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Contents1-1 RelationsandFunctions.....................1l-2 Composition of Functions ..................3l-3 Graphing Linear Equations ................5l-4 Writing Equations ............7

1-5 Writing Equations of Parallel and

Perpendicular Lines ........................9l-6 Modeling Real-World Data with

Linear Functions ......'..11l-7 Piecewise and Absolute Value

Functions ................'....131-8 Graphing Linear Inequalities ..........'15

Answers ..17'20

2-l Solving Systems of Equations inTwo Variables ............ ....................21

2-2 Solving Systems of Equations inThree Variables ............23

2-3 Using Matrices to ModelReal-World Data ........ ...................25

2-4 Modeling Motion with Matrices ......27

2-5 DeterminantsandMultiplicativeTnverses of Matrices ......................29

2-6 Solving Systems of Inequalities ......31

2-7 Linear Programming ........................33Answers ..35-38

Symmetry and Coordinate Graphs ...39

Families of Graphs.. ........41

Graphs of Nonlinear Inequalities......43Inverse Functions and Relations .......45

Continuity and End Behavior ...........47

Critical Points and Extrema ..............49Graphs of Rational Functions ..........51

Direct, Indirect, and Joint Variation..53Answers ..55-58

Polynomial Functions .....59

Quadratic Equations .......61

The Remainder and Factor Theorems63The Rational Root Theorem .............65Locating Zeros of a PolynomialFunction .....67

Rational Equations and PartialFractions... ....................69

4-7 Radical Equations and Inequalities..'71

4-8 Modeling Real-World Data withPolynomial Functions.....................73

Answers ..75'78

3-13-2J-J

3-43-s3-63-73-8

4-l4-24-34-44-5

4-6

5-8

5-45-55-05-7

6-l6-26-3

o-5

6-6

7-57-67-7

5-1 Angles and Degree Measure.............795-2 Trigonometric Ratios in Right

Triangles... ....................815-3 Trigonometric Functions on the

Unit Circle ....................83Applying Trigonometric Functions.....85Solving Right Triangles .................... 87The Law of Sines ............89The Ambiguous Case for the

Law of Sines......... ........91The Law of Cosines ........93

Answers ..95-98

Angles and Radian Measure .............99Linear and Angular Velocity ........... 101

Graphing Sine and CosineFunctions.. ..................103

Amplitude and Period of Sine andCosine Functions.. ......105

Horizontal and Vertical Shifts ofSine and Cosine Functions.. .........107

Modeling Real-World Data withSinusoidal Functions .................... 1 09

Graphing Other TrigonometricFunctions.. ..................111

Trigonometric Inverses and TheirGraphs ......113

Answers ................115-118

7-l Basic Trigonometric Identities........1197-2 Verifying Trigonometric Identities .121

7-3 Sum and Difference Identities ........1237-4 Double-Angle and Half-Angle

Identities... ......,.,.........125Solving Trigonometric Equations ...127

Normal Form of aLinear Equation 129

Distance from a Point to a Line......131Answers ................133-136

Geometric Vectors ........137

Algebraic Vectors...... ....139

Vectors in Three-DimensionalSpace ........141

Perpendicular Vectors ...143Applications with Vectors ............... 145

Vectors and Parametric Equations ...147

Modeling Motion Using ParametricEquations..................... ..................149

Transformational Matrices inThree-Dimensional Space ............ 151

Answers ................153-156

6-4

6-7

6-8

8-18-28-3

8-48-58-68-7

til

8-8

9-1o_t

9-39-49-59-6

9-7

9-8

Polar Coordinates...... ....I57Graphs of Polar Equations .............. 159Polar and Rectangular Coordinates ..161Polar Form of a Linear Equation ....163Simplifying Complex Numbers...... 165The Complex Plane and Polar

Form of Complex Numbers .........167Products and Quotients of Complex

Numbers in Polar Form......... .......169Powers and Roots of ComplexNumbers... .,................171

Answers ................173-176

Introduction to Analytic Geomehy ...17 7Circles .........I79Ellipses..... .....................181Hyperbolas ....................183Parabolas.. ...185Rectangular and Parametric

Forms of Conic Sections .............. l87Transformations of Conics .............189Systems of Second-Degree

Equations and Inequalities ...........191Answers ................193-1gf

Real Exponents ........... ........ :......... -.1g7Exponential Functions .-.................. l9!gThe Number e........... ....20fLogarithmic Functions....................1t3Common Logarithms "...:OsNatural Logarithms .."----ffiModeling Real-World Data u-ith

Exponential and LogarithmicFunctions.. .............. -D

Answers ................21l-tfl

l2-l Arithmetic Sequences and Series ...Zlsl2-2 Geometric Sequences and Series....217l2-3 Infinite Sequences and Series. ........219l2-4 Convergent and Divergent Series ...221l2-S Sigma Notation and the nth Term ...22312-6 The Binomial Theorem ......,............225l2-7 Special Sequences and Series .........227l2-8 Sequences and Iterations ............. ...229l2-9 Mathematical Induction............... ...231

Answers ................233-237

Permutations and Combinatio ns .....239Permutations nirh Repretitions and

C ircular P;rmulenr.n_i ............. .....241Probabrliqr an-j OJds ....243Probabilitis or C..mu:nd Erents ...245Conditirrnar Prfuunu-L*ia:

" ". ".. .. . . . .. ...2 47hrH mdmm .......249,h 251-2s3

l&'l UQ;ury Munmr:trrjon............255ISE lF+pdr"q;irilrinmn Ten*Jencv.. ......257f,}il il .-dniF m*mm:ri .........'..........25gl{*l Mffiftmrmnl,rrion ................261il"fi xtrffrm,umrnmm .....263, r .....265-267lfn liro .- ............269Er! ffinnmAmmnrfuriratir-es ......2716fl fmtlfrou6 e.--..-.. ................273Gil ffionnwl Tno.='rem of

ffi** ...............27sh 277-278

10-1t0-210-310-410-5t0-6

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aNAME

Study Guide

Retotions ond FunctionsA relation is a set of ordered pairs. The set of first elementsin the ordered pairs is the domain, while the set of secondelements is the range.

Example I State the domain and range of the following relation.l(5,2), (30,8), (15,3), (17,6), (14' 9)lDomain: 1.5,14,15, 17, 30)Range: 12,3,6,8,9)You can also use a table, a graph, or a rule torepresent a reiation.

Example 2 The domain of a relation is all odd positive integersless than 9. The range y of the relation is 3 more tham x,where r is a member of the domain. Write the relationas a table of values and as an equation. Then graph therelation.Table: Equation:

y:x*3

A function is a relation in which each element of the domain ispaired with exactly one element in the range.

Example 3 State the domain and range of each relation. Thenstate whether the relation is a function.a. l?2,1), (3, -1)' (2' 0)l

The domain is {-2,2, 3} and the range is {-1, 0, 1}.

Each element of the domain is paired with exactly oneelement of the range, so this relation is a function.

b. {(3, -1), (3, -2), (9, 1)}The domain is {3,9}, and the range is {-2, -1, 1}. In thedomain, 3 is paired with two elements in the range, - 1

and -2. Therefore, this relation is not a function.

Exampte 4 Evaluate each function for the given value.a. f(-l) if f(r) :ZxB + 4$2 - 5x

f(-1) :2(-1)B + n(-t)' - 5(-1) x: -7--2+4+5or7

b' g(+) if g(r) : x4 - 3x2 + 4

. g(4): (4)a - 3(4)2 + + x:4:256-48+4or2I2

DATE PERIOD

._J

,

x v1 4

3 b

5 8

7 10

O Glencoe/McGraw-Hill Advanced Mathematical Concepts

Retotions ond FunctionsStafe the domain and range of each relation.the relation is a function. Write yes or no.

1. {(-I,2),(3,10),(-2,20), (3, 11)} 2.

Then sfifth

{(0,2), t13,filUir,'lr0&,,$-" t$- -

3. {(1, 4),(2,9), (3, 24)} 4. Ie 1, -2ru rffifril1fl5lirt-S" - ].6,. (3, 81)]

5. The domain of a relation is all evennegative integers greater than -9.The range y of the relation is the setformed by adding 4 to the numbersin the domain. Write the relation asa table of values and as an equation.Then graph the relation.

Evaluate each function for the given value.6.f(-2)if f(x):4x3 + 6x2 + 3x 7. f(3)if frxr =W-&,-'i

8. h(t) if h(x) : 9xe - 4xa * 3x - 2 9. f(S + trillqilrurlll :# - g* - 1

lO. Climate The table shows record high andlow temperatures for selected states.a. State the relation of the data as a set of

ordered pairs.

b. State the domain and rangeof the relation.

c. Determine whether the relation is a funcrion-

h tiltilithm$rrlllllmmM lme ler:er

il

2

Hllrd |,.ffi Ternoeratures ( "F)n ttEtr Lowfriffi "2 -27M -':

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4flua*/::e.l t,{ athematical Concepfs

lDATE PERIOD

b. (f - eXr)(f - e)6): f(x) - s@):x2-x-6-(x+2):x2-2x-8

Study Guide

Composition of FunctionsOperations of I Two functions can be added together, subtracted,Functions I multiplied, or divided to form a new function.

Example 1 Given f(x):x2 -x -Gandg(r): x*Zrfindeachfunction.

a. (f + gXr)(f +e)@):f(x) +g(r)

:x2-x-6*x*2:x2-4

c. (f 'gXr)(f ' s)@): f(x) 'g(x)

:(x2-x-61(x-l 2):x3+x2-Bx-12

a. (ur)r"r

(r\rxt :\b/

:

J

f(x)g(x)x2-x-6

x -12(x - 3)(x+ 2)

x-l 2:x-3rx*-2

Functions can also be combined by using composition. Thefunction formed by composing two functions f andg is calledthe composite of / and g, and is denoted by f " g.lf 'gl(r) isfound by substituting g(r) for x in f(x).

Example 2 Given f(x) : 3x2 + ?.x - L andg(r) : 4x + 2,find tf " gl(r) and [9 " f](r).

lf"sl@):f(g(x)): f(4x + 2) Substitute 4x * 2 for g(x).:3(4x +2)2 +2(4x+ 2) - I Substitute4x + 2for xin f(x).: 3(16x2 * 16r + 4) + 8x + 4 - |: 48x2 + 56r + 15

ls"fl@):e(f@)): g(3x2 + 2x - l) Substitute 3f * 2x - 1 for f(x).: 4(3x2 * 2x - I) + 2 Substitute 3x2 * 2x - 1 for x in g(x).:12x2 -l 8x - 2

Adv anced M athe m ati cal Co nce?ts@ Glencoe/McGraw-Hill 3'

Practice

6. f(*) : 2xB - 3r2 -8(r) : 3-r

8, .f(x):3x,2 - 2x - 5g(x):2x - I

:== ODDA-E

Given f (x) :2x2 + 8 and g(x) :5x - 6, find each function,1. (f + gXr) 2. (f -sxr)

3. (f'sxr)

Find [f . g](x) and [g " f ](x) for each f (x) and g(x),

Composition of Functions

5.f(*'l:r*5g(x'): x - 3

7. f(*):2x2 - 5r * 1

g(x):2x - 3

+. (rr\r.t

9. State the domain of lf . Sliu-) for fk\ : f; - 2 andg(r) : 3.r.

Find the first three iterates of each function using the giveninitial value.lO. f@) : 2x - 6; xo: 7 Il, f(*) : x2 - L; xo: 2

t2. Fitness Tara has decided to start a walking program. Herinitial walking time is 5 minutes. She plans to double herwalking time and add 1 minute every 5 days. Provided thatTara achieves her goal, how many minutes will she be walkingon days 2l throug}'25? o

4@ Glencoe/McGraw-Hill Adv an ced A/l athe m ati cal Co n cepts

Study Guide

Grophing Lineor EquotionsYou can graph a linear equation Ax + By + C :0, whereAand B are not both zero, by using the x- andy-intercepts. Tofind the r-intercept,let y : 0. To find the y-intercept, let r : 0.

Example 1 Graph 4x +y - g:0usingther-andy-intercepts.Substitute 0 for y to find the r-intercept. Thensubstitute 0 for r to find the y-intercept.

r-intercept4x+y-3:04x*0-3:0

4x - 3:04x: B

o-_ o&-

4

y-intercept4x*y-3:0

4(0)+y-3:0y-3:0

^,-t.Y-.)

PERIOD

Adv an ced M athe m ati c al Conce pts

1ifr

t"1

The line crosses the r-axis .t (?, O) ana the y-axis at (0, B).

Graph the intercepts and draw the line that passes throughthem.

The slope of a nonvertical line is the ratio of the change inthe y-coordinates of two points to the corresponding change inthe *-coordinates of the same points. The slope of a line canbe interpreted as the ratio of change in the y-coordinates tothe change in the r-coordinates.

SlopeThe slope m ot a line through two points (x, y,) and E, y) is given by

v^-v.X^-X.

Example 2 Find the slope of the line passing throughA(-3,5) and 8(6,2),

lo - !.m: ' -xz- xt

2-5 r,: 6ft1 Let x,::+ot-+

: 5, fi2: 6, and y,

O Glencoe/McGraw-Hill

NAME

Practice

Grophing Lineor EquotionsGraph each equation using the x- and y-interceI.2x-y-6:0 2.4x*

\-

Graph each equation3'Y:5*-i

using the y-intercept and then. r: t*

Find the zero of each function.

5.f(x):4x-BThen graph the function.

6.f(*):2xi4

7. Business In 1990, a two-bedroom apartment at RemingtonSquare Apartments rented for $575 per month. In 1999, thesame two-bedroom apartment rented for $850 per month.Assuming a constant rate of increase, what will a tenant payfor a two-bedroom apartment at Remington Square in theyear 2000?

6 Advan ced M ath em at i cal C o n ce pts@ Glencoe/McGraw-Hill

.ffiffi*,, -,*im;flffi1i",

I

NAME DATE PERIOD

Study Guide

Writing Lineor EquotionsThe form in which you write an equation of a line dependson the information you are given. Given the slope andy-intercept, or given the slope and one point on the line, theslope-intercept form can be used to write the equation.

Example 1 Write an equation in slope-intercept form for each linedescribed.

a. a slope of f and a y-intercept of -5Substitut" ! to, m and -5 for b in the general

slope-intercept form.

y:mx+b ---> y:2{- s.

The slope-intercept form of the equation ofthe line is y : ?, - 5.

b. a slope of 4 and passes through the pointat (_2,3)Substitute the slope and coordinates of thepoint in the general slope-intercept form of alinear equation. Then solve for b.

!:mx*b3 : 4(-2) + b Substitute -2 for x, 3 for y, and 4 for m.

ll : b Add B to both sides of the equation.

The y,intercept is 11. Thus, the equation for the line is y : 4x +Ll.

When you know the coordinates of two points on a line,you can find the slope of the line. Then the equation of theline can be written using either the slope-intercept or thepoint-slope form, which is y - !1: rn(x - x).

Example 2 Sales In 1998, the average weekly first-quartersales at Vic's Hardware store were $SZSO. In 1999,the average weekly first'quarter sales were$10,100. Assuming a linear relationship, find theaverage quarterly rate of increase.

Since there are two data points, identify the twocoordinates to find the slope of the line.Coordinate 7 represents the fi,rst quarter of 1998and coordinate 5 represents the first quarter of1999.

(1, 9250) and (5, 10,100)!z- !trn:

-xz- xt

: 10.100 - 92505-1

: ry or 212.5

Thus,

@ Glencoe/McGraw-Hill

for each quarter, the average sales

'7increase was $212.50.

Adv an ce d M athemati cal C o n cePts

PracticeWriting Lineor EquotionsWrite an equation in slope-intercept form for

1. slope - -4,y-intercept : 3

3. slope - -4, passes through B(3, 8)

5. slope : 1, passes through D(-6,6)

7. slope = 3, y-intercept : ?

9. slope - - 1, passes through F(-L,7)

LL. Aaiation The number of active certifiedpilots has been declining since 1980, as shown inthe table.a. Find a linear equation that can be used as a

to predict the number of active certifiedpilots for any year. Assume a steady rate of

b. Use the model to predict the number of pilotsthe year 2003.

LA(-3,2)

c(-9,4)

E(3, -3)

G(3,2) t

\*,

8

h

@ Glencoe/McGraw-Hill Htematical Concepts

Study Guide

O Writing Equotions of Porottet ond Perpendicutor Lines. TWo nonvertical lines in a plane are parallel if and only if

their slopes are equal and they have no points in common.

. Graphs of two equations that represent the same line are

said to coincide.. Two nonvertical lines in a plane are perpendicular if and

only if their slopes are negative reciprocals'

Example I Determine whether the graphs of each pair ofequations are parallel, coinciding, ot neither'a. b - 3y:5 b. l?.x * 6Y:19

6x - 9Y :11 4x:-2yt6Write each pair of equations in slope-intercept form.

a. 2x - 3Y:5 6x - 9Y:21", _ 2"" _ 5 n, _ z* _ 7J-Et 3 t 3* 3

The lines have the same sloPe

but different y-intercePts, so

they are parallel.

b. l2x * 6y: 19

!:-2x+34x: -2y * 6

!:-2x*3The equations are identical, so thelines coincide.

Example 2


Write the standard form of the equation of theline that passes through the point at (3, -4) andis parallel to the graph of r + 3y - 4: O.

Any line parallel to the graph of x 1- 3y - 4 : 0

will have the same slope. So, find the slope of thegraphofx-rBy-4:0.

Am: -j__I a:t B:S- -5 r-L - -L)

Use the point-slope form to write the equation ofthe line.

! - !t: nl(x - x)

y - (-4): -]t" - e) x1: 3, ! t: -4, * : -!,/ly+4:-tx-t1-

3y + 12: -x t 3 Multiply each side by 3.*

x + 3y * 9:0 Writein standardform.

Advanced Mathematical ConcePts

Practice

5.4x't 8y :22xt4y:g

4.x!J,:63r -r':6

6. 3r -J':96r - 2r': 18

writing Equotions of Porottet ond perpendicutor LinesDetermine whether the graphs of each pair of equations areparallel, perpendicular, coinciding, or none of these.l. x't 3y : 18

3x-r9y:12 2'2x-4Y:Bx-2Y:{

3. -3r 1- 2y :62xI3y:12

write the standal fory of the equation of the line that is parailerto the graph of the given equatio,n and thatpasses through thepoint with the given coordinates.7.2xt-y-5:0; (0,4) 8. Br -y*B:0; (-L,-2) 9. Br _2y+g:0; (2,5)

write the standard form of the equation of the line that isperpendicular to the graph of the given equation and thatpassesthrough the point with the given coordinates.lO.2x-y+ 6:0; (0,-g) ll.2x-Ey-6:0; (- 4,2) 12.Sx+4y- 1S:0; (2,7)

13. Consumerism Marillia paid $1'80 for B video games and 4books. Three monthslater she purchased 8 books and 6 video gr-"..-Her brother g"";"a that shespent $szo. Assuming that the pricesof iideo j*". and bools did not change,is it possible that she spent $szb for the ,".orri set of purchases? Explain.

\, _t

@ Glencoe/McGraw-Hill 10 Advan ced M athe mati cal C o n ce pts

l_.

When real-world data are collected, the data graphed usually do notform a straight line. However, the graph may approximate a linearrelationship.

Example The table shows the amount of freighthauled by trucks in the United States. Usethe data to draw a line of best fit and topredict the amount of freight that will becarried by trucks in the year 2010.

Graph the data on a scatter plot. Use the year as theindependent variable and the ton-miles as the dependentvariable. Draw a line of best fit, with some points on the lineand others close to it.

o rge6 19BB 1990

Year

Write a prediction equation for the data. Select two points thatappear to represent the data. We chose (1990, 735) and (1993, 861).

Determine the slope of the line.

pr:!z-tt * ,991 -Tll^ :r?6 or421993 - 1990 3

IJse one of the ordered pairs, such as (1990, 735), and theslope in the point-slope form of the equation.

t - lt: m(x - xt)y - 735: 42(x - 1990)

!: 42x - 82,845

A prediction equation is y : 42x - 82,845. Substitute 2010 forr to estimate the average amount of freight a truck will haulin 2010.

!:42x-82,845y:42(2010)-82,845y :1575

According to this prediction equation, trucks will haul1575 billion ton-miles in 2010.


