 + 27x 2 + 28x

11. f(x) = x 2 (X 4 + 2X) 2 12. f(x) = (x2 + 3x)(x 2 + 2x) 210x9+ 28x6 + 16x3 6x 5 + 35x4 + 64x3 + 36x 2

x2 x2-- 3x + 113. f(x) -- x -- 1 14. f(x) -- 2X- 9

X2 -- 2x 2x2 - 18x + 25(x - 1)2 (2x - 9)2

15. f(x)=_+7x-8 16. f(x)=_+1

,4x + 7 ' 2x

2 _ +Tx- 8 3 _/(x 2 + 1)2


NAME DATE

17-3 Practice WorksheetArea Under a Curve

Write a limit to find the area between each curve and the x-axis for the giveninterval Then find the area.

1. y = x 2 from x = 1 to x = 6 2. y = x 3 from x = I to x = 5

3. y=x gfromx=ltox=3 4. y=x 5fromx=2tox=5


NAME DATE

17-3 Practice WorksheetArea Under a Curve

Write a limit to find the area between each curve and the x-axis for the giveninterval. Then find the area.

1. y = x 2 from x = 1 to x = 6 2. y = x 3 from x = 1 to x = 5

_ ,,_2 _o'/_o_- _=,,_2 _'/__/n--_i= 1 in/ i n ) -->_i= 1 In/ in/n #7

lim _ /i _2(, _. lim r(/_3(!_ •n_<_i= 1 \_) t--_)' n___>.i_= l t n) tn)'

2square units 156 square units71

3. y = x 4 from x = 1 to x = 3 4. y = x5 from x = 2 to x = 5

n / Z (_)3(5)A = Iim _ 3i/4(3/- A =/im -n-->_i= 1 --hi # \--hi # --->_i=1

n(n)(n) (i 4 1 2ilim, lim n ;n -->_ i 1 n --> OOl 1"z

482 square units 25931 square units

Tl16Glencoe Division, Macmillan/McGraw-Hill

NAME DATE

17'4 Practice WorksheetIntegrationFind each integral.

1. |8 dx 2. _(2x + 6)dxJ

3. f(6x2- 12x+ 8)dx 4. f(9x2+ 12x- 9)dx

7. f(5 - x) 6 dx 8. f(3x - 1)4 dx

9. fx--_dx 10. fx3__4dx

11. f3x2 N/-_x3+Sdx 12. fx _x2 - 5dx

flO f 413. -7 dx 14. fi dx


NAME DATE

17-4 Practice WorksheetIntegrationFind each integral

1. f8 dx 2. f(2x+ 6)dx

8x + C x2 + 6x + C

a. f(6x2- 12x+s)dx 4. f(ex2+12x- 9)dx

2x3 - 6x2 + 8x + C 3x3 + 6x2 - 9x + C

5. f(x -= 2) 10 dx 6. f2(2x - 3) 4 dx

11 5

7. f(5- X) 6dx 8. f(3X- 1) 4dx

_ (5-x)7 + C (3x-1)5 + C7 15

,. f;-_dx lO. fx3_4dx

In Ix + 21+ C 3 In Ix- 41+ C

11. f3x2x/fix3+5dx 12. fx _-_x2 - 5dx

2 _//(X 3 + 5)3 + C 3 _X 2 _ 5)4. + .C

flO , f413. z dx 14. -fi dx

lo +C -_-+C.X

Tl17GlencoeDivision,Macmillan�McGraw-Hill

NAME DATE

17-5 Practice WorksheetThe Fundamental Theorem of CalculusUse integration to find the area of each shaded region.

1. 2.

27

18

9

3 x

y= x 3

Evaluate each definite integral

3. 2x dx 4. (x + 1)adx9.

5o (4X 3 -- 3x 2) dx 6. (2x 3 - 6x 2 _- 7)dx1

7. _ (x - 1)(x +3)dx 8. (2x + 1)(2x - 1)dx2 3

29. fo 2x(x2 + 2)2 dx 10. 1 (3x2 + 2x)(x3 + x2)2 dx


NAME DATE

17-5 Practice WorksheetThe Fundamental Theorem of Calculus

Use integration to find the area of each shaded region.

1. 2.

27

18

9

_ 3x

y= x3 ,--27"

1 401 sq, units1-5 sq, unitsEvaluate each definite integral

s: s'3. 2x dx 4. (x + 1)3dx2

321 33

5. (4X 3 -- 3x 2) dx 6. (2x 3 - 6x 2 + 7)dx1

1072 32½

7. (x - 1)(x +3) dx 8. (2x + 1)(2x - 1) dx2 3

2-9 41-_

s: s:9. 2x(x 2 + 2)2 dx 10. (3x 2 + 2x)(x 3 + x2) 2 dx1

2691 2-_


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