NAME DATE PERIOD

Study Guide

Modeting Reot'Wortd Doto with Lineor Functions

Amount(billions ofton-miles)

Adv anced M athe m ati cal C o n ce pts

1 000

800

600

400

200

1 994

,11

U.S. Truck Freight Traffic

YearAmount

(billions ofton-miles)

1 986 oJz

1 987 663

1 988 700

1 989 716

1 990 735

1 991 758

1992 815

1 993 861'1994 908

1 995 921

Source: Ironsporlolian in Amerbo

Modeting Reot-Wortd Doto with Lineor Functions

Complete the following for each set of data.a. Graph the data on a scatter plot.b. Use two ordered pairs to write the equation of a best-fit line.c. lf the equation of the regression line shows a moderate or

strong relationship, predict the missing value. Explain whetheryou think the prediction is reliable.

U. S. Life Expectancy

BirthYear

Number ofYears

1 990 75.4

1 991 75.5

1992 75.8

1 993 75.5

1 994 75.7

1 995 75.8

2015 ?

Source: National Centerfor Health Statistics

75.9

75.875.7

75.6

75.575.4

PERIOD

Practice

t

a.1.

Numberof Years

Population(millions)

1 9951 990 1991 1992 1993 1994

Bidh Year

1 992

I

a.2.275

270

265

260

255

250

01 9961 990 1 994

Year

1 998

Population Growth

YearPopulation(millions)

1 991 252.1

1992 255.0'1993 257.7

1 994 260.3

1 995 262.8

1 996 265.2

1 997 267.7

1 998 270.3

1 999 272.9

2010 2

Source: U.S. Census Bureau

@ Glencoe/McGraw-Hill 12 Advanced Mathematical Concepts

{.-'

DATE

Study Guide

Piecewise Functions

Piecewise functions use different equations for differentintervals of the domain. When graphing piecewise functions,the partial graphs over various intervals do not necessarilyconnect.

Example I

First, graph the constant function fG) - - 1 fot x '< -3' Thisgraphis a horizontal line. Because the point at (-3, -1) isincluded in the graph, draw a closed circle at that point.

Second, graph the function f(x): 1 + x for -2 I x = 2'

Because x : -2 is not included in this region of the domain,draw an open circle aL (-2, - 1). The value of x : 2 is includedin the domain, so draw a closed circle at (2,3) since for

f(x):t+x,f(:2):3.Third, graph the line f (x) : 2x fot x ) 4. Draw an open circle

at (4,8) since for f(x) : 2K, f(4) : S.

A piecewise function whose graph looks like a set of stairs isruil"d a step function. One type of step function is thegreatest integer function. The symbol [x] means fhe

greatest integer not greater than r. The graphs of step

iunctions are often used to model real-world problems such as

fees for phone services and the cost of shipping an item of agiven weight.

The absolute value function is another piecewise function.Consider f(x) : lr | . The absolute value of a number isalways nonnegative.

Example 2 Graph f(x) : Zl"l - Z.

Use a table of values to determine points on the graph.

X zlxl - z (x, f (x))

,4 21,41 - 2 (-4, 6)

-J 2l-31 - 2 (-3, 4)

- t-J 2l-1.51 - 2 (- 1 .5, 1)

0 2l0l -2 (0, -2)1 2l1l -2 (1, 0)

2 2l2l -2 (2,2)


[-1 ifr=-BGraph f(x) :l 1 + x if -21x=2

lzx ifx>4

PERIOD

13 Adv an ced M athemati c a/ ConcePts

NAME @mm !E: OD

f-grr< -?2.f,*:f r-rff-fi<.r-l[r -sffs > 3

4. f(*t: hil - !

t_

6. f(x): [Zr - n]

7. Graph the tax rates for the different incomes by using a step function.

50

40

Tax 30Ratee/") 20

10

0

60 90 120 150 180 210 240 270 300

Taxable lncome(in thousands)

I

PracticePiecewise FunctionsGraph each function.

Ititx=zl.fk):lrif-1<x<2L-"-3ifr<-2

3. f{x): l"l - g

5. fk) : Slxl - Z

lncome Tax Rates for aGouple Filing Jointly

Limits of TaxableIncome

TaxRate

$o to $41,200 15%

$41,2o1to $99,600 28%

$99,601 to $151,750 31%

$151 ,751 to $271 ,050 36%

$271,051 and up 39.60/o

Source: lnformation please Almanac


NAME DATE

Study Guide

Grqphing Lineor lnequotitiesThe graph of y : -ly + 2 is aline that

separates the coordinate plane into tworegions, called half planes. The line

described by y : -], + 2 is called the

boundary of each region. If the boundaryis part of a graph, it is drawn as a solid line.A boundary that is not part of the graph isdrawn as a dashed line.

The graph of y > -fy + 2 is the region above

the line. The graph of y < -ly + 2 is the region

below the line.

You can determine which half plane to shade by testing apoint on either side of the boundary in the original inequality.If it is not on the boundary, (0, 0) is often an easy point totest. If the inequality is true for your test point, then shade

the half plane that contains the test point. If the inequality is

false for your test point, then shade the half plane that does

not contain the test point.

Example Graph each inequalitY.

PERIOD

a.x-y+2=Ox-y+2=0

-^r< -x-2J-y>x*2

The graph doesinclude the boundary,so the line is solid.Testing (0, 0) in theinequality yields afalseinequality,0>2.Shade the half planethat does not include(0, 0).

Reuerse the inequalitywhen you diuide ormultiply by a negatiue.

b. y > l' - r I

Graph the equation with adashed boundary. Thentest a point to determinewhich region is shaded.The test point (0, 0) yieldsthe false inequality 0 > 1,

so shade the region thatdoes not include (0, 0).

O Glencoe/McGraw-Hill Advanced M athe m ati cal C o n ce Pts

6.y=-|x+a

8. -4<r(-2y<6

PERIODDATE

PracticeGrophing Lineor lnequotitiesGraph each inequality.

l.x> -2 2.y<-2x-4

3.y>3x+2

5.y>l*-21

1.-3<y <fx+4

4.y<lr+31

@ Glencoe/McGraw-Hill 16 Adv an ced M ath e mati cal Con ce pts

I

DATE PERIOD

y:3x-L0,:-L;r+3

Example 2Use the elimination method to solvethe system of equations.2x - 3Y: -2L5x * 6Y :15To solve this system, multiply each sideof the first equation bY 2, and add thetwo equations to eliminate y. Then solve

the resulting equation.2(2x - 3y):2(-2I) --> SP - 6Y : -424x-6Y:-425x*6y: 15

9x : -27c)

-t - -uNow substitute -3original equations.

5x * 6y :155(-3) -t 6y :15

6y:30y:5

The solution is (-3, 5).


The solution to the system is @,2').

for r in either of the

o:X. - -a)

Study Guide

I Sotving Sustems of Equotions in Two VoriobtesOne way to solve a systern of equations in two variables is bygraphing. The intersection of the graphs is called the solutionto the system of equations.

Example 7 Solve the system of equations by graphing.3r -Y:19x*4Y:12First rewrite each equation of the system inslope-intercept form by solving for y.

3r-Y:16x+4y:12

A consistent system has at least one solution.A system having exactly one solution is independent.lf a system has infinitely many soltttions, the system is dependent.Systems that have no soluiion are inconsistent.

Systems of linear equations can also be solved algebraicallyusing the elimination method or the substitution method.

Example 3Use the substitution method tosolve the system of equations.x:7y * 32x-y--lThe first equation is stated in terms of r,so substitute 7y * 3 for r in the secondequation.

2x*Y:-72(7y+3)-y--7

13Y: -13!:*I

Now solve for x by substituting - 1 fcrr yirr either of the original equations.x:7y-f 3

x:7(*1)+3 !:-7^L--'t

The solution is'(-4, -.1).

21 Advanced Mathematical Conce?ts

PracticeSotving Sgstems of EquStafe whether each sysfem r,s consisfentaconsisfent and dependent, or inconsrbtentl. -xty:

3x-3y: -472

So/ye each system of equations by graphittgt3.r*y:6

2x-l 3y : 12

So/ye each system of equations algebraically.5.x*y:4

3r-2y:76. 3x - 4y: I0

-3x I 4Y :8

9.2x f 3y:197x-y:9

8.4x-t5y:3x-2y:

11

-9

Ll' Real Estate AMC Homes, Inc. is planning to build thc-dih.{edroomhomesinahousingdevelopment.uiludChe"stnutIIilIs.hdqrr',,6indicates a need for three iitttut as many four-bedroomhcrrrfuthree-bedroomhomes' The net profit from each thre"-bldroo- ho-" is *$,md frrom eachfour-bedroom home, $17,000. If AMC Homes must net a tdrl Fffit d*l3.a millionfrom this development, how many homes of each type shm5 ttr., 6ft;ru'-

r#'rtt-/rJ

u

M llffirrl:crtical ConceptsO Glencoe/McGraw-Hill 22

.."

DATE PERIOD

Study Guide

Sotving Sgstems of Equotions in Three VoriobtesYou can solve systems of three equations in three variablesusing the same algebraic techniques you used to solvesystems of two equations.

Example Solve the system of equations by elimination.

4x+y*22--8Lr-BY-42:4-7x*2Y*32:2

Choose a pair of equations and then eliminate one of thevariables. Because the coefficient of y is 1 in the firstequation, eliminate y from the second and third equations.

To eliminate y using the first and secondequations, multiply both sides of thefirst equation by 3.

3(4x+y+22):3(-8)I2x + 3y + 6z: -24Then add the second equation to thatresult.

: -24:4,

To eliminate y using the first and thirdequations, multiply both sides of the firstequation by -2.-2@x-r y -r 2z): *2(-8)-8r - 2y - 4z: 16

Then add the third equation to thatresult.

-8r - 2y - 42 :16-7x-t2Y-f 3z: 2

The value of r is -1.Finally, use one of the original equationsto find the value of y.

4x-fY+22:-84(*l) -t y + 2(-3): -8 x: -7, z: -3

y:2The value of y rs 2.

l2x*3y+622x-3y-42

L4x -l 2z: -20 -l1x *z: 18

Now you have two linear equations in two variables. Solvethis system. Eliminate z by multiplying both sides of thesecond equation by 2.Then add the two equations.

2(-l5x - z) :2(18) I4x + 2z : -20-30r - 2z :36 -30r - 2z : 36

-l6x : 16x: -L

By substituting the value of r into one ofthe equations in two variables, we cansolve for the value of z.

l4x+22:-2014(-1) * 2z : -20 x: -7

z:-3The value of e is -3.The solution is x : -1, ! : 2, z : -3. This can be written as

the ordered triple (-1,2, -3).

@ Glencoe/McGraw-Hill 23 Adv an ced M ath e m ati cal Co n cePts

Practice

Sotving Sgstems of Equqtions in Three VoriobtesSolve each system of equations.

l.x'ry-z--1x*y*z:33x-2Y-z:-4

3.3r-5y+z:84y - z:10TxtY:4

5.2x-Y-fz:-lX-Y*z:Ix-2Y-fz:2

7.x-z:5y*32:122x*Y:7

Business The president of Speedy Airlines has discovered that hercompetitor, Zip Airlines, has purchased 13 new airplaines from CommuterAviation for a total of $15.9 million. She knows that Commuter Aviationproduces three types of planes and that type A sells for $1.1 million, type Bsells for $1.2 million, and type C sells for $t.Z mittion. The president ofSpeedy Airlines also managed to find out that Zip Airlines purchased 5

more type A planes than type C planes. How many planes of each typedid Zip Airlines purchase?

2.x*y:53x * z:24Y - z:8

4. 2x + 3y + 3z:210r - 6Y -f 3z:04x-BY-62:2

4x+4y-22--B-Gx-6Y*62:52x-3y-42:2

8. 2x + 4Y - 2z:94x-GY*22:-9)c-y*32:-4


re s.uorcu*".:**qili,#

Modeting Reot'Wortd Dqto with MotricesA matrix is a rectangular array of terms called elements.A matrix with m rows and n columns is an m X n matrix.The dimensions of the matrix are m and n.

Example 7 Find the values of r and y for which

[sr +;] :

[ '-]]is true'

Since the corresponding elements mustbe equal, we can express the equalmatrices as two equations.

3r+5:y-9!:-4x

Solve the system of equations by usingsubstitution.

To add or subtract matrices, the dimensionsmust be the same.

DATE PERIOD

3r+5:y-93r + 5 : (-4x) - I Substitute -4x for y.

o

Y : -4x,y : -4(-2) Substitute -2 for x.y:8The matrices are equal 7f x : -2 and y : 8.

of the matrices

Advanced Mathematical ConcePts

SYLMAR HIGH MAGNET

Example 2

To multiply two matrices, the number of columns in the first matrixmust be equal to the number of rows in the second matrix. Theproduct of two matrices is found by multiplying columns and rows.

Exampre 3 Find each product n x :l! ;] "'u " :

[3 -3]

- 18]

-12l]

Find c-Dtf c:[-; l] "'u ":l-+C-D:C+eD)

I 3 6.1 l-- 1 -5]:l-, 4l *L z -81

_f 3+(-1) 6+t-5tl-l -2+7 4+(-8)l

_12 1l-lr -4)

a. XY

xY :til3l I ;[3] i[s] i ;[-3]l * [;3b. YX

YX : [,,-if I ].3[3] n*if I lu3 :,711 * I'Z ?:'l


DATE PERIOD

PracticeModeting Reot-Wortd Doto with MotricesFind the values of x and y for which each matrix equation istrue,

' Bl : lu -r1,1

Use matrices A, B, and C to find each sum, difference, orproduct.

[-r b 6l t 2 B

A:l 2 -7 _zl B:l-r 1l+ 4 il Ls-23.4+B

5.8_A 6. -24

7. CA 8. AB

9. AA LO. CB

rt. (cA)B 12. c(AB)

,.1'. -i] : [r;]

i] " : [-3 ig r?]

4.4-B

13. Entertainment On one weekend, theGoxfield Theater reported the followingticket sales for three first-run movies, asshown in the matrix at the right. If theticket prices were $6 for each adult and$4 for each child, what were the weekendsales for each movie.

Adults ChildrenMovier Irozt szslMovie z I zs+z zss I

Movie s lsosz z+sa)

@ Glencoe/McGraw-Hill 26 Advan ced M ath em ati cal Co n ce pts

DATE PERIOD

Study Guide

Modeting Motion with MotricesYou can use matrices to perform many transformations, such as

translations (slides), reflections (flips), rotations (turns), anddilations (enlargements or reductions).

Example 1 Suppose quadrilateral EFGH with verticesE(-1,5), F(3, 4), G(4,0), and I/(-2' 1) istranslated 2 units left and 3 units down.

The vertex matrix for the quadrilat"rd i, [-l t^

t -?)

rhe translation matrix t. L 3 3 -? _?)

Adding the two matrices gives the coordinates of thevertices of the translated quadrilateral.

l-r 3 4-21 -l-2-2-2-2) l-3 | 2-41I s 4 o 1l l-s-B-B -sl-l z r-3-21Graphing the pre-image and the image of thetranslated quadrilateral on the same axes, wecan see the effect of the translation.

The chart below summarizes the matrices needed to produce specificreflections or rotations. All rotations are counterclockwise about the origin.

Reflections &"".: [; ?]Ry-uri" [-1 ol-t o 1l

t0R -ly:v 11

1lol

Rotationst0 -1-lRotno:1.,

O]not-^^:l t ol

rdu I 0 _1 I

Rotrro- i-? l]Example 2 A triangle has vertices A(-I,2), B(4,4), and C(3, -2).

Find the coordinates of the image of the triangle aftera rotation of 90" counterclockwise about the origin.

The vertex matrix i, [- 1 4 3lt 2 4-21

Multiply it by the 90" rotation matrix.

[o -1] l-r 4 3l:l-2 -4 ?lIt ol I 2 4-21 :L-t 4 3l

Example 3 TbapezoidwWz has vertices W'(2, 1),X(1, -z),Y(-L, -2), and Z(-2,1). Find thecoordinates of dilated trapezoid W'XYZ' fota scale factor of 2.5.

Perform scalar multiplication on the vertexmatrix for the traPezoid.

t2 1 -1 -21 -l 5 2.5 -2.5 -51

'ult -; -; il -lr u -; -; z;l

@ Glencoe/McGraw-Hill Advanced Mathematical Concepts

NAME

1. triangle with verticesA(1,2), B(2, - 1), andC(-2,0); scale factor 2

2. quadrilateral withvertices E(-2, *7),F(4, -3), G(0, 1),and H(-4, *2); scalefactor 0.5

4. triangle with verticesJ(3, t), K(2, -4), andL(0, *2) translated4 units left and2 units up

6. a rhombus withvertices 8(2,3),R(4. - 1r. Sr -7, -2t,and T(-3,2)reflected over theIineY:v

8. PentagonVWXYZwith vertices V(1, 3),W(4,2t, X(3, -2),Y(-L, -4), Z(-2,1)rotated 180"

use matrices to determine the coordinates of the vertices of eachtranslated figure. Then graph the pre-image and the image on thesame coordinate grid.3. square with vertices

w(r, -3), x(*4, -2),Y(-3,3), and Ze,2)translated 2 unitsright and 3 unitsdown

use matrices to determine the coordinates of the vertices of eachreflected figure. Then graph the pre-image and the image on thesame coordinate grid.

5. AMNP with verticesM(-3, 4), N(3,1), andP(-4, -3) reflectedover the y-axis

Use matrices to determine the coordinates of the vertices of eachrotated figure. Then graph the pre-image and the image on thesame coordinate grid.7. quadrilateral CDFG

with vertices C(-2, 3),D(3,4), F(3, -1), andG(-3, -4) rotaLed 90'

PracticeModeting Motion with Motrices

use sca/ar multiplication to determine the coordinates of thevertices of each dilated figure. Then graph the pre-image and theimage on the same coordinate grid.


Det

DATE PERIOD

Study Guideerminonts ond Muttipticotive lnverses of Motrices

Each square matrix has a determinant. The determinant of

a 2 x 2 matrixis a number denoted bv I ot ?'l o, a.t[ Z;i:]Its value ts arbr- orbr. "

1 "' o'l

Exampte I Find the value "t l3 -?l

l5 -rll; il :s'1r-3( -2tot71

The minor of an element can be found by

deleting the row and column containingthe element.

The minor of a, is

Example 2 Find the value of

-+'r+rq&, b, co

+; b; ,i

nl I ol1 ,o D

+l - ol-rll

11) or -90ed as follows.

::i:i-z:-3ll^ 'l

b, crlb'^ t'"1', ,l

893357

-1 24893357

-1 2 4

: tl;: 8(6)

of a m

]'"u

rnverse

lo, bL

lo, b2

il-,1_ e(1e)

atrix is

la, btlI o, b,lt"-l

D

-1+3(

defin

+0,

-tbtzl

The multiplicative

If A: thenA 1:

Example 3 Solve the system of equations by using matrix equations.5x * 4Y: -$3x - 5Y: -24Write the system as a matrixequation.

tl -il t;l: l-;1^lTo solve the matrix equation,first find the inverse of thecoefficient matrix.

1

-t)

Now multiply each side of the matrixequation by the inverse and solve.

r I -5 -41 l5 4l l"l-s7l -g s,l 'le -sl Iyl:-+L_3 -il

t _,:^l

lxl t -slltl : I gl

The solution is (-3, 3).l5 +t[-s -41 - -1[-5l; -;iL-3 sl- -a7l -3

O Glencoe/McGraw-Hill 29 Adv an ced M at h e m at i cal C o nce Pts

DATE PERIOD

PracticeDeterminonts ond Muttipticotive

Find the value of each determinant.

Inverses of Motrices

t. -2 sla -nl 2. ll -sl

t't 9l

lz 34. i-s -1l1-4

6' l'3

lr -1 ols. lz r +l

ls -s sl

Find the inverse of each matrix, if it exists.

D.Bsl

-1 sl

So/ve each system by using matrix equations.

7. 2x - 3y :173x-fy:9

zl+l

8. 4x - 3y:2x-t5y:

f-r11 1uL-r

-1618

So/ye each matrix equation.

I z -1 sl [r] l-slo. I t 2 rl . lyl : I sl, irthe inverse is -f-r -B -zl L,) L-zl

-11-1

7

-Tl

sl

tzlzl

t4l

[s-2 +] l"l l-21 I zro. ls -4 zl . lyl : I o l, if the inverse is -1 I - rLr-s rl L,l Lil 'L-;

-101

13

Il. Landscaping TWo dump truck have capacities of 10 tons and 12 tons.They make a total of 20 round trips to haul 226 tons of topsoil for alandscaping project. How many round trips does each truck make?

@ Glencoe/McGraw-Hill 30 Advanced M athematical Concepts

L

NAME DATE

Study Guide

Sotving Sgstems of Lineor lnequotitiesTo solve a system of linear inequalities, you must find theordered pairs that satisfy all inequalities. One way to do thisis to graph the inequalities on the same coordinate plane. Theintersection of the graphs contains points with ordered pairsin the solution set. If the graphs of the inequalities do notintersect, then the system has no solution.

Example 1 Solve the system of inequalities -x * 2y = 2bygraphing. x<4The shaded region represents the solution to thesystem of inequalities.

A system of more than two linear inequalities can have asolution that is a bounded set of points called a polygonalconvex set.

Exampte 2 Solve the system of inequalities r > 0by graphing and name the Y > 0coordinates of the vertices of 5x I 8y < 40the polygonal convex set.

The region shows points that satisfy all threeinequalities. The region is a triangle whosevertices are the points at (0, 0), (0, 5), and (8, 0).

Example 3 Find the maximum and minimum values off (x, y7 : x * 2y + I for the polygonal convexset determined by the following inequalities.x>O Y>0 ?"x*Y<4 x*Y<3First, graph the inequalities and find thecoordinates of the vertices of the resultingpolygon.

The coordinates of the vertices are (0, 0),(2, 0), (I, 2), and (0, 3).

Then, evaluate the function f(x, y) : x + U + L

at each vertex.

/(0,0):0+2(0)+l-:1.f(2,0):2+2(0)+1:3Thus, the maximum value of the function is 7,

and the minimum value is 1.

f(1,2):I+2(2)*1:6f(0,3):0+2(3)+L:7

PERIOD

O Glencoe/McGraw-Hill Adv anced M athe m ati cal Co n ce Pts

DATE

Practicesotving Sgstems of Lineor lnequotitiesSo/ye each sysfem of inequatities by graphing.1. -4x * 7y = -27 3x 1- 7y < 28 Z.

PERIOD

so/ve each system of inequalities by graphing. Name thecoordinates of the vertices of the polygonal convex set.3. r > 0;y > 0;y > x - 4;7x + 6y < E4 4. x>0;y+2=0;Ex+6y < 1g

Find the maximum and minimum values of each function forthe polygonal convex set determined by the given system ofinequalities.5.3x-2y>0 y>-0

3x+2y<24 f(x,r;:7y-3x

7. Business Henry Jackson, a recent college graduate,plans to start his own business manufacturing bicycletires. Henry knows that his start-up costs are going tobe $3000 and that each tire wil cost him at teast $"2 tomanufacture. In order to remain competitive, Henrycannot charge more than $5 per tire. Draw a graplr-toshow when Henry will make a profit.

x=3;y=5;x-fy>I

6.y<-r*8 4*-By>-Br*8y>8 f(x,rS:4x-5y

@ Glencoe/McGraw-Hill 32 Advan ced M ath em ati cal C o n cepts

DATE

Study Guide

Lineor ProgrommingThe following example outlines the procedure used to solvelinear programming problems.

Example The B & W Leather Company wants to addhandmade belts and wallets to its product line. Eachbelt nets the company $18 in profit, and each walletnets $12. Both belts and wallets require cutting andsewing. Belts require 2 hours of cutting time and6 hours of sewing time. Wallets require 3 hours ofcutting time and 3 hours of sewing time. If thecutting machine is available t2 hours a week and thesewing machine is available 18 hours per week, whatmix of belts and wallets will produce the most profitwithin the constraints?

Let b : the number of belts.Let w : the number of wallets.

PERIOD

Defineuariables.

Write anequation.

Substituteualues.

Answer theproblem.

Write b >- 0inequalities. w - 0

2b + 3w < 12 cutting6b + 3w < 1-8 sewing

Graph thesystem.

Since the profit on belts is $18 and theprofit on wallets is $12, the profit functionis B(b, w) : l8b -t l2w.

B(0, 0) : 18(0) + 12 (0) : 0B(0, 4): 18(0) + 72(4) : 48B(t.s,3) : 18(1.5) + 12(3) : 63B(3, 0) : 18(3) + I2(0): 54

The B & W Company will maximize profitif it makes and sells 1.5 belts for every3 wallets.

When constraints of a linear programming problem cannot besatisfied simultaneously, then infeasibility is said to occur.

The solution of a linear programming problem is unbounded if theregion defined by the constraints is infinitely large.

O Glencoe/McGraw-Hill Adv an ce d M athe m ati cal C o n cepts

NAME

2.2x+2y>102x-fy>8f(x,r1 :x+y

PERIOD

PracticeLineor Progromming

Graph each sysfem of inequalities. ln a problem asking youto find the maximum value of f (x, y), state whether thesituation is infeasible, has alternate optimal solutions, or isunbounded. In each system, assume that x > O and y > Ounless stated otherwise.1. -2y=2x-36

x+y>30f@,11 :3x*3y

Solve each problem, if possible. If not possible, sfatewhether the problem is infeasible, has alternate optimalsolutions, or is unbounded,3. Nutrition A diet is to include at least 140 milligrams of

VitaminA and at least 145 milligrams of Vitamin B. Theserequirements can be obtained from two types of food. Tlpe Xcontains 10 milligrams of VitaminA and 20 milligrams ofVitamin B per pound. Tlpe Y contains 30 milligrams ofVitamin A and 15 milligrams of Vitamin B per pound. Iftype X food costs $12 per pound and type Y food costs $8 perpound how many pounds of each type of food should bepurchased to satisfy the requirements at the minimum cost?

4. Manufacturing The Cruiser Bicycle Company makes twostyles of bicycles: the TYaveler, which sells for $200, and theTourester, which sells for $600. Each bicycle has the same frameand tires, but the assembly and painting time required forthe Tbaveler is only t hour, while it is 3 hours for theTourister. There are 300 frames and 360 hours of laboravailable for production. How many bicycles of each modelshould be produced to maximize revenue?

@ Glencoe/McGraw-Hill 34 Advan ced M ath e m ati cal Con cepts

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Study Guide

Summetrg ond Coordinote CIqPhsOni type of syirmetry a graph may have is point symmetry. Acommon point of symmetry is the origin. Another type ofsymmetry is line symmetry. some common lines of symmetryare the r-axis, the y-axis, and the lines y : x and y : -x:.

rcoint r'axis Y-axis Y : r

Example I Determine whether f(x) : r3 is symmetric withresPect to the origin.

If f(-x) : -f(x), the graph has point symmetry.

Find f(-x).f(_x): (_r)B

f(-x) : -x3The graph of f(x): 13 is symmetric with respectto the origin because fGx) : -f(x).

Example 2 Determine whether the graph of x2 * 2: y2 issymmetric with respect to the r-axis, the y'axis,the line ! : frs the line ! : -fi, or none of these'

Substituting (a, b) into the equation yieldsa2 + 2 : b2. Check to see if each test produces

an equation equivalent to a2 + 2: b2'

PERIOD

Find -/(r).-f(x') : -x3

.x-ax1s

y:x

!: -x

y-axis ?a)2 * 2: b2

a2+2:b2

Substitute (a, -b) into the equation.Equiualenttoa2*2:b2Substitute (-a, b) into the equatiort.Equiualenttoa2*2:b2Substitute (b, a) into the equation.Not equiualent to a2 + 2 : b2

Simplify.I{ot equiualent to a2 + 2 : b2

(-b)2 t 2: (-a)2 Substitute (-b, -a) into the equation.

a2 + 2: (-b)2a2+2:b2

(b)2 + 2: (a)2

d2-2:b2

b2+2:a2a2-2:b2

Therefore, the graph of x2 t 2 : y2 is symmetric with respect to

the r-axis and the Y-axis. * '

point !:-x

@ Glencoe/McGraw-Hill 39 Advanced M athematical ConcePts

PERIOD

Study Guide

SUmmetrg ond Coordinote GrophsOne type of symmetry a graph may have is point symmetry. Aeornrr\or\ point of symrnetry is the origin. Nnother \ype otsymmetry is line symmetry. Some common lines of symmetryare the r-axis, the y-axis, and the lines y : x and y : -x.

Y'axrs y : x

Example 1 Determine whether f(x): rs is symmetric withrespect to the origin.If f(-x) : -f(x),Find /(-r).f(-x) : (-x)3f(-x) : -x3

the graph has

The graph of f(x): 13 is symmetric with respectto the origin because f(-x) : -f(x).

Example 2 Determine whether the graph of x2 I 2: y2 issymmetric with respect to the r-axis, the y-axis,the line ! : fis the line ! = -fis or none of these.

Substituting (a, b) into the equation yieldsa2 + 2: b2. Clteck to see ifeach test producesan equation equivalent to a2 + 2: b2.

r-axis a2 + 2 : (-b)2 Substitute (a, -b) into the equation.a2 + 2: b2 Equiualent to a2 I 2 : b2

y-axis (*a)2 * 2 : b2 Substitute (-cL, b) into the equatioru.a2 + 2 : b2 Equiualent to a2 -f 2 : b2

y - x (b)2 + 2 : (a)2 Substitute (b, a) into the equation.a2 - 2:b2 Notequiualenttoa2 + 2:b2

! : -x (-U2 t 2 : (-a)z Substitute (-b, -a) into the equation.b2 + 2: a2 Simplify.e2 -2:b2 Notequiualenttoa2 + 2:b2

Therefore, the graph of x2 r 2 : y2 is symmetric with respect tothe r-axis and the y-axis.

point symmetry.

Find -./(r).-f(x) : -x3

point tr-axrs !:-x

O Glencoe/McGraw-Hill 39 Adv an ced M ath e m ati cal Co n c epts

NAMEPERIOD

PracticeSgmmetrg qnd Coordinote Grophs

7:l;:{:i;#:ether the graph or each runction is symmetric with respect

r.f(x)- -r2x 2' f(x): x5 - 23.f(x):x3-4x 4.f(x):

s_*L

Determine whether the graph of each equation is symmetric with respect tothe x-axis, the y-axis, tne me y : X, the iine y : 1r, or none of these,5.x*!:6 6.x2ty:27.xy:39.y:4*

8. 13 t y2:4

lO.Y:x2-111' Is f (x) : jrj an even function, an odd function, or neither?

Refer to the graph at the right for Exercises 12 and 13.12. Complete the graph so that it is the graph

of an odd function.

13. Complete the graph so that it is the graphof an even function.

14' Geometry cameron tord her friend Juanitathat the graphof lyl:_A _ l3rl has the shapeof a geometric figure. Determine wheth"" ih"graph of lyi : 6 - l3tri is symmetric withrespect to the r_axis, the y_axis, both, orneither. Then make a sketch ofthe graph.Is Cameron correct?

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O Glencoe/McGraw-Hill 40 Advanced Mathematical Concepts

Study GuideFomities of @rophsA parent graph is a basic graph that is transformed to create othermembers in a family of graphs. The transformed graph may appearin a different location, but it will resemble the parent graph.

A reflection flips a graph over a line called tlte axis of symmetry.A translation moves a graph vertically or horizontally.A dilation expands or compresses a graph vertically orhorizontally.

Example I Describe how the graphs of f (x) : 16 and,9(x\ : f -x - 1 are related.

The graph of g(r) is a reflection of the graph off(x) over the y-axis and then translated down1 unit.

Example 2 Use the graph of the given parent functionto sketch the graph of each related function.a. f(x): x3;!: x3 + 2

When 2 is added to the parent function, thegraph of the parent function moves up 2 units.

b. f@): [rn;y : 3[rn

The parent function is expanded vertically bya factor of 3, so the vertical distance betweenthe steps is 3 units.

c. f(x): lrl;y: o,llxlWhen lr I is multiplied by a constant greaterthan 0 but less than 1, the graph compressesvertically, in this case, by a factor of 0.5.

d. f(*): x2;y: l*'- 4l

The parent function is translated down 4 unitsand then any portion of the graph below ther-axis is reflected so that it is above the r-axis.

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PracticeFomities of Grophs

Describe how the graphs of f (x) and g(x) are related.l. f (x) : x2 and g(.x) : (r + 3)2 - 1 2. f(x): lrl andg(r): -lz*l

Use the graph of the given parent function to describe the graph of eachrelated function,3. f(x) : x3

d. y :2x34. f (*): \G,

a.Y:fx+g+t

b.y:-0.5(r-2)3 b.y:f-x-2

c.y:l(r+1)31 c.y:fuzsx-+

Sketch the graph of each function.5. f(*): -(x - 1)2 + 1

7. Consurner Costs During her free time, Jill baby-sitsthe neighborhood children. She charges $4.b0 for eachwhole hour or any fraction of an hour. Write and grapha function that shows the cost ofr hours of baby-sitting.

fk):2lx+21

O Glencoe/McGraw-Hill 42 Advan ced M ath emat i cal C o n ce pts

Study GuideGrophs of Nonlineor lnequotitiesGraphing an inequality in two variables identifies all orderedpairs that satisfy the inequality. The first step in graphingnonlinear inequalities is graphing the boundary.

Example 7 Graph y <x6- S +2.The boundary of the inequality is the graph of y :\/G= + 2.To graph the boundary curve, start with the parent graphy :\,G,. Analyze the boundary equation to determine how theboundary relates to the parent graph.

y:xG1 +ztt

moue 3 units right nLoue 2 units up

Since the boundary is not included in the inequality, thegraph is drawn as a dashed curve.

The inequality states that the y-values of the solution areless than the y-values on the graph of y :V x - 3 + 2.Therefore, for a particular value of r, all of the points in theplane that lie below the curve have y-values less thanf x - 3 + 2. This portion of the graph should be shaded.

To verify numerically, test a point not on the boundary.

Y;lf--z * z0< V4 - 3 + 2 Replace (x,y) with (4,0).0<3 ,/ Tlue

Since @, 0) satisfies the inequality, the correct region isshaded.

Example 2 Solve lr - 3l - 2> 7.

Two cases must be solved. In one case, x - 3 is negative, andin the other, x - 3 is positive.

Casel If al},then lol: -o. Case2 If a> 0,then lol:o.-(x - 3) - 2> 7 x - 3 - 2> 7

-x+3-2>7 x-5>7-x) 6 x> 12x1-6

The solution set is {r I x < -6 or x > I2l.

PERIOD


DATE PERIOD

PracticeGrophs of Nontineor lnequotitiesDetermine whether the ordered pair is a solution for the given inequatity.Write yes or no.r. v>(x+2)2+3,(-2,6) 2. y <(x *3)3 +2,(4,5) 3. y<12*-41-r,(*4,r)

Graph each inequatity.4.y=2lx-Il 5.y>2(x-r)2

6.y<xG-2+t 7.y>(r+3)3

So/ve each inequality.8.14x-101<6 9. l* + 5l+ 2> 6 LO.i2x-2i-t<7

II. Measurement Instructions for building a birdhouse warn that theplatform, which ideally measures 14.78 cm2, should not vary in sizeby more than 0.30 cm2. If it does, the preconstructed roof for thebirdhouse will not fit properly.a. write an absolute value inequality that represents the range of

possible sizes for the platform. Then solve-for r to find the iange:

b. Dena cut a board 14.42 cm2. Does the platform that Dena cut fitwithin the acceptable range?

O Glencoe/McGraw-Hill Advan ced M ath em ati cal C o n ce pts

Study Guide

lnverse Functions ond RetotionsTWo relations are inverse relations if and only if one relationcontains the element (b, a) whenever the other relationcontains the element (a, b).If f(.x) denotes a function, then

,-r(/) denotes the invers e of f @).

Example I Graph f (x) : I"u - 3 and its inverse'

To graph the function, let , : f (.x)' To graph

f -1(x), interchange the r- and y-coordinates ofthe ordered Pairs of the function'

f(x):F"-"x v

J -9.75-2 -5-1 -3.25

0 _J

1 -2.752 -1

3.75

f -'(xlx v9.75 3-^

-3 a

-3.25 1

a 0I

-2./c I

-1 I

3.75 3

You can use the horizontal line test to determine if theinverse of a relation will be a function. If every horizontal lineintersects the graph of the relation in at most one point, thenthe inverse of the relation is a function'

You can find the inverse of a relation algebraically-, First, lety : f(x). Then interchange r and y. Finally, soive the

resulting equation for Y.

Example 2 Find the inverse of f (x\ : @ - 1)2 * 2. Deterrnineif the inverse is a function'

vx

x-2!\,G4

v.f

"r(x)yT4 Solue fo, y.

lx - 2 Replacey with f 1(x)-

Since the line y : 4 intersects the graph af f(x)at more than one point, the function fails thehorizontal line test. Thus, the inverse of /(r) is

not a function.

Let y : f(.x).Interchange x and y.

Isolate the expression containing y.

Take the square root of each side.

:(:r-:(!*:(y--v*:1-+-:1+

l)2+2r)'+2D2

1

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@ Glencoe/McGraw-Hill 45 Advanced M athemati cal C a n c e P :.

Practice

lnverse Functions ondGraph each function and its inverse.l.f(x):(x-1)3+1

Retotions

2. fkr : Sl"rl + z

Find f-l(x). Then sfafe whether f -t(x) is a function.3. f (x): -4x2 + r a. f@): {x6 _ 1 5. f(x)- 4

Q-3)z

Graph each equation using the graph of the given parent function.6.y: -f.+ 3- L,p(x):y2 7. y:Z+'Vii,p(x):xE

Fire Fighting Airplanes are often used to drop water on forest fires in an effort tostop the spread of the fire. The time / it takes ttre water to travel from height h tothe ground can be derived from the equation 6 : Lrgt2 *tr"r" g ir irr"-".."r.rationdue to gravity (32 feet/second2).

a. write an equation that will give time as a function of height.

b. suppose a plane drops water from a height of l024feet. How many seconds willit take for the water to hit the ground?

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Study GuideContinuitg ond End Behovior

A function is continuous at x : c if it satisfies the following three conditions.(1) the function is defined at c; in other words, /(c) exists;

(2) the function approaches the same y-value to the left and right of x - c; and(3) the y-value that the function approaches from each side is /(c).

Functions can be continuous or discontinuous. Graphs thatare discontinuous can exhibit infinite discontinuity, jumpdiscontinuity, or point discontinuity.Example I Determine whether each function is continuous

at the given r-value. Justify your answer usingthe continuity test.

a.f(x):2lxl +g;x:2(1) The function is defined at x : 2;

f(2) : 7.

(2) The tables below show that yapproaches 7 as.r approaches 2from the left and that y approaches7 as x approaches 2 from the right.

PERIOD

b./(r):fr1;x:LStart with the first condition in thecontinuity test. The function is notdefined at x : 1 because substituting 1for r results in a denominator of zero.So the function is discontinuous at x :I.

This function fails the second part of thecontinuity test because the values of/t.r't approach 1 as tr approaches 2 fromthe left, but the values of f(x) approach5 as x approaches 2 from the right.

x v: fVl1.9 6.8

1.99 6.98

1.999 6.998

x v: fkl2.1 7.2

2.01 7.02

2.001 7.002

c.f(x): {?_.rt ;tr ::_;

(3) Since the y-values approach 7 astr approaches 2 from both sidesand f(2) : 7 , tlte function iscontinuousatx:2.

The end behavior of a function describes what the y-values doas lr lbecomes greater and greater. In general, the end behavior ofany polynomial function can be modeled by the function made upsolely of the term with the highest power of r and its coefficient.

Example 2 Describe the end behavior of p(x) : -x5 + zxB - 4.

Determine f@) : an.xn where r" is the term in p(r) withthe highest power of r and o,, is its coefficient.

f(x):-x5 xn:x5 ar:-7Thus, by using the table on page 163 of your text, youcan see that when o" is negative and n is odd, the.endbehavior can be stated as p(x') --> -oo as rc --+ * andP@) --= oo as ff ---> -co.

47O Glencoe/McGraw-Hill Advanced Mathematical Concepts

NAME DATE PERIOD

PracticeContinuitg ond End BehoviorDetermine whether each function is continuous at the given x-value. Justifyyour answer using the continuity test.

L' y: #,*: -1

3.y:x3-2x*2;x:7 4. y : #;*:"tT+ -4

Describe the end behavior of each function.5.y:2x5-4x 6.y:-2x6+4xa-2x-tl

7.y:xa-2x3+x 8.Y: -4x3+5

Given the graph of the function, determine the interuat(s) for which the functionis increasing and the interval(s) for which the function is decreasing.

9.

LO. Electronics Ohm's Law gives the relationship between resistance R, voltageE, and current,I in a circuit as E : ? ttthe voltage remains constant but thecurrent keeps increasing in the circuit, what happens to the resistance?

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Study Guide

Criticot Points ond ExtremoCritical points are points on a graph at which a line drawntangent to the curve is horizontal or vertical. A critical point may be

a maximllm, a minimum, or a point of inflection. A point ofinflection is a point where the graph changes its curvature. Graphscan have an absolute maximllm, an absolute minimum, a

relative maximum, or a relative minimum. The general term formaximum or minimum is extremum (plural, extrema).

Example I Locate the extrema for the graph of y : f (x)-Name and classify the extrema of the function.

The function has an absolute maximumat (0, 2). The absolute maximum is thegreatest value that a function assumesover its domain.

The function has an absoluteminimum at (- 1, 0). The absoluteminimum is the least value that a

function assumes over its domain.

PERIOD

The relative maximum and minimumma,v not be the greatest and the leasty-r'alue for the domain, respectively, butthe;,' are the greatest and least y-value onsome intervai of the domain. The functionhas a relative maximum at (*2, -3) and arelatir.e minimum at (0, -5). Becausethe graph indicates that the functionincreases or decreases without bound as

r increases or decreases, there is neitheran absolute maximum nor an absoluteminimum.

c.

b.

By testing points on both sides of a critical point, you can

determine whether the critical point is a relative maximum, a

relative minimum, or a point of inflection.

Exampte 2 The function /(r) : 2x6 * 2xa - 9x2 has a criticalpoint at x :0. Determine whether the criticalpoint is the location of a maximum, a minimum,or a point of inflection.

x x-0.1 x*0.1 f (x - o.1l f(xl f (x + o.1lType of

Critical Point

0 -0.1 0.1 -0.0899 0 -0.0899 maxrrTu.J.t't]

Because 0 is greater than both /(r - 0.1) and

f(x + 0.1), r : 0 is the location of a relative maximum.

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Practice

Criticot Points qnd Extremo

Locate the extrema for the graph of y : f (x).extrema of the function.

Name and classify the

Determine whether the given critical point is the location of a maximum, aminimum, or a point of inflection,

5, y: x2 - 6x + 1,r:3 6. y:x2 -2x- 6,x: I 7. y:xa + 3x2 - 5,x:0

8. y : x5 - 2x3 - 2x,2rx:0 9. y : xB + x2 - x,x: -1 10.

I1-. Physics Suppose that during an experiment youlaunch a toy rocket straight upward frorn a heightof 6 inches with an initial velocity of 32 feet persecond. The height at any time / can be modeled bythe function s(t) : -I6t2 + 32t + 0.5 where s(r) ismeasured in feet and t is measured in seconds.Graph the function to find the maximum heightobtained by the rocket before it begins to fall.

y:2xBl4,x:o

2.1.

@ Glencoe/McGraw-Hill 50 Advanced M athe m ati cal C o n ce pts

The values in the table confirmthat fk) -> -oo as ff --+ -3 fromthe right, so there is a verticalasymptote at x: -3.

Example 2 Determine the slant asymptote for

f(x) :First use division to rewrite the function'

3x-t 7

* - tE*':ii +i,3x2 - 3x

x+Zx-l

PERIOD

the graph of

One way to find the horizontalasymptote is to let f(x) : y andsolve for r in terms of y. Thenfrnd where the function isundefined for values of y.

",-2x-LJ - xi3.\'(.t -r 3) : 2x - 1

't1,+3y--2x-Lt.t, - 2x: -3y - 1

rrl' - 2): -3y * |-- 3v-1n- v 2

The rational expressior, -.31J I itJa

undefined for y :2. Thus, thehorizontal asymptote is the lineJ'-2.

NAME

Study Guide

Grophs of Rotionot Functions

Example 1 Determine-the asymptotes forf ("1 : !-* ].Since .f(-3) is undefrned, theremay be a vertical asYmPtote atx : -3. To verify that x : -3 isa vertical asYmPtote, checkto see that f@) -+ oo or

f(x) '--' -oo as x ' -3 fromeither the left or the right.

x ft6l,2.9 -68-2.99 -698-2.999 -6998-2.9999 ,69998

---> f(x):3r + 1 + =,r- I

DATE

J

As r - *,:1* 0. Therefore, the graph of f(.x) .

will approach that of y :3r t 1' This means thatthe line y : 3x + 1 is a slant asymptote for thegraph of f@).

A rational function is a quotient of two polynomial functions.

The linex: a is a vertical asymptote for a function t'(x) tf f (x) + F or f E) - -@ asx + a

from either the left or the right.

The linelz: b is a horizontal asymptote for a function f (x) il f (x) '+ b asx + @ or as

x+-m.A slant asymptote occurs when the degree of the numerator of a rational {unction is exactly

one greater than that of the denominator

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PracticeOrophs of Rotionot FunctionsDetermine the equations of the vertical and horizontal asymptotes, if any, ofeach function.

2. f (*): 2xt1xi_l

use the parent graph f (x) : f, to graph each equation. Describe thetransformation(s) that have taken place. Identify the new locationsof the asymptotes.4.y:+ - -2" ,r+l

Determine the slant asymptotes of each equation,G.y : w .+T- 7, y : *'- +" r+ I

8. Graph the function.y : 't't ' l' - 6" x*l

9. Physics The illumination l from a light source is given by theformula I : 4, where k is aconstant and d,is distance. As thed2,distance from the light source doubles, how does the illuminationchange?

L.f(x)- 4" x2+7

5.y:-/r+s

@ Glencoe/McGraw-Hill 52 Adv an ced M athe mati cal C o n ce pts

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Study Guide

Direct, lnverse, ond loint VoriotionA direct variation can be described by the equation ! : hx".The k in this equation is called the constant of variation.To express a direct variation, we say that y varies directly as

x". An inverse variation can be described by the equation

y : + or x"y: fr. When quantities are inverselyprofrortional, we say they uary inuersely with each other.Joint variation occurs when one quantity varies directly as

the product of two or more other quantities and can be

described by the equation ! : l?x;'z'

Example 1 Supposeyvaries directly asr and !:14 whenr:8.a. Find the constant of variation and write an

equation of the form ! : kx".b. Use the equation to find the value of y when

x:4.a. The power of r is 1, so the direct variation

equationrsY:lzx.Y:kx

14:l?(8) y:14,x:81.75 : lz Diuide each side b1' 8.

The constant of variation is 1.75. Theequation relating r and y is y : 1.75'r-.

b' Y : t'75xy : t.75(4) x: 4y:7When x: 4, the value of Y is 7.

Example 2 If y varies inversely as r and y :102 when x:7,findrwhen!:12.Use a proportion that relates the values.xr' _ x2n

!2 lt

# : & Substitute the lznown ualues.

l2x : 714 Cross multiply.

* : ry or 59'5 Diuide each side bY 12'

When ! : L2, the value of r is 59.5'

PERIOD

@ Glencoe/McGraw-Hill 53 Adv an ced M athe m ati cal Con cePts

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Practice

Direct, lnverse, ond loint VoriotionWrite a statement of variation relating the variables of each equation.Then name the constant of variation.1. *f :3

v

PERIOD

2.8:IR

3'Y : 2* 4. d:6t2

Find the constant of variation for each relation and use it to write anequation for each statement. Then solve the equation.

5. Supposey varies directly asr and3l : 35 when x:5. Findy when x:7.

6. lf y varies directly as the cube ofr andy : 3 when x : 2,frnd r when ! : 24.

7.If y varies inversely asr and y :3 when x:25, findr when.l: 10.

8. Supposey varies jointly as r and z, and! :64 when x: 4 andz : 8.Findywhen x:7 andz:11.

9. Suppose Vvaries jointly as /z and the square of ry and V: 45r when r: 3 andh : 5.Find r when V : I75r and h : 7.

10. If y varies directly as r and inversely as the square of z, and ! : -5 when r : 10 andz:2, findy when x:5 andz :5.

Il. Finances Enrique deposited $200.00 into a savings account. The simpleinterest / on his account varies jointly as the time i in years and the principal PAfter one quarter (three months), the interest on Enrique's account ls $Z.ZS.Write an equation relating interest, principal, and time. Find the constant ofvariation. Then find the interest after three quarters.

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Study Guide

Potgnomiot FunctionsThe degree of a polynomial in one variable is the greatest exponent ofits variable. The coefficient of the variable with the greatest exponentis called the leading coefficient. If a function f(x) is defined by apolynomial in one variable, then it is a polynomial function. Thevalues of x for which f(x) : 0 are called the zeros of the function.Zeros of the function are roots of the polynomial equation whenf(x) : 0. A polynomial equation of degree n has exactly n roots in theset of compiex numbers.

Example I State the degree and leading coefficient of the polynomialfunction f (x) : 6x5 + 8rs - 8r. Then determine whethet nf i" "zero of f(x).6xs + 8r3 - 8r has a degree of 5 and a leading coefficient of 6.

Evaluate the function for, : ,/3.

That is, find f(rE)

f(ft): u(€)' *'(,8)' -'(€t It\/3

:Trt|*fv3-tv$:0

Since f(ft):0,ftisazero of f(x):6x5+8x3 -8r.

Exampte 2 Write a polynomial equation of least degree with roots O, fri,and, -fzi.The linear factors for the polynomial are r - 0, x - f2i, and r + fEt.Find the products of these factors.

(r _ oXr _ :,ED@ + xEi): ox(x2 * 2i21: g

x(x2 + 2): 0 *2i2 : -2(-1) or 2x3 + 2x:0

Example 3 State the number of complex roots of theequation 3x2 + l-Lx * 4: 0. Then find the roots.

The polynomial has a degree of 2, so there are twocomplex roots. Factor the equation to find the roots.3x2+ILx-4:0

(3r - 1Xr + 4): 0To find each root, set each factor equal to zero.3r-1:0 xt4:0

3r:1 x: -4*:!The roots are -4 and $.

/

@ Glencoe/McGraw-Hill Advan ced M athe m ati cal Co nce pts

DATE

PracticePotgnomiot Functions

state the degree and leading coefficient of each polynomial.L.6aata3-2a 2. 3p'- 7pu - 2p3 + 5

write a polynomial equation of least degree for each set of roofs.3. 3, -0.5, 1 4. 3,3,7,L, -2

PERIOD

5. -r2i,3, -3 6. -1, 3 * i,2 -r 3i

state the number of complex roots of each equation. Then findthe roots and graph the related function,7.3x - 5:0 8.x2+4:0

9.c2+2cl-I:0 10.13+2x2-15r:0

lL. Rea.l Estate A developer wants to build homes on arectangular plot of land 4 kilorneters long and B kilometers wide.In this part of the city, regulations require a greenbelt of uniformwidth along two adjacent sides. The greenbelt must be 10 timeilthe area of the development. Find the width of the greenbert.

@ Glencoe/McGraw-Hill 60. Advanced Mathematical Concepts

Study Guide

Quodrotic EquotionsA quadratic equation is a polynomial equation with a degree of 2.Solving quadratic equations by graphing usually does not yieldexact answers. Also, some quadratic expressions are not factorable.However, solutions can always be obtained by completing thesquare.

Example 1 Solve x2 - L2x * 7x2-12x+7:0

x2 * I2x: -7

: 0 by completing the square.

x2-lLx+36:*7+36Subtract 7 from each side,Contplete tnn ,quorn"ii

"aai"s [tf -n1',or 36. to each side.

(x - 672: 29 Factor tl-Le perfect square trinomial.x - 6: *f 29 Tahe the square root of each sid.e.

x: 6 * \/29 Add 6 to each side.

The roots of the equation are 6 * \,8.Completing the square can be used to develop a general formula forsolving any quadratic equation of the form o,x2 - bt - c : 0. Thisformula is called the Quadratic Formula and can be used to findthe roots of any quadratic equation.

QuadraticFormula If u^'*bx c:Owitha-O,x -b trG

In the Quadratic Formula, the radicand b2 - 4ac is called thediscriminant of the equation. The discriminant tells thenature ofthe roots of a quadratic equation or the zeros oftherelated quadratic function.

Example 2 Find the discriminant of 2x2 - 3r : 7 amd.describe the nature of the roots of the equation.Then solve the equation by using the QuadraticFormula.Rewrite the equation using the standard form ax2 + bx * c : 0.2x2 - 3x - 7 : 0 d : 2,b : -3,andc : -7The value of the discriminant b2 - 4ac ts(-3)2 - 4(2)(-7), or 65.Since the value of the discriminant is greater thanzero, there are two distinct real roots.

Now substitute the coefficients into the quadraticformula and solve.

*(-3) t V(-3'12, 4(2)(-7) -b -t \@ - +dcx:-._ B tr4sL-

4

The roots are


2(.2)

*/@u"aL/@

61

rAdvanced Mathematical Concepts

DATE

2. -4x,2 - Ilx:'l

PERIOD

Practice

Quodrotic Equotions

So/ve each equation by completing the square.L.x2-sx-!:o

Find the discriminant of each equation and describe the nature ofthe roots of the equation. Then solve the equation by using theQuadratic Formula.

3.x2+r-6:0 4.4x,2-4x-15:0

5.9x2 - I2x * 4:0 6.3x2t2x*5:0

Solve each equation.

7.2x2+5x-12:0 8. 5x2 - l4x * 11:0

9. Architecture The ancient Greek mathematicians thought thatthe most pleasing geometric forms, such as the ratio of thL heightto the width of a doorway, were created using the golden section.However, they were surprised to learn that the golden section isnot a rational number. one way of expressing the golden section isby using a line segment. In the line segment shown ,# : # ,,AC :1 unit, find the rafio ffi.

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PERIOD

Study Guide

Example / Dividex4 - 5x2 - L7x - 1^2by r * B.

x3-Bx2+ 4x-2g Find the value of rin this division.x - r: r * 3

_-- oI

-.)o/ - -(f

According to theRemainder Theorem,P(r) or P(-g) shouldequal 75.

xl S)xa + Ors * Ex2 - l7x - 12xa + SxB

^aFo- Jx" - Dx.'

-BxB - gx2

4x2 - I7x4x2 + 72x

-29x - 12

-29x - 8775 - renzainder

use the Remainder Theorem to check the remainder found bylong division.

P(x):xa-5x2-L7x-t2P(-3) : (-3)a - 5(-3)2 - r7(-3) - 12:81-45+5I-12or7E

The Factor Theorem is a special case of the RemainderTheorem and can be used to quickly test for factors ofa polynomial.

t+ilTt,it I te oinoriarx - r is a factor of the porynomiar p(x) if and onry if p(r) : o.

Example 2 Use the Remainder Theorem to find theremainder when ZxB + Exz - l4n _ g is divided byx - 2. State whether the binomial is a factor ofthe polynomial. Explain.Find f(2) to see if r - 2 is a factor.

f(x):2x3+5x2-r4x-8f(2) : 2(2)3 + 5(Z7z _ 14(2) _ 8

16*20-28-8-U

Since f(2) :0, the remainder is 0. So thebinomial x - 2 is a factor of the polynomialby the Factor Theorem.

The Remoinder ond Foctor Theorems

The RemainderTheorem

lf a polynomial P(x) is divided by x - r the remainder is a constant p(r),and P(x) : (x - r).Q(x) + p(r)

where Q(x) is a polynomiar with degree one less than the degree of p(x).

@ Glencoe/McGraw-Hill Advanced M at h e m ati cal C o n cepts

Practice

The Remoinder ond Foctor Theorems

Divide using synthetic division.

l.(3x2 + 4x- L2)+ (r + 5) 2. (x2 - 5x-12)+ (r - 3)

3.(xa - 3x2 + 12) - (r * 1) 4. (zxs + 3x2 * 8r * 3) - (r + 3)

Use the Remainder Theorem to find the remainder for each division.State whether the binomial is a factor of the polynomial.

5,(2xa + 4x3 - x2 + 9) - (r + 1) 6. (2x3 -3x2 - llx + 3) + (r - B)

7. (3t3 - l0t2 + t - 5) + (t - 4) 8. (1013 - ILx2 - 47x + 30) - (x 't 2)

9. (xa + lx3 - I4x2) - (x - 2) lO. (2xa + l4x3 - 2x2 - l4x)+(x+7)

tt. (y3 -t y, * 10) + 1y + 3) L2, (na - n3 - I0n2 + 4n + 24) + (n + 2)

13. Use synthetic division to find all the factors of 13 + 6x2 - 9x - 54if one of the factors is r - 3.

14. Manufacturing A cylindrical chemical storage tank must havea height 4 meters greater than the radius of the top of the tank.Determine the radius of the top and the height of the tank if thetank must have a volume of 15.71 cubic meters.

@ Glencoe/McGraw-Hill 64' Adv an ced M athe m ati cal C o n ce pts

DATE

Study Guide

The Rotionol Root TheoremThe Rational Root Theorem provides a means of determiningpossible rational roots of an equation. Descartes' Rule of Signscan be used to determine the possible number of positive real zerosand the possible number of negative real zeros.

Example 1 List the possible rational roots ofxs - 5x2 - tZx - $ : 0. Then determinethe rational roots.p is a factor of 6 and g is a factor of 1

possible values ofP: + l, *2, -3. -6possible values of q: +L

possible rational roots, *t *1,-2. -3. -6'qTest the possible roots using synthetic division.

PERIOD

There is ct root at x : -2.The depressed polynomial ts x,2 - 7x * 3.You can use the Quadratic Formula tofind the trvo irrational roots.

1

-12

-2L)

3

6

-6

A

-6_e

a

2

-81

-11

-274

-520

75

-27-72

-300

-21-11-23

L)

z\)7

-1149

Example 2 Find the number of possible positive real zerosand the number of possible negative real zerosfor flx) : {s4 - 1313 - 2lx2 + 38r - 8.

According to Descartes'Rule of Signs. the number ofpositive real zeros is the same as the number of signchanges of the coefficients of the terms in descendingorder or is less than this by an even number. Count thesign changes.

f(x) : 4xa - I3x3 - 2Ix2 + 38r - 8

4 *13 -2t 38 -BThere are three changes. So, there are 3 or 1 positivereal zeros.

The number of negative real zeros is the same as thenumber of sign changes of the coefficients of the termsof f(-x), or less than this number by an even number.

f(-v; : 4(-x)a - 13(-r)3 - 21(-x)2 + 38(-r) - 8

4 13 -2r -38 -8There is one change. So, there is 1 negative real zero.

RationalRootTheorem

LeIaox'+ arxn 1+'..+ an I + an:0 representa polynomial equationof

degree n with integral coefficients. lf a rational number P where p and q

have no common factors, is a root of the equation. then p is a factor of an

and q is a factor of ao.

J

@ Glencoe/McGraw-Hill Advanced Mathematical Concepts

DATE

PracticeThe Rotionol Root Theorem

List the possrb/e rational roots of each equation. Then determinethe rational roots.l.x3-x2-8x+12:0

2.2x3-Bxz-2x*3:0

3.36x4-13x2*1:0

4.x3+3x2-6r-8:0

5. xa - 3r3 - Ilxz + 3x f 10 : 0

6.xa+x2-2:0

7.3x3+x2-8r*6:0

8.13+ 4x2-2x-t15:0

Find the number of possible positive real zeros and the number ofpossrb/e negative real zeros. Then determine the rational zeros.9.f(x):x.3 -2x2 -I9x+20 1.O, f(*):x4 +x3 -7x2 -xt6

ll. Driaing An automobile moving at 12 meters per secondon level ground begins to decelerate at a rate of - 1.6 metersper second squared. The formula for the distanceanobjecthastraveledisd(t):uot*to',,whereuoistheinitial velocity and a is the acceleration. For what value(s) of/ does d(t) :40 meters?

PERIOD

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Study Guide

Locoting Teros of o Potgnomiot FunctionA polynomial function may have real zeros that are not rationalnumbers. The Location Principle provides a means of locatingand approximating real zeros. For the polynomial function o : f(x'),if o and b are two numbers with.f(o) positive and /(b) negative,then there must be at least one real zero betu.'een 0 and b. Forexample, if f(x): x2 - 2,f(0): -2 and/t2t:2, Thus, azeroexists somewhere between 0 and 2.

The upper Bound Theorem and the Lolver Bound Theoremare also useful in locating the zetos of a function and in determiningwhether all the zeros have been found. If a pol1-nomial function P(r)is divided by x - c, and the quotient and the lemainder have no

change in sign, c is an upper bound of the zeIOS of P(r). If c is an.tppui bound of the zeros of P(-x), then -c is a lower bound of thezeros of P(x).

Example I Determine between which consecutive integers thereal zeros of /(r) : xB - 2x2 - Lt * 5 are located.According to Descartes'Rule of Sig:ls. there are two or zetopositive real roots and one negatir-e real root. Use syntheticdivision to evaluate f (x) for consecutir-e integral values of r.

There is a zero at 1. The changesin sign indicate that there arealso zeros between *2 and -land betrveen 2 and 3. This resultis consistent with Descartes'Ruleof Sig:rs.

Example 2 Use the Upper Bound Theorem to show that 3 is anupper bound and the Lower Bound Theorem toshow tlrrat -2 is a lower bound of the zeros off(x):x3-3x2*x-l'-synthetic

division is the most efficient rvay to test potentialupper and lower bounds. First, test for the upper bound.

Since there is no change in the signs in the quotientand remainder, 3 is an upper bound.Now, test for the lower bound of f @) by showing that 2

is an upper bound of fGx).

f(-x):1-r)3 - 3(-x)2 + (-r) - 1- -xB - 3x2 -x - Irt-1 -3 -1 -12l -1 -5 11 l-23

since there is no change in the signs, -2 is a lower bound of f@).

PERIOD

2011

41

-4-5-4-1

-4-3-2-1

01

2J

-b-5-4-J

-z-1

01

-75-28-3

b

5n

2

rl1 -3 1 -1slr 0 1l z

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O Glencoe/McGraw-Hill Adv an ced M athe m ati c al C o n ce Pts

PracticeLocoting Zeros of o potgnorniqt FunctionDetermine between which consecutive integers the real zeros ofeach function are located.l.f(x):3x3 -I0x2 +22x- 4 Z. f(*):2x3 +Ex2 _7x_B

3. f (x) :2xB - 73x2 + I4x - 4 a.f@):x3-I2x2+I7x-g

5, f(x) : 4x4 - l6x3 - 2Ex2 + Lg6x - 146

6..fG):x3-9

Approximate the real zeros of each function to the nearest tenth.7.f(x):3x4+4x,2-7 8. f(xS:3f3 - x t- 2

9.f(*;-4x4-6x2+l LO.f(*):2x3*x2-I

Ll-. f(x): x3 - 2x2 - 2x + B 12. f (*): x3 - 5x2 + 4

use the upper Bound rheorem to find an integral upper bound andthe Lower Bound rheorem to find an integratlower bound of thezeros of each function.L3. f(*):3x4 - x3 - Bx2 - Bx- 20 14. f(*):2xB - x2 + x_ 6

15. For f {x) : xB - Bx2, determine the number and type ofpossible complex zeros. use the Location principle todetermine the zeros to the nearest tenth. The graph has arelative maximum at (0, 0) and a rerative minimum at(2, -4). Sketch the graph.

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Study GuideRotionot Equotions qnd PortioI FroctionsA rational equation consists of one or more rational expressions.one way to solve a rational equation is to multiply each side of theequation by the least common denominator (LCD).Ary possiblesolution that results in a zero in the denominator must be excludedfrom your list of solutions. In order to find the LCD, it is sometimesnecessary to factor the denominators. If a denominator can befactored, the expression can be rewritten as the sum of partialfractions.

Exampte / Solve #, : + * ;5.6(x - z)G@-t+l : o{, - 2,(t - ..5 ) YyPply each.sid.e by the

LCD, 6(x - 2).2(x + 1) : (r - 2)(5x) - 61 1t

2x+2:5rz - tOr * 6 SimpLify.5x2*12x*4:0

(5x - 2)G - 2): 05x-2:0 x-2:

2x:E x:

Write in standard form.Facton

Exampte3 Solve ** hr1Reu.rite the inequality as the related

^ 1-' 3 -rtunctron J(t): i, * i, - r.

Find the zeros of this function.

4tr!' + +ri9\ - 4ttD: 4t(o)"'2t t -"y+t )

5 - 4t:0t: I.25

The zero is 1.25. The excluded valueis 0. On a number line, mark thesevalues with vertical dashed lines.Testing each interval shows thesolution set to be 0 < t < 1.25.

llll#

-2 -1 O , ,'.ru '

02

JSince r cannot equal 2 because a zerodenominator results, the only solution is ].

Example 2 Decompo"" #- x2+2x-3into partial fractions.

Factor the denominator and expressthe factored form as the sum of twofractions using A and B as numeratorsand the factors as denominators.

x2 + 2x - 3: (r - 1Xr + 3)

2x-__l- AJ-B*r*2* S -x-1-"t3

2x-I:A(x+3)+B(x-I)LeL x: L LeL x: -3.2(1) - 1:{(1 + 3) 2(-3) - 1- B(-3 - 1)

L: 4A -7 -- *48t-L p-jt1.- 4 _ D - 4

LI2x*l 4 4 1 _L 7

7T B: r---l + x 3 orZG:1, + 4,*-r3,

/


Rotionot Equotions ond portiot FroctionsSo/ve each equation,1.15-m+8:10

m.

o I ' 6n-9 -2at.:-T---2nJnn

s.ffi-#,:

t, -3x - 29" rz-a*-2t

So/ve each inequality.g.f * Br?

11. 1 + -3J >2L-y

6.2p*t:p+7 p-l

q 4 -3 _ -2bb 3 b b-3

PERIODDATE

Practice

4.t-f:s

15-ppr_1

Decompose each expressio n into partial fractions.g. 71x-7

2x2-3x-2

to.##=#

12.+-5rJ1;3

13. commuting Rosea drives her car B0 kilometers to the trainstation, where she boards a train to complete her trip. Thetotal trip is 120 kilometers. The average speed of the train is20 kilometers per hour faster than that of the car. At what speedmust she drive her car if the total time for the trip is less than

*"

2.5 hours?

O Glencoe/McGraw-Hill 70 Advan ced M ath e m ati cal Co n ce pts

PERIODNAME

Study GuideRodicot Equotions ond lnequotitiesEquations in which radical expressions include variables are knownas radical equations. To solve radical equations, first isolate theradical on one side of the equation. Then raise each side of theequation to the proper power to eliminate the radical expression.This process of raising each side of an equation to a power oftenintroduces extraneous solutions. Therefore, it is important tocheck all possible solutions in the original equation to determine ifany of them should be eliminated from the solution set. Radicalinequalities are solved using the same techniques used for solvingradical equations.

Example 1 Solve g:fyz -21* a 1- 1.g:fp-2*-t1-14: \lr*z - 2* 1 1

64:x2-2x-fl0:x,2-2x-630: (x - 9)(x + 7)

x-9:0 x,+v-O&-u

check both solutions to make sure they are not extraneous.V;r=rc+I_ If(EF-2(e)+1-1ile4-t

Isolate the cube root.Cube each side.

Factor.7:0x: -7

4-73r'

x : -7?,3 :qa.):o,()=t1?):

oJ_

4-r3r'

.tr:9:3:taO=.Jao!t2.):o_d-

Square each side.Diuide each side by 4.

W,-2x+r- rv(-7)2-2e7)+I-Il/e+ - r

Example2 Solve ZfSx + S>2.zfex + s> z4(3x + 5)> 4

3r+5>13x> -4r > -1.33

In order for f 3x + 5 to be a real number, Br + 5 mustbe greater than or equal to zero.3r-l5>0

3x> -5x > -L.67

Since -1.33 is greater than -LG7, the solution is r > -1.88.check this solution by testing values in the intervals definedby the solution. Then graph the solution on a number line.

II

-2 i-r o 1 2

-1 .33

@ Glencoe/McGraw-Hill 71 Advan ced M ath e m ati cal Co n ce pts

DATE

PracticeRodicot Equotions ond lnequotitiesSo/ve each equation.L.fx-2:6 z.</Gr-r:B

s.l/7r + E - -3 +.fax+n-f4yag:1

5.\/x- - s{x + n: -11 G.vEn-s:l[a7n

7.5+2x:\,Fl 2x+I 8.3- Vr+r:14-r

So/ve each inequality.g.r,€r+S>f to.fzt-s<E

tt.fzm+s>b tz. x€il+ s <g

13. Engineering A team of engineers:rrrust design a fuel tankin the shape of a cone. The surface area qfg_gqAe (excludingthe base) is given by the formula g : nf ,z a 62. Find the "*'radius of a cone with a height of 2L meters and a surface areaof 155 meters squared.

!\,. i

iII

I

1

l

l

l

1

l

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PERIOD


Study Guide

J Modeting Reot-wortd Doto with Potgnomiot FunctionsIn order to model real-world data using polynomial functions, youmust be able to identify the general shape of the graph of each typeof polynomial function.

Example f Determine the type of polynomial function thatcould be used to represent the data in eachscatter plot.

The scatter plot seems tochange direction three times,so a quartic function wouldbest fit the scatter plot.

The scatter plot seems tochange direction two times,so a cubic function would bestfit the scatter plot.

I Example 2 An oil tanker collides with another ship andstarts leaking oil. The Coast Guard measuresthe rate of flow of oil from the tanker andobtains the data shown in the table. Use agraphing calculator to write a poll,nomialfunction to model the set of data.Clear the statistical memory and input the data.Adjust the window to an appropriate setting andgraph the statistical data. The data appear to changedirection one time, so a quadratic function wil fit thescatter plot. Press , g.J4li, highlight CALC. and choose5:0uadRes. Then enter E-q_l tl1l E J!g__ trzl ,E,rttl.Rounding the coefficients to the nearest tenth,flx) : -0.4x,2 + 2.8x + 16.8 models the data. Sincethe value of the coefficient of determination 12 is veryclose to 1, the polynomial is an excellent fit.

J

'--{=EH r *b:+*ta=-. 4BFEr'lu'_rB! tb=r lt,I4:4I41c.=18. ?EEEE,EEFEr=. ?FIf51f,5ft,

Time(hours)

Flow rate(100s of liters

per hour)

1

z

3

4A

b

7

8

o

10

18.0

20.5

21.3

21.1

19.9

17.8

15.9

11.3

7.6

3.7

LI-F

a

\rEI

LltlI =

1EI n.5J1.it1.115.51F.E15.9

LJ LITiIIE

6t

Ltitl = I

1EI'I.5t1.3t 1.11.5.51I.Et 5.9

[0, 10] scl:1 by [0.25] scl:5

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DATE PERIOD

PracticeModeting Reot-Wortd Doto with Potgnomiqt FunctionsWrite a polynomial function to model each1. The farther a planet is from the Sun, the

complete an orbit.

set of data.

longer it takes to

source: Astronomy: Fundomenrols ond Fronriers, by.rostrow, Robert, ond rtilt."r, H. r6ffi

2. The amount of food energy produced by farms increases as moreenerg'y is expended. The following table shows the amount ofenergy produced and the amount of energy expended to producethe food.

3. The temperature of Earth's atmosphere varies with altitude.

water quality varies with the season. This table shows theaverage hardness (amount of dissolved minerals) of water in theMissouri River measured at Kansas City, Missouri.

Source: The Encyclopedio of Environmentol Sri"rrs 1g7A

\-

4.

Source: NSTA Energy-Environment Source Book.

Living in ke Environment, by Miller G

@ Glencoe/McGraw-Hill 74 Advan ce d M athe m ati cat Co n c epts

PERIOD

Study Guide

Angtes ond Degree MeosureDecimal degree measures can be expressed in degrees("),minutes('), and seconds(").

Example I a. Change 12.520" to degrees, minutes, andseconds.12.520. : !2" + (0.520 ' 60)' Multiply the decimal portion of

: !2o + 31.2' the degrees by 60 to find minutes.: l2o + 31' + (0.2 ' 60)' Multiply the decimal portion of: 12" + 31' + 12" the minutes by 60 to find seconds.

12.520" can be written as 12o 3]-' 12".

b. Write 24" 15'33" as a decimal rounded to thenearest thousandth.

24o 15', 33" : 24" + 15'(#) + 33" (r.*-):24.259"

24" 15' 33" can be written as 24.259'.

An angle may be generated by the rotation of one raymultiple times about the origin.

J Example 2 ff;rrff.

angle measure represented bv each

a. 2.3 rotations clockwise2.3 x -360 : -828 Cl.oclzwise rotations haue negatiue measures.

The angle measure of 2.3 clockwise rotations is -828''

b. 4.2 rotations counterclockwise4.2 x 360 : 1512 couruterclockwise rotations haue positiue

MECLSUTCS.

The angle measure of 4.2 counterclockwise rotations is 1512".

If a is a nonquadrantal angle in standard position, its referenceangle is defined as the acute angle formed by the terminal side ofthe given angle and the r-axis.

ReferenceAngle Rule

For any angle a, 0o < a < 360", its reference angle a' is defined by

a. a, when the terminal side is in Quadrant l,b. 180' - a, when the terminal side is in Quadrant ll,

c. (r - 180', when the terminal side is in Quadrant lll, and

d. 360' - a, when the terminal side is in Quadrant lV.

Example 3 Find the measure of the reference angle for 22A".

Because 220" ts between 180' and 270',theterminal side of the angle is in Quadrant III'220" - 180o:40oThe reference angle is 40'.

DATE

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@ Glencoe/McGraw-Hill Adv an ced M ath e m ati cal C o ncePts

PracticeAngtes ond Degree

PERIOD

Angtes ond Degree Meosure

Change each measure to degrees, minutes, and seconds.r.28.955" 2. -57.327"

Write each measure as a decimal degree to the nearestthousandth.3.32" 29',L0" 4, -73" r4',35"

Give the angle measure represented by each rotation.5. 1.5 rotations clockwise 6. 2.6 rotations counterclockwise

ldentify all angles that are coterminal with each angle. Then findone positive angle and one negative angle that are coterminalwith each angle.7.43" 8. -30'

lf each angle is in standard positionn determine a coterminal anglethat is between o" and 360", and state the quadrant in which theterminal side /ies,9.472' 10. -gg5'

Find the measure of the reference angle for each angle.I1.227' 12. 640"

L3. Na,oigation For an upcoming trip, Jackie plans to sail fromSanta Barbara Island, located at 33' 28' 82" N, 119" 2, 7,, W, toSanta Catalina Island, located at 33.386" N, 118.480" W. Writethe latitude and longitude for santa Barbara Island as decimalsto the nearest thousandth and the latitude and longitude forSanta Catalina Island as degrees, minutes, and seconds. ,l

J

O Glencoe/McGraw-Hill 80 Adv an ced M ath e mati cal C o n cepts

PERIOD

Study Guide

J Trigonometric Rotios in Right TriongtesThe ratios of the sides of right triangles can be used to definethe trigonometric ratios known as the sine, cosine, andtangent.

Example I Find the values of the sine, cosine, and tangentfor LA.First find the length of BC.(AC)2 + (BC)2 : (AB)2 Pythagorean Theorem

B

k10 cm

102 + (BC)2 :202(BC)2: 300

side opoositesrrt l{ : =-------nyPoten use

sinB: A orffisxffi "' %

hvoolenusecsc ll : --'j--side oppostte

csc4:*orf26

Substitute 10 for AC and 20 for AB. c

BC : V300 or 10VB Take the squdre root of each side.Disregard the negatiue root.

Then write each trigonometric ratio.. A sideopposit? --- I siclectdjacent

o.ll.. -

--

vvDlr - )h.ypotenuse t1.\'Potenuse

sinA:Wor* cosA:#o.iTligonometric ratios a.re often simplified but neter tt'ritten as

mixed numbers.

Three other trigonometric ratios, called cosecant, secant,and cotangent, are reciprocals of sine, cosine, and tangent,respectively.

Exampte 2 Find the values of the six trigonometricratios for LR.

First determine the length of the hypotenuse.(Rflz + (SZ)2: (RS)2

''E2 + 32: (RS)2

(RS)2 : 234

PJ'thagorean TheoremRT:15,57:3

tanA- si4eoppositesrde adlacent

tan A: # o, \./5

R 15cm T

_ side oppositeside adjacent

:*ori_ side adjecent

side opposite

RS : \Ei4 or 3f 26 Disregard the negatiue root.

L1

cos R - si4e adjacent tan .Rn.ypote n u se

cos R : t#b o'+# tan R

secR : lY.Po'9,!"t, cot,Rstde 1dJclcent

secR:# orP cotB: for5

g@ Glencoe/McGraw-Hill 81 Advanced M athematical Concepts

DATE

PracticeTrigonometric Rotios in Right TriongtesFind the values of the sine, cosine, and tangent for each /_8.

PERIOD

\l

1.8 2.n

3. If tan 0 :5, find cot g. 4. If sin g : $, fina csc g.

Find the values of the six trigonometric ratios for each .1-s.

21 m

o

7 in.

R

R 7m S

7. Physics suppose you are traveling in a car when a beam of lightpasses from the air to the windshield. The measure of the angleof incidence is 55o, and the measure of the angle of refraction it35" 15'. Use Snell's Law, ## : n, tofind the index of refraction nof the windshield to the nearbst thousandth.

6.o. \-

O Glencoe/McGraw-Hill 82 Advan ced M ath e mati cal Co ncepts

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Study GuideTrigonometric Functions on the Unit Circte

Example 1 Use the unit circle to find cot (-27O'\.

The terminal side of a -270o angle instandard position is the positive y-axis,which intersects the unit circle at (0, 1).

By de{inition, cot (-270"): i ot t.Therefore, cot (-27 0') : 0.

Example 2 Find the values of the six trigonometric functionsfor angle 0 in standard position if a point withcoordinates (-9, 12) lies on its terminal side.

We know that x: -9 and y : 12. We need tofind r.

,: {*' + y',: {Gg)\ rz,r: f225 or 15

sing:#*tcsco:#*t

Example 3 Suppose 0 is an angle in standard position whoseterminal side lies in Quadrant I. If cos 0 : f,, findthe values of the remaining five trigonometricfunctions of 0.

ooor': x' + \)"P9 ng , " 9it' : ;5' -f \t'16: !2

*A: ctJ

Since the terminalbe positive.

Istnd:T5

tan"A,:

cot0:csc0: sec0:

Pythagorean TheoremSubstitute 5 for r and 3 for x.

Take the square root of each side.

side of 0 lies in Quadrant I, y must

PERIOD

sine:l coso:f ane:f,

csc0-l sec0:I cot0:{yxy

TrigonometricFunctions ofan Angle inStandard Position

Pythagorean Theorem

Substitute -9 for x and 12 for y.

Disregard the negatiue root.

cosg:#"r -i tano:3o, -+secg:5"r-* cot0:#*-t

4J

C)

45DL)

54


DATE PERIOD

3. sin (-90')

8. (-3, -4)

PracticeTrigonometric Functions on the Unit circteUse the unit circle to find each value.1. csc 90' 2, tan270'

use the unit circle to find the values of the six trigonometricfunctions for each angle,4. 45'

5. 120'

Find the values of the six trigonometric functions for angle 0 instandard position if a point with the given coordinates /ies on itsterminalside.6. (-1,5) 7. (7,0)

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O Glencoe/McGraw-Hill 84' Advan ced M ath e m ati cal Co n cepts

t

DATE

Study GuideApptUing Trigonometric FunctionsT?igonometric functions can be used to solve problemsinvolving right triangles.

Example I

sin 7: Zu

sin 45" : #

cos 30o - 2'6 cos :r

. side oppositeSLN:.+

h.ypotenuse

Substitute 45' for T and 20 for u.

PERIOD

If T :45" and u :2O, find f to the nearest tenth.From the figure, we know the measures of anangle and the hypotenuse. We want to know themeasure of the side opposite the given angle. Thesine function relates the side opposite the angle rand the hypotenuse.

20 sin 45" : t Multiply each side by 20.14.14213562: t Use acalculator.

Therefore, / is about 14.1.

Example 2 Geom.etry The apothem of a regular polygon is themeasure of a line segment foom the center of thepolygon to the midpoint of one of its sides. Theapothem of a regular hexagon is 2.6 centimeters.Find the radius of the circle circrrmscribed aboutthe hexagon to the nearest tenth.

First draw a diagram. Let a be the anglemeasure formed by a radius and its adjacentapothem. The measure of o is 360" + \2 or30o. Now we know the measures of an angleand the side adjacent to the angle.

side adjacenthypotenuse

r cos 30o : 2.6 Multiply each side by r.

r: -#0. Diuide each sid.e by cos 30'.

r: 3.0022214 Use a calculator.

Therefore, the radius is about 3.0 centimeters.

@ Glencoe/McGraw-Hill 85 Adv anced M ath em ati cal Co n ce pts

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PracticeApptging Trigonometric FunctionsSo/ye each problem. Round to the nearest tenth.l.If A: 55o 55'and c: t6. find o.

2. If a: 9 and B : 49o. find b.

3. If B : 56o 48' and c : 63.1, find b.

4. If B : 64" andb : 19.2, find a.

5. If b : 14 and A : 16", find c.

6. Construction A 30-foot ladder leaning againstthe side of a house makes a70" 5' angle with theground.a. How far up the side of the house does the

ladder reach?

b. What is the horizontal distance between thebottom of the ladder and the house?

7. Geornetry A circle is circumscribed about a regular hexagon with anapothem of 4.8 centimeters.

a. Find the radius of the circumscribed circle.

b. What is the length of a side of the hexagon?

c. What is the perimeter of the hexagon?

8. Obseraation A person standing 100 feet from the bottomof a cliff notices a tower on top of the cliff. The angle ofelevation to the top of the cliff is 30'. The angle of elevationto the top of the tower is 58'. How tall is the tower?

PERIOD

\

\*

trtrnnnft ntrn

@ Glencoe/McGraw-Hill 86 Adv an ced M ath e m ati cal Co n ce pts

DATE

Study GuideSotving Right TriongtesWhen we know a trigonometric value of an angle but not thevalue of the angle, we need to use the inverse of thetrigonometric function.

Trigonometric Function lnverse Trigonometric RelationY:slnxY:cosxy:lanx

x : sin 1 y or x: arcsin yx : cos 1 y or x: arccos yx:tan-1 yorx:arclany

Example I Solve tam x: \,6.If tan *: {5, then r is an angle u,hose tangentis r,6.

x: atctart f,6From a table of values, you can determine thatr equals 60o, 240", or any angle coterminal withthese angles.

Example 2 If c : 22 and b : 12, find B.

In this problem, we know the side opposite theangle and the hypotenuse. The sine functionrelates the side opposite the angle and thehypotenuse.

sinB:6c

sinB:#n: sin-r(S)

cos A

cos A

A

A

PERIOD

J. side oonositesln: --- 1-l--

hypotenuse

Substitute 12 for b and 22 forDefinition of inuerse

c.

B - 33.05573115 or about 33.1'.

Example 3 Solve the triangle where b :20 and c : 35, giventhe triangle above.

a2+b2:c2a2 + 202 :352

a : {828a: 28.72281323

55.15009542 + B: 90B: 34.84990458

Therefore, d : 28.7, A : 55.2, and B : 34.8o.

_bc

_2035

: ^^.-ll 20 \""" \gb /- 55.15009542

J

@ Glencoe/McGraw-Hill 87 Advanced M ath e m ati cal C o n ce pts

NAME DATE PERIOD

Practice

Sotving Right Triongles

So/ve each equation if O" = x < 360".

^, [ol.cos*:t 2. tanx: I 3. sinr:

Evaluate each expression. Assume that all angles are in Quadrant I.

4. tan(r"" 'f) 5. tan (."'-'3) 6. cos (arcsi" S)

Solve each problem. Round to the nearest tenth.7.If q : 10 and s : 3, find S. s

8. Ifr : 12 and s : 4,findR.

9,If q : 20 and r : 15, find S.

Solye each triangle described, given the triangle at the right.Round to the nearest tenth, if necessary.

10. o :9,8:49"

l1-. A: 16o, c : 14

L2.a:2,b:7

13. Recreation The swimming pool at Perris Hill Plunge is 50 feetlong and 25 feet wide. The bottom of the pool is slanted so thatthe water depth is 3 feet at the shallow end and 15 feet at thedeep end. What is the angle of elevation at the bottom of thepool?

\--

t,

\-

\

@ Glencoe/McGraw-Hill 88 Advan ced M ath e m ati cal Concepts

JStudy Guide

The Low of SinesGiven the measures of two angles and one side of a triangle,we can use the Law of Sines to find one unique solution forthe triangle.

Example 1 Solve LABC if A :30o, B : 100o, and o : 15.

First find the measure of tC.c : l-80. - (30" + l-00') or 50"

Use the Law of Sines to find b and c.

a-bsinA sinB15_b

sin 30' sin 100"

15 sin 100' - z-

.it-30" - "29.54423259: b

a -__p_:___.csin A sin B sin CLaw of Sines

c_a.*C - sin Ac:15

sin 50' sin 30o

.. - 15 sin 50o' sin 30o

c: 22.98133329

Therefore, C : 50o, b : 29.5,and c : 23.0'

The area of any triangle can be expressed in terms of trvosides of a triangle and the measure of the included angle.

Area (K) of aTriangle I X: lOcsinn K: |acsinB K: labsinC

Example 2 Find the area of LABC if a : 6.8, b : 9.3, and C : 57".

K:+"bsinCx: !{a.ax9.3) sin 57'

A-K: 26.5L876336

The area of AABC is about 26.5 square units.

6.8

r

/

@ Glencoe/McGraw-Hill 89 Adv an ced M athe m ati cal C on cePts

DATE PERIOD

PracticeThe Low of Sines

So/ve each triangle. Round to the nearest tenth.l.A:38o,8:63o,c:15 2. A:33', B : 29", b : 4l

3.A: 150', C :20o,a:200 4.A:30',8:45",o:10

Find the area of each triangle. Round to the nearest tenth.5. c : 4,A: 37", B: 69o 6. C:85",a:2,8:19o

7.A:50",b:12,c:14 8. 6 : 14, C: 110o, B :25o

9.b: L5,c:20,A: 115' lO. a: 68, c : 110, B : 42.5o

LI. Street Lighting A lamppost tilts toward the sunat a2" angle from the vertical and casts a2l-footshadow. The angle from the tip of the shadow to thetop of the lamppost is 45". Find the length of thelamppost.

\_

L

O Glencoe/McGraw-Hill 90 Advan ced M athe mati cal C o n cepts

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DATE

Study Guide

The Ambiguous Cose for the Low of SinesIf we know the measures of two sides and anonincluded angle of a triangle, three situationsare possible: no triangle exists, exactly one

triangle exists, or two triangles exist. A trianglewith two solutions is called the ambiguous case.

Example Find all solutions for the triangleif a: 20, b: 30, and A : 40o. If nosolutions exist, write none.

Since 40o < 90o, consider Case 1.

b sin A : 30 sin 40'bsinA-19.28362829Since 19.3 < 20 < 30, there are trvo solutions forthe triangle.Use the Law of Sines to find B.

20:30c--sin 40o sin B sinA

sinB: e%g/ "lein lg'\B: sin ,l+ 20 l

B:74.61856831

So, B - 74.6o. Since we know there ale t\\-o solutions,there must be another possible measurement for B.In the second case, B must be less than 180' andhave the same sine value. Since we knorv that ifcr ( 90, sin a: sin (180 - ct), 180' - 74.6'or 105.4o

is another possible measure for B' Norv solve thetriangle for each possible measure of B.

PERIOD

bsin B

J

Solution I

C : 180' - (40' + 74.6") or 65.4"

CT-CsinA sin C20:c

sin 40' sin 65.4'20 sin 65.4'

.* 40'

c: 28.29040558

One solution is B : 74.6",C : 65.4", and c :28.3.

Solution ll

C : 180' - (40' + 105.4') or 34.6'AC

sir.A - sitl C20

sin 40' sin 34.6'

^ - 20 sin 34.6't' - "nl 40'

c: 17.66816088

Another solution*is B : I05.4o,C:34.6", and c: l7i.

Case 1:A < 90" for a. b. andAa<bsinA no solutiona:bsinA one solutiona>b one solutionbsinA<a<b two solutions

Case2:A>90'a<b no solutiona>b one solution

./

O Glencoe/McGraw-Hill 91 Advan ced M athe m ati cal C o n ce Pts

NAME DATE PERIOD

PracticeThe Ambiguous Cose for the Low of Sines

Determine the number of possible solutions for each triangle.L.A:42",a:22,b:12 2. a,: 15,b:25,A:85o

3.A:58o,a:4.5,b:5 4.A:110o,o:4,c:4

Find all solutions for each triangle. lf no solutions exist, writenone. Round to the nearest tenth.

5. b :50, a: 33, A:132" 6. a: I25,A:25", b : 150

7.o:32,c:20,4: Il2' 8. a: 12,b: t5,A: 55o

9.4:42",a:22,b: 12 IO. b: 15,c:13, C:50'

Il. Property Maintenq.nce The McDougalls plan to fence a triangular parcel of theirland. One side of the property is 75 feet in length. It forms a 38'angle with anotherside of the property, which has not yet been measured. The remaining side of theproperty is 95 feet in length. Approximate to the nearest tenth the length of fenceneeded to enclose this parcel of the McDougalls'lot.

\-

t

t,


Study GuideThe Low of Cosineswhen we know the measures of two sides of a triangle andthe included angle, we can use the Law of cosines to findthe measure of the third side. often times we will use boththe Law of cosines and the Law of sines to solve a triangle.

lz : f Bg2*4878k:19.81003534

The area of the triangle is about 19.8 square

11.5(11.5 - 5X11.5 - 8X11.5 - 10)

PERIOD

J

a2:b2+c2-2bccosALaw of Cosines ', 62 = g2 -r c2 - 2ac cos B

c2:a2+b2-2abcosC

k:

Example 1 Solve LABC if B:4Oo, a:12, and c :6.b2 : a2 I c2 - 2ac cos B Law of Cosinesb2:122 + G2 - 2(12)(6) cos 40"b2: 69.68960019b - 8.348029719

So, b - 8.8.

--fu : *fe Law of Sines

8.3_6sin 40o sin C

sinC:6sil:CI'6.3

C : sin rf 0 sin-+o'i\s.alC : 27.68859159

So, C : 27.7".A - 190. - (40" + 27.7") - 1L2.3"

The solution of this triangle is b : 8.8, A - lr2.Bo, and c : 27.7"

Example 2 Find the area of LABC if a: 5, b : g, and c : 10.

First, find the semiperimeter of AABC.t:!{o+b+c)

":jts+8+10)s : 11.5

Now, apply Hero's Formulalz : \/ s(s - oXs - b)(s - cj

units.)

T


NAME DATE PERIOD

PracticeThe Low of Cosines

So/ve each triangle. Round to the nearest tenth.L.a:20,b:L2,c:28 2.a:10,c:8,8:100o

3.c:49,b:40,A:53o 4. d.:5,b:7,c:I0

Find the area of each triangle. Round to the nearest tenth.5.a:5,b:12,c:L3 6. e.: Il,b:13, c : 16

7.d,:L4,b:9,c:8 8. a:8rb:7,c:3

9. The sides of a triangle measure L3.4 centimeters,18.7 centimeters, and26.5 centimeters. Find the measureof the angle with the least measure.

lO. Orienteering During an orienteering hike, two hikersstart at point A and head in a direction 30' west of southto point B. They hike 6 miles frorn pointA to point B. Frompoint B, they hike to point C and then from point C back topointA, which is 8 miles directly north of point C. Howmany miles did they hike from point B to point C?

lv

B

94'@ Glencoe/McGraw-Hill Advan ced M ath e mati cal Co n cepts

v

NAME

Study Guide

Ii, ii;1;:,::a!t1_

Angles ond Rodion MeosureAn angle of one complete revolution can be represented eitherby 360'or by 2rr radians. Thus, the following formulas can be

used to relate degree and radian measures.

Degree/Radian I t raOian : $ O"gt"es or about 57'3"

Conversion Formulas I t Oegree : ffi radians or about 0.017 radian

Example 1 a. Change 36'to radian measure in terms of n.

b. Chang " -+- radians to degtee measure.

b. -+: -tlj, *: -1020'

Example 2 Evaluate sin f.The reference angl e for ff is f,. Sinc e t : 45",

the terminal side of the angle intersects the unitcircle at a point with coordinates "f (+.+)Because the terminal side of + lies in Quadrant II,

the r-coordinate is negative and the y-coordinate is

positive. Therefore, sin f : +.

Example 3 Given a central angle of 147", find the lengthof its intercepted arc in a circle of radius10 centimeters. Round to the nearest tenth.First convert the measure of the central anglefrom degrees to radians.

a.36o:36'xl180':a5

180"'E

147":147"x1h;49li,60

Then find the length of the arc.

s:r0 Formula for the length of an arc

1 degree : dn

s:ro(ff) ':70,u:#fs:25.65634The length of the arc is about 25.7 cm.

@ Glencoe/McGraw-Hill 99 Adv an ced M ath e mati cal C o n cePts

PracticeAngtes ond Rodion Meosurechange each degree measure to radian measure in termsof n.l. -250" 2. 60

4.970" 5. 18'

3

change each radian measure to degree measure. Round tothe nearest tenth, if necessary.7.4n s.ff

tt. -2.56

14, cos !

3. -145.

6. -920.

9. -1

12.-+

15. sin f

1s. sin f

10.#

Evaluate each expression.t3. tan f

Given the measurement of a central angle, find the lengthof its intercepted arc in a circle of radiis 1o centimeters,Round to the nearest tenth.le.ff 20.+

16. lutr l-14 t7. cos !

2t. t

Find the area of each sector, given its central angle 0 andthe radius of the circre. Round to the nearest teitn.22.0:ft,r:14 zg. o:+,r:4

O Glencoe/McGraw-Hill 100 Advan ce d M ath e m ati cal Co n cepts

J

NAME

Study Guide

Lineor ond Angutor VetocitgAs a circular object rotates about its center, an object at theedge moves through an angle relative to the object's startingposition. That is known as the angular displacement, orangle of rotation. Angular velocity c,.r is given by , : !,where 0 is the angular displacement in radians and t is time.Linear velocity u is given by u : r!, where f .ep"e."nts the

angular velocity in radians per unit of time. Since , : f, thisformula can also be written as u : r@.

Example 7 Determine the angular displacement in radiansof 3.5 revolutions. Round to the nearest tenth.

Each revolution equals 2n radtans. For3.5 revolutions, the number of radians is3.5 x 2r, or 7rr. 7n radians equals about22.0 radians.

Example 2 Determine the angular velocity if 8,2 revolutionsare completed in 3 seconds. Round to the nearesttenth.

The angular displacement is 8.2 x 2ir. or 76.4n radians.ea:: tL6.4rr@-

3a - 17.17403984

The angular velocity is about 17.2 radtans per second.

Example 3 Determine the linear velocity of a point rotatingat an angular velocity of 13zr radians per secondat a distance of 7 centimeters from the center ofthe rotating object. Round to the nearest tenth.

u:r0)u :7(LBn)u :285.8849315

0:76.4rr,t:3Use a calculator.

f:7, a: 73rUse a calculaton

The linear velocity is about 285.9 centimeters per second.

101

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@ Glencoe/McGraw-Hill Adv an ced M ath e m ati c al Co ncepts

NAME

Determine each angular velocity.7. 2.6 revolutions in 6 seconds

DATE PERIOD

3. 85 revolutions

6. 17.8 revolutions

PracticeLineor ond Angutor VetocitgDetermine each angular displacement in radians. Round tothe nearest tenth.

1. 6 revolutions 2. 4.3 revolutions

4. ll.5 revolutions 5. 7.7 revolutions

Round to the nearest tenth.8. 7.9 revolutions in 11 seconds

9. 118.3 revolutions in 19 minutes 10. 5.5 revolutions in 4 minutes

12. 14 revolutions in 2 minutes11.22.4 revolutions in 15 seconds

Determine the linear velocity of a point rotating at the givenangular velocity at a distance r from the center of the Lrotating object. Round to the nearest tenth.13. ro : 14.3 radians per second, r : 7 centimeters

14. a:23radians per second, r:2feet

15. a : 5.4r radians per minute, r : 1.3 meters

16. o : 41.7 n radians per secon d, r : 18 inches

17. o : 234 radians per minute, r : 31 inches

18. Clocks Suppose the second hand on a clock is 3 inches long.Find the linear velocity of the tip of the second hand.

@ Glencoe/McGraw-Hill 102 Adv an ced M ath e m ati cal Co nce pts

NAME

Study Guide

J Orophing Sine ond Cosine FunctionsIf the values of a function are the same for each giveninterval of the domain, the function is said to be periodic.Consider the graphs of y : sin r and y : cos r sh6wn belowNotice that for both graphs the period is 2n and the range isfrom - 1 to 1, inclusive.

Example 7 Find sin $ by referring to the graph of the sinefunction. -

The period of the sine function is 22. Since T , ,r.rewrite T ""

a sum involving 2n.

+:2n,t) + ! This is aform"f + * 2rn.

So, sin +: sin f or -1.

Example 2 Find the values of 0 for which cos 0 : 0 is true.Since cos 0 : 0 indicates the r_intercepts of thecosine function, cos 0: 0 if 0: t t rrn, where ruis an integer.

Example 3 Graphy : sin x for Gtr < x < 8r.The graph crosses the r-axis at 6n,7r, and 8rr. Its maximum value of 1

is at r : +, and its minimum valueof _1is atr : +. Use thisinformation to sketch the graph.

Properties of the Graph of y = s|n, Properties of the Graph of y = s.. ,The x-intercepts are located at z.n. where n is aninteger.

The x-intercepts are located at f + rz.n, where n isan integer.

The y-intercept is 0. They-intercept is 1

The maximum values are y : 1 and occur when*: i * 2Tn, when n is an integer.

The maximum values are y : 1 and occur whenx : Tn. where n is an even integer.

The minimum values are y : - 1 and occur whent : + + 2rn, where n is an integer.

The minimum values are y : - 1 and occur whenx : Tn, where n is an odd integer.


NAME DATE

PracticeGrophing Sine ond Cosine FunctionsFind each value by referring to the graph of the sine or thecosine function.

1. cos zr

Find the values of 0 for which each equation is true.4. sin 0 :0 5. cos 0: I

Graph each function for the given interval.7.y: sinx; -f =r=t

2. sin !

PERIOD

B. sint+)

6.cos0:-I

\_

8.y:cosrc; 7r<x39n

\*

Determine whether each graph is y : sin x, y : cos x, orneither.

9.

lI. Meteorotogy The equationy :70.5 + 19.5 sin l*r - Olmodelsthe average monthly temperature for Phoeni", Ril"or", irl ;"g"*,Fahrenheit. In this equation, t denotes the number of months, witht : I representing January. what is the average monthly temperaturefor July?

10.

v

@ Glencoe/McGraw-Hill 104. Advanced Mathematical Concepts

NAME DATE PERIOD

Study Guide

J Amptitude ond Period of Sine ond Cosine FunctionsThe amplitude of the functions y : A sin 0 and y : A cos 0 is

the absolute value of A, or lA | . The period of the functions

y : sin Iz0 and,y : cos k0 is ff, where k > 0.

Example 1 State the amplitude and period for the functiony: -z "o" f;.The definition of amplitude states that the amplitudeof y : A cos g is lA | . Therefore, the amplitude of!:-2cosf is l-2 1,or2.

The definition ofperiod states that the period ofy : cos ku,:" +. Since -2 cos f equals *2 cos (10), ttt.

zTtperiod is 1 or 8zr.

Example 2 State the amplitude and period for the functionI:3 sin 20. Then graph the function.

Since A: 3, the amplitude is l3 | or 3.

Since h : 2, the period is ! or n.

Use the amplitude and period above and thebasic shape of the sine function to graph theequation.

Example 3 Write an equation of the sine function withamplitude 6,7 and period 3zr.

The form of the equation will be y : A sin le?.

First find the possible values ofA for anamplitude of 6.7.

lAl : 6.7A: 6.7 or -6.7

Since there are two values ofA, two possibleequations exist.

Now find the value of ft when the period is 3zr.

T: t, The period' of the sine function is !k:!" or!

The possible equations are y : 6.7 sin fA or

y : -6.7 sin ]0.

Y=3sin20

J


DATE PERIOD

PracticeAmptitude ond Period of Sine ond Cosine FunctionsSfate the amplitude and period for eachgraph each function.L. Y: -2sino

\-

2.

function. Then

y:4cos$

3. y : 1.5 cos 49 4.y:

Write an equation of the sine function with each amplitudeand period.5. amplitude : 3, period :2n 6. amplitude : 8.5, period : Gn

Write an equation of the cosine function with eachamplitude and period.7. amplitude : 0.5, period :0.2rr 8. amplitude : f, neriod : ln

9. Music A piano tuner strikes a tuning fork for note A above middte Cand sets in motion vibrations that can be modeled by the equationy : 0.001 sin 8802r/. Find the amplitude and period for the function.

-! sin$

L

@ Glencoe/McGraw-Hill 106 Advan ced M athe m ati cal C o n cepts

Study GuideTrqnstqtions of sine ond cosine FunctionsA horizontal translation of a trigonometric function is called aphase shift. The phase shift of the functions y : A sin (h,0 + c)andy:A cos (lzg + c)is -ft, where t?> 0.If c > 0, the shift isto the left. If c < 0, the shift is to the right. The vertical shift ofthe functionsy:A sin (k0 + c) + h andy:A cos (kg + c) + hish.rf h > 0, the shift is upward. rf h < 0, the shift is downward. Themidline about which the graph oscillates ts y : h.

Example I State the phase shift for y: sin (40 + rr).Then graph the function.The phase shift of the function is -f, o, -t.To graph y : sin (40 + zr), consider the graphof y : sin 49. The graph of y : sin 4g has anamplitude of 1 and a period of ,L. Graph thisfunction, then shift the graph -f.

Example 2 state the vertical shift and the equation of themidline for y: B cos 0 + 2. Then graph the function.The vertical shift is 2 units upward. The midlineis the graph of y : 2.

To graph the function, draw the midline.Since the amplitude of the function isl3l, or 3, draw dashed lines parallel tothe midline which are 3 units above andbelow y : z.That is,y : 5 and ! : -1.Then draw the cosine curve with aperiod of 2n.

Example 3 write an equation of the cosine function withamplitude 2.9, period !, Vh^re shift -fi, and,vertical shift -3.The form of the equation wilt be ! : A cos (k0 + c)+ h. Find the values of A, le, c, and h.

:2.9 cz:2.9 or -2.9

? The period is !.5

hz

The possible equations are y : -rL.g "or (So * +) - t

,lr

A|

hz

lAlA

9*I?

k:

_ L _ ltk- z_c _ _1r52

^_Sn(- -: 2

h: -3

TJ3cose +z

The phase shift is -f.k:b

O Glencoe/McGraw-Hill 107 Advan ced M ath e m ati cal C o n ce pts

DATE PERIOD

PracticeTronstqtions of sine ond cosine Functionsstafe the vertical shift and the equation of the midline foreach function. Then graph each function.l.Y:4cos0+4 2. Y : sin2g - 2

sfate the amplitude, period, phase shift, and vertical shiftfor each function.,Then graph the function.B.y:2sin (o+;)-s i:;:f cos (zs_ n)+2

write an equation of the specified function with eachamplitude, period, phase shift, and vertical shift.5. sine function: amplitude : 15, period : 4ir, phase shift: ff,vertical shift: -10

6. cosine function: amplitude : f, perio d : t, phase shift : -f, vertical shift : b

7. sine function: amplitude : 6, perio d : n, phase shift : 0, vertical shift : _32


T@ Glencoe/McGraw-Hill 109 Advanced M athe m ati cal C o n ce pts

DATE PERIOD

Study Guide

J Modeting Reot-Wortd Doto with Sinusoidot Functions

Example

ta: ff:tzL-7tIL_

Cr

h: h : *+- or 57 h is half the sum of the greatest ualueand the least ualue.

The period is 12.

values into the general form of

+h !:27 sin:[r -c)+StTo compute c, substitute one of the coordinatepairs into the equation.

^,- 27 sin (4t + c\ + 57r-H'",-'\6" '") ''

"n - 27 sin l*tl) + cl + 57 f1se (t, y) : (1,30).t)u- rb r

The table shows the average monthlytemperatures for Ann Arbor, Michigan. Writea sinusoidal function that models the averagemonthly temperatures, using t : I to representJanuary. Temperatures are in degreesFahrenheit ("F).

These data can be modeled by a function of theform ! : A sin (h,t * c) + h, where f is the time inmonths.

First, find A, h, and k.

Az A : *;- or 27 A is hatf the difference betweenthe greotest temperature andthe least temperature.

Substitute thesethe function.

y:Asin(kt+c)

nn-27sin/4+c)_zt_ \b )

-27 : sin 14 + c\27 ""'\6 '-lsin-1 (-1):t*"

sin-r (*1) - 4: ct)

-2.094395702: c

Jan. 30'Feb. 34'Mar. AJ

Apr. cvMay 71'-

June 80'July 84'Auq. 81"

Sept. 74',

Oct. 62"

Nov. 4g'Dec. 35"

Subtract 57 from each side.

Diuide each side by 27.

Definition of inuerse

Subtract ff fro^ each side.

Use a calculaton

J

The function y : 27 sin (tt - 2.09) * 57 is one

model for the average monthly temperature inAnn Arbor, Michigan.

J

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PracticeModeting Reot-Wortd Doto with Sinusoidot Functionsl' llet-eorology -\I1e

average monthly temperatures in degrees Fahrenheit (.F)for Baltimore, Maryland, are given below.

a. Find the amplitude of a sinusoidal function that models themonthly temperatures.

b. Find the vertical shift of a sinusoidal function that models themonthly temperatures.

c. what is the period of a sinusoidal function that models themonthly temperatures?

d. write a sinusoidal function that models the monthlytemperatures, using t : ! to represent January.

e. Acco_rding to your model, what is the average temperature inJuly? How does this compare with the actuir average?

f. According to your model, what is the average temperature inDecember? How does this compare with the actuai average?

Boating A buoy, bobbing up and down in the water as wavesmove past it, moves from its highest point to its lowest point andbu* to its highest point every 10 seconds. The distance betweenits highest and lowest points is 3 feet.a. What is the amplitude of a sinusoidal function

the bobbing buoy?

b. what is the period of a sinusoidal function that models thebobbing buoy?

write a sinusoidal function that moders the bobbing buoy,using t : 0 at its highest point.

According to your model, what is the height of the buoy att : 2 seconds?

e. According to your model, what is the height of the buoy att : 6 seconds?

that models

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Study Guide

Grophing Other Trigonometric FunctionsThe period of functions y : csc k0 and,y : sec k0 is ff, wherel? > 0. The period of functions y : tan k0 and y : cot k0 is f,,where l? > 0. The phase shift and vertical shift work the sameway for all trigonometric functions. For erample, the phaseshift of the function y : tan (k0 + c) * h ts -lfi, and itsvertical shift is h.

Example 7 Graph y : tant x.

To graph y : tan r, first draw theasymptotes located at tc : fn, wheren is an odd integer. Then plot thefollowing coordinate pairs and drawthe curves.

(-+'*t), {-,' u, (-+,1), (-?, -1),(0, o), (i,r),(+, -t), G, o), (+, 1)

Example 2 Graphy : sec (20 + n) + 4.

Since k : 2, the period j" T or z. Sinc e c : n,the phase shift is -f,.Tll'e vertical shift is 4.

Using this information, follow the steps forgraphing a secant function.

Step 1 Draw the midline, whichis the graph of y : 4.

Step 2 Draw dashed lines parallel tothe midline, which are 1 unitabove and below y : 4.

Step 3 Draw the secant curve witha period of a'.

Step 4 Shift the graph f units tothe left.

Notice that the range values for the interval -+ = * = -trepeat for the intervals -S = * = ; and, ! = * = *.So, the tangent function is a periodic function with a period of f, or n.

PERIOD

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PracticeGrophing Other Trigonometric FunctionsFind each value by referring to the graphs of the trigonometricfunctions.r. tan (-+) z. *t(!)

3. sec 4n 4. csc ?+)

Find the values of 0 for which each equation is true.5. tan 0 :0 6. cot 0:0

7. csc 0: I 8.sec0:-L

PERIOD

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Graph each function.9.y:tan(20+n)+7

I1-.y:csc0*3

10. v : cot (9 - +\ - t--'J """\2 2l 4

L2.y:sec($+.) -t


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Study GuideTrigonometric lnverses ond Their GrophsThe inverses of the Sine, Cosine, and Tangent functions arecalled Arcsine, Arccosine, and Arctangent, respectively. Thecapital letters are used to represent the functions withrestricted domains. The graphs of Arcsine, Arccosine, andArctangent are defined as follows.

Arcsine Function Given y : Sin x, the inverse Sine function is defined by theequation/ - Sin-t x or y : Arcsin x.

Arccosine Function Given y : Cos x, the inverse Cosine function is defined by theequationy : Cos 1 x or y : Arccosx.

Arctangent Function Given y - Tan x, the inverse Tangent function is defined by theequation y - fan 1 x or y: Arctan x.

Example I Write the equation for the inverse of y : Arcsin 2r.Then graph the function end its inverse.

y: Arcsinr: Arcsin

Sin x: 2y

2x2y Exchange x and y.

J f si" x:yNow graph the functions.

Example 2 Find each value.

a. Arctanf-Var\ 3/Let o: Arctan (- Ft

Tan 0: -fa- 7Tlr-

6

b. Cos-l lsir -\\ liJIf y: stnf, theny:1.Cos 1(ti";) :Cos r1

J , ",:,

Definition of Arcsin functionDiuide each side by 2.

- 1. t \

Arctan l-i" ) means that

angle whose tan is -*Definition of Arctan function

Replace sin 4 with 1.'z


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PERIOD

PracticeTrigonometric lnverses ond Their Grophswrite the equation for the inverse of each function. Then graphthe function and its inverse.I. y : tan2x 2.y:$+Arccosx

Find each value.3. Arccos (- 1)

a. si"-'f 7. cos-l ("" f) 8. tan (si"-' r - cor-'*)

9. weo.ther The equation y : 10 sin (t, - +) .57 models the averagemonthly temperatures for Napa, California. In this equation, / denotesthe number of months with January represented by t-:1. Duringwhich two months is the average temperature 62o?

4. Arctan 1 b. Arcsin (-+)

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Study Guide

Bosic Trigonometric ldentitiesYou can use the trigonometric identities to help find thevalues of trigonometric functions.

Example 1 lf sin 0 : fi, fittd tan 0.

Use two identities to relate sin 0 and tan 0.

sin2 g + cos2 0 : I Pythagorean identity

(g)' _r cos2 g : 1 substitutn ! fo, sin o.

cos2o:#cos d : * l+or*!

',1 zb 5

Now find tan 0.

tan 0: #+ Quotient id.entityorl5tanO: *+-b

tan 0 : -+3^

PERIOD

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To determine the sign of a function value, use the symmetryidentities for sine and cosine. To use these identities with radianmeasure, replace 180'with rr and 360'with 2r.

Case 1 sin (A + 360k") : s;p 4 cos (A + 360K) : ses 4Gase 2: sin [A + 180'(2k - 1)]: -sinA cos [A + 180"(2k - 1)]: -cosACase 3: sin (360k" - A) : -sin A cos (360k" - A) : cos A

Case 4: sin [180'(2k - 1) -A]: sinn cos [180"(2k - 1)- A] : -cosA

Case 3, with A: ff and k : 2

Quotient identity

Example 2 Express tan S as a trigonometric function of an angle

in Quadrant I.

The sum of # and f, which it # or 4zr, is amultiple of 2n.

+:2en) - !. l1nstn 5tan ff : lllrcos

3

,i"lzrza - tf: cos lzrzo - !]

-sin f"os f

- -tan I

Symmetry identities

Quotient identity

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Practice

Bosic Trigonometric ldentities

Use the given information to determine the exact trigonometricvalueifO"<0<90".

1. If cos e : !, find tan O. 2. lf sin 0 : f, fina cos O.

3. If tan o : $, find sin g. 4. If tan 0 : 2, find cot 0.

Express each value as a trigonometric function of an angle inQuandrant I.

5. cos 892o 6. csc 495o

I1.. Kite Flying Brett and Tara are flying a kite. When the stringis tied to the ground, the height of the kite can be determined bythe formulu h : csc 0, where Z is the length of the string and g isthe angle between the string and the level ground. What formulacould Brett and Tara use to find the height of the kite if theyknow the value of sin 0?

7. sinff

10. sin2 0 cos2 0 - cos2 0

Simplify each expression.

8. cosr * sin xtanx 9. #h

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Study Guide

Verifging Trigonometric ldentitiesWhen verifying trigonometric identities, you cannot add orsubtract quantities from each side of the identity. Anunverified identity is not an equation, so the properties ofequality do not apply.

Example 1 Verify tfrat fffr;a : sin2 r is an identity.

Since the left side is more complicated,transform it into the expression on the right.

secz{-13 sin2rsec" x

(tan2r-1)-It ^'-r..'*'-";- -:Slll-fsec" .r

tart2x, ^' ,#:LSln"fsec' .T -sin2 rcos2 x ,

6;';sin?r.coszxacos- tr

sin2 r :

.,sln'f

sec2 x: t0,n2 x * 7

Simplify.

tan2x:#,sec2x:

PERIOD

1;;?;

J The techniques that you use to verify trigonometric identitiescan also be used to simplify trigonometric equations.

Example 2 Find a numerical value of one trigonometricfunction of r if cos r csc r : 3.

You can simplify the trigonometric epression on the leftside by writing it in terms of sine and cosine.

cos f csc r: 3

1^COS .T --;:-: Jsln tr

cos.r _ D

--

r)sln tr

cot r: 3

sin2 rsin2 r Multiply,

,r, *: -J-ELn tc,

Multiply.

cotx:4&sln tc

Therefore, if cos tr csc x : 3, then cot x : 3.

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PracticeVerifging Trigonometric ldentities

Verify that each equation is an identity.

l. ' cslf :cosr

cot.f + tan f,

2.--1 - --=-1 _ --2sec21,srny-I stny+I

B. sin3r - cos3 x : (L* sintr cosrXsin* _ cosr)

4. tan o * ,f#a : sec g

Find a numerical value of one trigonometric function of x.

5. sinxcotx:l 6. sinx:Scosfr 7. cos x: cot x

8. Physics The work done in moving an object is given by the formulaW : Fd cos 9, where d is the displacement, F is the force exerted, andg is the angle between the displacement and the force. Verify thatW : Fd. # tr an equivalent formula.

&.,

PERIOD

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Sum ond Difference ldentitiesYou can use the sum and difference identities and thevalues of the trigonometric functions of common angles tofind the values of trigonometric functions of other angles.Notice how the addition and subtraction symbols are relatedin the sum and difference identities.

Sum and Difference ldentities

Cosine function cos (a + B) : cos a cosP + sin a sin B

Sine function sin (a + B) : sin a cosF * cos a sin F

Tangent functiontana+tanFtan(a+F): l-+"*t"r.n

Example 1 Use the sum or difference identity for cosine tofind the exact value of cos 375'.

375" :360'+ 15o

cos 375o : cos 15o Symmetry identity, Case 1

cos 15o : cos (60" * 4-or 60" and 45o are two commonr t angles thot differ by 15".

cos 15o : cos 60'cos 45" + sin 60o sin 45o Dffirence identity for cosine

cos JKo - 1 vE + \,tr . Vr,.," l4 : r 6Ld -t 2 ' 2 2 vr 4

Example2 Findthevalue of sin (r+y; if 0< * <E,O <y <t,sinr:f,andsiny:#.In order to use the sum identity for sine, youneed to know cos .tr and cos y. Use a Pythagoreanidentity to determine the necessary values.

sin2 a + cos2 cv: | + cos2 a: L - sin2 a. Pythagorean identity

Since it is given that the angles are in Quadrant I,the values of sine and cosine are positive. Therefore

cos a : \/1 - tin3[.

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I t,-t,2cosy:,lI-(#lv \ull

: rE o't :ffio*#Now substitute these values into the sumidentity for sine.

sin (r * y): sin r cosy * cos r siny: 13-1135 i + t4\t 12\ ^r 153- \s/\37/ ' \5/\szl "' 185

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Practice

Sum ond Difference ldentities

2. sin (*165')

DATE PERIOD

LUse sum or difference identities to find the exact value of each trigonometricfunction.

t. cos $

4. csc 915o 5. tan l#) 6. sec f

Find each exact value if O <x < [ and 0 <y < Z.7. cos (x + y)if sin *: $and sin y : +

8. sin (* - r)ifcos * : fiand cos y : +

9. tan (* - r)ifcsc * : land cot y : +


10. cos (180" - 0) : -cos 0 11. sin (360" * 0) : sitr I

12, Physics Sound waves can be modeled by equations of theform ! : 20 sin (3t + 9). Determine what type of interferenceresults when sound waves modeled by the equations! : 20 sin (3/ + 90") and y : 20 sin (3r + 270') are combined. *-(Hint: Refer to the application in Lesson 7-3.)

3. tan 345'

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j Doubte-Angte ond Hotf-Angte ldentities

Example I It sin 0: f and 0 has its terminal side in the firstquadrant, find the exact value of sin 20.

To use the double-angle identity for sin 20, we mustfirst find cos 0.

sin20+cos20:L. .ot1\L 1 ^.-^n- 7l+l +cosz0:I sin0---\4) 4

cos'g : #cos g

4

Now find sin 20.

sin 20 : 2 sin 0 cos 0 Double-angle identity for sine

: e/ i \\45'\+) 4

: \,458

sing:!,cosu:+

Example 2 IJse a half-angle identity to find the exact valueof sin ft.

PERIOD

T.6stn .,7: Sln t

[Jse sin t : * 1ry. since fi is in

Quadrant I, choose the positiue sine ualue.

-

lz-t/s-l n

-

Yz-r/s2

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PracticeDoubte-Angte qnd Hotf-Angte tdentitiesuse a half-angle identity to find the exact varue of each function,

1. sin 105' 2. tan ff 3. cos ff

Use fhe given information to find sin 20, cos 20, and tan 20.

DATE

4.sin u:#,oo<o<g0o

6.seco:-*,t.g.n

5. tan 0:i,n10.+

7. sin 0 : t,0 .0 .1


8. 1+ sin2x: (sin x * cosx)2

g.cosrsin*:S!J]2s-2

lO. Baseball A batter hits a ball with an initial velocity uo of100 feet per second at an angle 0 to the horizontal. An outfieldercatches the ball 200 feet from home plate. Find 0 if the rangeof a projectile is given by the formula O: #ro, sin20.

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Study GuideSotvi ng Trigonometric Eq uotionswhen you solve trigonometric equations for principal values ofx, x is in the interval -90' 3 x 3 90o for sin r and tan r. Forcos ff, r is in the interval 0o < r < 180o. If an equation cannotbe solved easily by factoring, try writing the expressions interms of only one trigonometric function.

Example 1 Solve tam x cos .r - cos r : 0 for principal valuesof r. Express solutions in degrees.

tanx cosrc - cos x:0cosfr(tanx-1):0 Facton

cos r : 0 or tan x - 1 : 0 Set each factor equal to 0.r:90o tanx:l

x:45oWhen x, : 90o, tan x is undefined, so the only principalvalue is 45'.

Example2 Solve 2tanzx - secz r*3: I- - 2tanxfor 0 <r 12n.This equation can be written in terms of tan r only.

2tan2 x - sec2 x -f 3: 1 - 2tanx2tan2 x - (tan2 x + l)* 3 : I - 2 tanx sec2 x : td.n2 x * 1

tan2 x * 2: | * ztanx Simptify.tan2x*2tanr*1:0

(tanx * 1)z: g

tanx * 1:0tanx: -1

PERIOD

FactonTah,e the square root of each side.

*:!or*:iWhen you solve for all values of r, the solution should berepresented as x * 360"h or x I 2nh, for sin I and cos r andx -f I80"k or x * nle for tan x, where k is any integer.

Exampte 3 Solve sin r + VE : -sin r for all real values of x.sinr*f5:-sinr

2sinxifS:02sinx:-f3

^16sinr:-\-D*: ! i 2nk or.r : ! + 2nk, where fr is any integer

The solutions are { + 2nlz and { + Znn. $s?

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Practice

Sotving Trigonometric Equotions

Solye each equation for principal values of x. Expresssolutions in degrees,

1. cos x:3 cos x - 2 2.2sin2r-1:0

So/ve each equation for O" < x < 360",3. sec2 x*tanr - 1:0 4. cos 2x+3 costr - 1:0

So/ve each equation for O < x 12tr.5. 4sin2 x - 4sinr * 1 : 0 6. cos 2x * sinx : I

Solve each equation for all real values of x.7.3cos2x-lcosff:1 8. 2 sin2 x - 5 sinr f 2:0

9.3sec2x-4:0 10. tanr (tanr - 1) : 6

LL. Aaiation An airplane takes off from the ground and reachesa height of 500 feet after flying 2 miles. Given the formulaH : d tan 0, where I/ is the height of the plane and d is thedistance (along the ground) the plane has flown, find the angleof ascent 0 at which the plane took off.

PERIOD

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Study Guide

J Normol Form of o Lineor EquotionThe normal form of a linearequation isxcos d +ysin 6 - p : O,

Normal Form I where p is the length of the normal from the line to the origin and @

is the positive angle formed by the positive x-axis and the normal

You can write the standard form of a linear equation if youare given the values of @ andp.

Example 7 Write the standard form of the equation of a linefor which the length of the normal segment tothe origin is 5 and the normal makes an angle of135'with the positive r-axis.

tr cos 6 + Y sin@ - P : 0 I'{ormal formrccos 135" +y sin 135'- 5:0 6:735'andp:5

- vE * -. \'8,- 2**-Ty-5:0{Z* - \,Ey + 10 : 0 Muttipty each sid.e by -2.

The standard form of the equation is f1x - {Zy * 10 : 0.

The standard form of a linear equation, Ax I By + C : 0, can bechanged to the normal form by dividing each term of the equationby -r!'42 1 62. The sign is chosen opposite the sigrr of C. You canthen find the length of the normal, p units, and the angle @.

Example 2 Write 3r * 4y - 1-O: 0 in normal form. Then findthe length of the normal and the angle it makeswith the positive r-axis.

Since C is negative, use vE + B' to determinethe normal form.

vF*Bz:fB'+4'or5The normal form is fr * ty - + : 0 or 3* * fu - 2 : 0.

Therefore, cos @ :

$, sin 6 : +, and, p : 2.

Since cos @ and sin Q are both positive, $ mustlie in Quadrant I.

sin Stan$:

0:The normal segment has length 2 units andmakes an angle of 53' with the positive r-axis.

5t3 0r;d

5

530

tan. $ : cos $

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Normol Form of o Lineor Equotion

Write the standard form of the equation of each line, givenp, the length of the normal segment, and Q, the angle thenormal segment makes with the positive x-axis.

L.p:4,6:30"

3.p:3,d:6oo

5.P:2\,8,0:Z-

2.p:2\O,,0:T

4.p:8,0:+

6. p:15,6:225o

PERIOD

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Write each equation in normal form. Then find the length ofthe normal and the angle it makes with the positive x-axis.

7.3x-2y-1:0

8. 5:1 4- y - 12:0

9.4x+3y-4:0

10.y:x*5

lL.2x'ty-1 1:0

12.x+y-5:0

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J Distonce from o point to o LineThe distance from a point at (xr, !r) to a line with equationAx + By + C : 0 can be determined by using the formula

I uv. t vH-LVI 4'J #!=9*. The sign of the radicat is chosen opposite+y@1gz'the sign of C.

Example I Find the distance between p(8, 4) and the linewith equation 4x * 2y: I0.First, rewrite the equation of the line in standard form.

4x+2y-10:0Then, use the formula for the distance from a point to a line., y'-yt+ Byr-l Cu : +v Ai--

d - 4(3) + 2(4)_ te-+f 4z 'r 2z A:4' B : 2, C : -70, x1: 3, lt: 4

6 : -J9_ or lE2V5 Since C is negatiue, use afp a gz.

d : 2.24 units

Therefore, P is approximately 2.24 units from the linewith equation 4x * 2y : 10. since d is positive, p is onthe opposite side of the line from the origin.You can also use the formula to find the distancebetween two parallel lines. To do this, choose any pointon one of the lines and use the formula to find thedistance from that point to the other iine.

Example 2 Find the distance between the lines withequations 2x - 2y : Sandy : x - L.

Sincey : x - 1is in slope-intercept form, youcan see that it passes through the point at(0, - 1). Use this point to find the distance tothe other line.

The standard form of the other equation ts 2x - 2y - b : 0.

"-fur*BYrtCLZ ---_:--rf lz :r gz

,t_2(0)_21_It_scL: *\m (-2y A: 2, B : -2, C : -5, xr: 0, lt: -Ivt.

d: -;:^ o, -3Y Since C is negatiue, use alA-1fr.2V2 4: - 1.06

The distance between the lines is about 1.06 units.

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5.2x*3y+4:09-\t:+x+b.,

6.4x-y*1:04x*Y-8:0

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Distonce From o Point to o Line

Find the distance between the point with the given coordinatesand the line with the given equation.

1.(-I,5),3x-4y-1:0 2.(2,5),5x-I2y*1:0

3.(1,-4),12x+5y-3:0 4.(-1,-3),6x+8y-3:0

Find the distance between the parallel lines with the givenequations.

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Find equations of the lines that bisect the acute and obtuseangles formed by the lines with the given equations.

7.x-t2Y-3:0x-Y-l 4:0

8.9r + L2Y * 10 :03x+2Y-6:0


Study Guide

Geometric VectorsThe magnitude of a vector is the length of a directed linesegment. The direction of the vector is the directed anglebetween the positive r-axis and the vector. When adding orsubtracting vectors, use either the parallelogtam or thetriangle method to find the resultant.

Example 7 Use the parallelogram method to find thesum of i and vi.Copy ? and w-, placing the initial pointstogether.Form a parallelogram that has i and w-: as twoofits sides.Draw dashed lines to represent the other twosides.

The resultant is the vector from the vertex of iand w--- to the opposite vertex of theparallelogram.

Use a ruler and protractor to measure themagnitude and direction of the resultant.

The magnitude is 6 centimeters, and thedirection is 40'.

Example 2 Use the triangle method to find 2v - 3w-.

2i-3vi:2i+(-3w)Draw a vector that is twice the magnitude of ito represent 2i . Then draw a vector with theopposite direction to w---- and three times itsmagnitude to represent -3w-. Place the initialpoint of -3vi on the terminal point of 2i.Tip-to-tail method.

Draw the resultant from the initial point of thefirst vector to the terminal point of the secondvector. The resultant is 2v - 3w.

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PracticeGeometric Vectors

use a ruler and a protractor to determine the magnitude(in centimeters) and direction of each vector.

PERIOD

2.1. 3. \--+z\

Find the magnitude and direction of each resultant.4.x+t 5.i-t

6.2x+t 7.y+3A

Find the magnitude of the horizontat and vertical components ofeach vector shown in Exercises i-S,

8.x e.t fi.4

ll. Aoiation An airplane is flying at a velocity of 500 miles per ffiur .due north when it encounters a wind blowing out of the west at50 miles per hour. What is the magnitude of the airplane's resultantvelocity?

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J Atgebroic VectorsVectors can be represented algebraically using ordered pairsof real numbers.

Example 7 Write the ordered pair that represents the vectorfrom X(2, -g) toY(-4,2). Then find themagnitude of XY_.

EUrt represent XY as an ordered pair.XY : (x, - xt, lz - lrl: <-4 - 2,2 - (-3))

: (-6, 5)

Then determine the magnitude of .F.lxrl :@:@

:.l t,l4)2+52:fGL

.F ls represented by the ordered pair (-6, 5)and has a magnitude of V61 units.

Example 2 Let s : (4,2) and t : (-1, 3). Find each of thefollowing.a.S+t b.S-t

S+?: (4,2) + (-7,3) s -T: (4,2)- (-7,8): (4+ (-1),2+3> : (4 - (-1),2 - 3>: (3,5) : (5, -1)c.4s d. Bs+T4s: 4(4,2) Bs + i : B(4,2) + (_1, 3): <4. 4,4.2> : (t2,6) + (_1, 3): (16, 8) : (11; g)

A unit vector in the direction of the positiver-axis is represented by i, and a unit vector inthe direction of the positive y-axis is representedbyj. Vectors represented as ordered pui.r can bewritten as the sum of unit vectors.

Example 3 Write FP as the sum of unit vectors for M(2,2)and, P(5, 4).

Er-rst write [iP as an ordered pair.*':l1,;r''4-2>

Tbur write MP as the sum of unit vectors.MP :3i + 2j


PracticeAtgebroicAtgebroic Vectors

write the ordered pair that represents EE. rnen find themagnitude of AB.

l. A(2, 4), B(-r,3) 2. A(4, -2), B(5, _5) 3. A(-3, -6), B(9, -1)

Find an ordered pair to represent i in each equation if i : (2, -1,andfr:(_9,5),4.d:3i 5.u--:W-2v

6.d:4i+sF 7.il:sfi-3F

Find the magnitude of each vector, and write each vector as thesum of unit vectors.8. (2, 6)

"l'o. Gardening Nancy and Harry are lifting a stone statue andmoving it to a new location in their garden. Nancy is pushing thestatue with a force of r20 newtons (N) at a 60o angle with thehorizontal while Harry is pulling the statue with a force of180 newtons at a 40o angle with the horizontar. what is themagnitude of the combined force they exert on the statue?

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9. (4, -5)

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Study Guide

Vectors in Three-Dimensionot Spoceordered triples, like ordered pairs, can be used to represent

vectors. Operations on vectors respresented by ordered triplesare similar to those on vectors represented by ordered pairs.

For example, an extension of the formula for the distancebetween two points in a plane allows us to find the distance

between two points in sPace.

Example 1 Locate the point at (-1,3' 1).

Locate - 1 on the r-axis, 3 on the y-axis,and 1 on the z-axis.

Now draw broken lines for paralielograms torePresent the three Planes.

The Planes intersect at (-1, 3, 1)'

Example 2 write the ord.ered triple that represents thevector from X(-4,526) to Y(-2,6, 3)' Then findthe magnitude of W.u

=\i7,:!;n'''u

!'i"'u)- u)

:m: fi4 or 3.7

Exampte S Find an ordered triple that represents 23 + 3T ifs : (5, -Lr 2l and t : (4r 3, -21.

zB + sI : 2(5, -I.2> + 3(4, 3, -2>: (10, -2,4> + (12,9, -6): (22,7, -2>

Example 4 Write E r" the sum of unit vectors for A(5, -2r 3)

and B(-4, 2, l).First express AF ut an ordered triple' Then writethe sum of the unit vectors i, j, and k'

B : (_4,2,1) - (5, -2,8): <-4 - 5,2 - e2),1 - 3): (-Q, 4, ,2):-9i+4j-2k

PERIODl

,lDATE

(xz- xr)2 + (yr- !r)2 + (2, - zr)2

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PracticeVectors in Three-Dimensionot SpoceLocate point B in space. Then find the magnitude of a vector fromthe origin to B.l. B(4,7,6) 2. B(4, -2,6)

write the ordere! triple that repre.senfs ne. rnen find themagnitude ot Te.

3. A(2,1, 3), B(-4,5,7) 4. A(4,0, 6), B(7, r, -3)

5. A(-4,5, g), B(7,2, -g) 6. A(6,9, -5), B(7, -3,12)

E.tl:v-w

fO.il:5i-ZF

PERIOD

.j*lfs

l

I

':

Find an ordered triple to represent i in each equation ifi : (2, -4, 5, and fr = (4 -8, 9),

ll. Physics suppose that the force acting on an object can beexpressed by the vector (85, 95, 110), where each measure inthe ordered triple represents the force in pounds. what is themagnitude of this force?

7.d:?+ni

9..d:4i+3fr


DATE PERIOD ii

JStudy Guide

Perpendicutor VectorsTwo vectors are perpendicular if and only if their innerproduct is zero.

Example 7 Find each inner product if fr : (5, 1), v : (-3, 15),and fr : (2, -1). Is either pair of vectors perpendicular?

b. v.wn'w

a. u'vtl 'v

fr and i are perpendicular. i and fr are not perpendicular.

: 5(-3) + 1(15):-15f15-0

:11i-oj+ot<: 11i or (11, 0, 0)

Find the inner products.(11,0,0).(0,4, 1)11(0)+0(4)+0(1):o

Example 2 Find the inner product of i and S if i : (3, -1, 0)and s : (2,61 4>. Are i and s perpendicular?

i. s : (3X2) + (-1X6) + (0X4)

-9*(-6)+0-U

i and 3 are perpendicular since their innerproduct is zero.

Unlike the inner product, the cross product of two vectors isa vector. This vector does not lie in the plane of the givenvectors but is perpendicular to the plane containing the twovectors.

Example 3 Find the cross product of i and W if v : (0, 4, 1)and fr : (0, 1, 3). Verify that the resulting vectoris perpendicular to i and G.

iiklvxfr: 0 4 1l

0131: li ilt - 13 lF . l3 il* Expand by minors

(11,0,0).(0, 1,3)11(o)+0(1)+o(3):0

Since the inner products are zero, the crossproduct i x fr is perpendicular to both i and ri.

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PracticePerpendicutor Vectors

Find each inner product and state whether the vectors areperpendicular. Write yes or no.1. (3, 6> . (-4,2> 2. (-r,4> . (3, -2) 3. (2,0) . (-1, -1)

4.(-2,0, 1). <3,2,-3> 5. (_4, _1, 1) .<1,_3,4> 6. (0,0, 1) .(I, _2,0>

Find each cross product. Then verify that the resulting vector isperpendicular to the given vectors.7. (I,3, 4) x (-1, 0, -1) 8. (3, 1, -6) x (-2, 4,31

9. (3, t,2l x (2, -3, r> (4, -I,0) x (5, -3, -1)

11. (-6, 1, 3) x <-2, -2,1> L2. (0,0, 6) x (3, -2, -4>

L

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10.

13. Physics Janna is using a force of 100 pounds to push a cart upa ramp. The ramp is 6 feet long and is at a 80" angle with thehorizontal. How much work is Janna doing in the verticaldirection? (Hint: use the sine ratio and the formula w : F . d.l

xi,

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NAME

Study GuideAppticotions with VectorsVectors can be used to represent any quantity that hasdirection and magnitude, such as force, velocity, and weight.

Example Suppose Jamal and Mike pull on the ends of arope tied to a dinghy. Jamal pulls with a forceof 60 newtons and Mike pulls with a force of50 newtons. The angle formed when Jamal andMike pull on the rope is 60'.

a. Draw a labeled diagram that representsthe forces.

Let F, and F, represent the two forces.

b. Determine the magnitude of theresultant force.First find the horizontal (r) and vertical (y)components of each force.Given that we place F, o.r the r-axis, the unitvector is 1i + 0j.Th-erefo-re , the x- and y-components of F, areqoi + gi.F'r: xi + yicos 60" : # sin 60. : #

x : 50 cos 60" .y : 50 sin 60":25 - 43.3

Thus, F, : 25i + 43.3j.

Then add the unit components.

(6oi + ojl + ebi ++s.sj) : 8bi + 43.8j

fi-1ggz*4g82- \6099^89: 95.39

The magnitude of the resultant force is95.39 newtons.

c. Determine the direction of the resultant force.tan 0: n#

g:tantfra0:27"

The direction of the resultant force is 2T' withrespect to the vector on the r-axis.

Use the tangent ratio.

J


PracticeAppticotions with VectorsMake a sketch to show the given vectors,1. a force of g7 newtons acting on an object while a force of Bg newtons

acts on the same object at an angre of 20" with the first force

2. a force of 85 pounds due north and a force of 100 pounds due westacting on the same object

Find the magnitude and direction of the resultant vector for eachdiagram,

what would be the force required to push a 200-pound object up aramp inclined at 30'with the ground?

Nadia is pulling a tarp along level ground with a force of 2Epounds directed alons the tarp. If the tarp makes an angle of 50.yith the ground, find the horizontal and vertical components ofthe force.

Auiation A pilot flies a plane east for 200 kilometers, then 60o "'-

south of east for 80 kilometers. Find the plane's distance anddirection from the starting point.

\-3.

5.

6.

'1.

T

250N

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Study Guide

J Vectors ond Porometric EquotionsVector equations and parametric equations allow us tomodel movement.

Example I Write a vector equation describing a line passingthrough P1(8, 4) and parallel to i : (6, -1). Thenwrite parametric equations of the line.

Let the Line ( through Pr(8, 4) be parallel to d.For any point Pr(x, !) on (, PrPr(r - 8, y - al.Since fJ is on { and is parallel to d, ffi: td,,

for some value t.By substitution, we have(x - 8,y - a): /(6, -1).Therefore, the equation (r - 8, y - 4l : t(6, -L>is a vector equation describing all of the points (x, y)on ( parallel to d through Pl9, 4).

Use the general form of the parametric equationsof a line with (a' ez) : (6, - 1)and (rr, !rl : (8, 4).

x:xrlta,x: 8 + ,(6)x:8-l 6t

Parametric equations for the line are ir : 8 * 6/andY:4-t'

Example 2 Write an equation in slope-intercept form of theline whose parametric equations are x: -3 * 4tandY:3*4t'Solve each parametric equation for t.

x: *3 -f 4tx-f 3:4tx*3 -,4"

y:V._tta^" "l :y:4+t(-7)Y:4-t

Y:3+4tv-3:4tv-3

4 -L

PERIOD

Use substitution to write an equation for the linewithout the variable i.

x-l 3:Y-344

(r+3)(4):4(y-3)4x+12:4y-12

y:x*6

Substitute.

Cross multiply.Simplify.Solue fo, y.

-,

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PracticeVectors ond Pqrometric Equotionswrite a vector equation of the line that passes through point pand is parallel to d. Then write parametric equations of the line.

L. P(-2,1), A : $, -4> 2. P(3,7),d: (4,5>

3. P(2, -4),d: (1, 3) 4. P(5, -8), a- : (9,2)

write parametric equations of the tine with the given equation.5.y:3r -8 6.y:-x*4

7.3x - 2y:6 8. 5x * 4y:29

write an equation in slope-intercept form of the tine with thegiven pa ra metric equations.

10.r:t-1 5y: -3t

PERIOD

\

9.x:2t+3y:t-4

1-r. Physical Education Brett and chad are playing touch footballin gym class. Brett has to tag chad before he reaches a 2O-yardmarker. Chad follows a path defined by (" * I, y - 19) : /(0, 1),and Brett follows a path defined by (* - I2,y - 0) : t(-11, 1g).*write parametric equations for the paths of Brett and chad. willBrett tag Chad before he reaches the 2}-yard marker?

li

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Study Guide

r Modeting Motion Using Porometric EquotionsWe can use the horizontal and vertical components of aprojectile to find parametric equations that represent thepath of the projectile.

Example I Find the initial horizontal and vertical velocitiesof a soccer ball kicked with an initial velocity of33 feet per second at an angle of 29" with theground.

l%l : lil cos o srl : l$l sinoir l : 33 sin 29"

li"l :16The initial horizontal velocity is about 29 feetper second and the initial vertical velocity isabout 16 feet per second.

The path of a projectile launched from the ground may bedescribed by the parametric equations ff : f li I cos g forhorizontal distance andy: tltl sin 0 - tsrt for verticaldistance, where / is time and g is acceleration due to gravity.Use g : 9.8 m/s2 or 32 ftls2.

Example 2 A rock is tossed at an intitial velocity of50 meters per second at an angle of 8o withthe ground. After 0.8 second, how far hasthe rock traveled horizontally and vertically?

First write the position of the rock as a pair ofparametric equations defining the postition ofthe rock for any time f in seconds.

PERIODi

i

r1

0, I : 33 cos 29"

i*l :29

x:tlil cosg

x - t(50) cos 8o

x : 50t cos 8o

Then find r and y when f : 0.8 second.

y:tlnl sin o-lstzy : r(50) sin 8o - |fs.alt' It I -- 5o mls

y : 50t sin 8" - 4.9P

r : 50(0.8) cos 8': 39.61-

y : 50(0.8) sin 8' - 4.9(0.8)2:2.43

After 0.8 second, the rock has traveled about39.61 meters horizontally and is about 2.43 metersabove the ground.J

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PracticeModeting Motion Using porometric EquotionsFind the initial horizontal and vertical velocity for each situation.1. a soccer ball kicked with an initial velocity of 39 feet per second at

an angle of 44'with the ground

2. a toy rocket launched from level ground with an initial velocity of63 feet per second at an angle of 84' with the horizontal

3. a football thrown at a velocity of 10 yards per second at an angleof 58' with the ground

4. a golf ball hit with an initial velocity of r02 feet per second at anangle of 67" with the horizontal

5. Model Rocleetry Manuel launches a toy rocket from groundlevel with an initial velocity of 80 feet per second at an angle of80" with the horizontal.a. write parametric equations to represent the path of the rocket.

b. How long will it take the rocket to travel 10 feet horizontallyfrom its starting point? what will be its vertical distance atthat point?

6. sports Jessica throws a javelin from a height of b feet with aninitial velocity of 65 feet per second at an angle of 45" with theground.

a. write parametric equations to represent the path of thejavelin.

b. After 0.5 seconds, how far has the javelin traveled horizontallyand vertically?

PERIOD

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Study Guide

J Tronsformotion Motrices in Three'Dimensionot Spoce

Example 1 Find the coordinates of the vertices of thepyramid and represent them as a vertex matrix.

A(-2, -2, -2)B(2, -2, -2)c(2,2, -2)D(-2,2, -2)E(0,0, 2)

The vertex matrix for the pyramid is

D-2

2

-2

Exampte 2 Let M represent the vertex matrix of the pyramidin ExamPle 1' l.r o ora. Find TM if r:lo -r ol.

loorlb. Graph the resulting image and describe the

transformation represented by matrix ?.

a. First findTM. C' D'2-2

-2 -2-2 -2

The transformation matrix reflects the imageof the pyramid over the xz-Plane.

E0l0l2l

AB Cxl-2 2 2

,l-z -z 2, l-2 -2 -2

J

fr o ol l-z 2 2

TM:10-1 0l.l-2-2 2

Lo o 1l l-z -z -zb. Then graph the points

represented by the resultingmatrix.

AB'-2 ol l-z 2; ol:l 2 2

-, il l-z-zL

E'0lol2l

J

E@ Glencoe/McGraw-Hill 151 Advanced M athematical ConcePts

PracticeTrqnsformotion Motrices in Three-Dimensionqt SpoceWrite the matrix for each figure,

Translate the figure in Question 1 using the given vectors. Graph each image andwrite the translated matrix.4. b <-1,2, -2>

T_ransform the figure in euestion 2 using each matrix. Graph eachimage and describe the result.

3. a (1,2,0)

'[3SS]loozJ

6. l- 1 0 0lI o r olL o o-rJ

@ Glencoe/McGraw-Hill 152 Advanced M athe m ati cal Co n cepts

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