PRECALCULUS CHAPTER 2 STUDY GUIDE f dEd yoti learn?

PRECALCULUSCHAPTER 2 STUDY GUIDE

f dEd yoti learn?

n i"3to analyze graphs of quadratic functions

to write quadratic functions in standard form and sketch their graphs

to use quadratic functions to model and solve real-life problems

3..2

to use transformations to sketch graphs of polynomial functions

to use the Leading Coefficient Test to determine the end behaviorgraphs of polynomial functions

to use zeros of polynomial functions as sketching aids

to use the lntermediate Value Theorem to help locate zeros of

How to use long division to divide polynomials by other polynomials

Haw to use synthetic division to divide polynomials by binomials

:,flow to use the Remainder Theorem and the Factor Theorem

'llow to use polynomial division to answer questions about real-life problems

::;Sgctie$ ?"5'E How to use the Fundamental Theorem of Algebra to determine thef numbers of zeros of polynomial functions

irF Uow to find rational zeros of polynomial functions':'_Ll How to find conjugate pairs of complex zeros

il ttow to find zeros of polynomials by factoring and using Descartes's Rules of Signs

.Seefion 2.6i,'! How to find the domains of rational functions.:'tr How to find the horizontal and vertical asymptotes of graphs of rational functionsL-l How to analyze and sketch graphs of rational functions

' n How to sketch graphs of rational functions that have slant asymptotes

"fl How to use rational functions to model and solve real-life problems

Rerriew Exercises1-6

7-18

19-22

23*28

29-32

33-38

39-42

43-48

49-56

57-60

61

80-85

86*93

94,95

96-1 0i

1 02-1 05

1 06-1 09

11A-121

122-125

126*129

SELF TEST

Take this test as you wourd take a test in crass. After you are done, check your workagainst the answers given in the back of the book.

1. Describe how the graph ofg differs from the gtaph of f(x) : az.(a)s(") :2-xz (b) g(x) : (, - 1)'

2. write the equation in standard form of the parabora shown in the figure.3. Thepathof aballisgivenby.y: _**, * 3x * S,whereyistheheighr(in

feet) of the ball and r is the horizontai distance (in feet) from where the barlwas thrown.

(a) Find rhe maximum height of the ball.(b) Find the disrance the ball travels.

4. Determine the right-hand and left-hand behavior of the graph of the functionh\t) : -;t' + 2tz.Then sketch its graph.5. Divide by long division.

3x3+4x-lx"+1

6. Divide by synthetic division.

2x4-5x2-3x-2

7. Use synthetic division to show that x : -6 ls u solution of the equation4x3 - x2 - 12x * 3 : 0. Use the result to factor the polynomialcompletely and list all the real solutions of the equation.

H ln Exercises 9 and l',.rist at the possibre rSti-gnar zeros of the function. use agraphing utility to graph the fun.tiJn unJfina aU the rational zeros.9. sQ) :2ta - k3 + l6t-24 10. h(x):3x5+ 2xa _ 3x_ z

11. Findall zeros of f(x) : x4 * x3 * 2x2 _ 4x _ g giventhatf(Zl) :0.

il"TJff::J"1and 13' find a polvnomial function with inteser coefficienrs that has

12.0,3,3+i.3_i 13. 1+ Jit,t_.4i,2,2

ln Exercises 14-16,find the domain of the function and identify any asymptotes.M.r:-z- ?--2

4 _ x ls. f(x): ff rc. ge) _.u2 ! 2x - 3

x_2ln Exercises 17 and 1g, graph the function. rdentify any intercepts and asymptotes.

17.hG\:4 -,x2 18. s(*) : *' + 2' x-1


learn?

$,.q, wh"ri .i ti9 Ue.tWeeofu,v.3riabl€s,arefunqtion5,use function notation and evaluate functionsnnO the ooOAins:o.f:fun.ffi!.,'.',:' ,1:.'..: ',' '''.

usef unttig,$.tl@e!gnd9qJv9req,t-Jlfe:p.robtems

15 At

SELF TEST

Take this test as you wourd take a test in crass.After you are done, check your workagainstthe answers given in the backofthe book.

ln Exercises 1-3, use intercepts and symmetry to sketch the graph of the equation.

7.v:a-1, 2.y:4-il-l 3.y:4-(x-))zln Exercises 4 and 5,find an equation of the rine passing through the given points.

4. (2, -3),(-4,e) s. (3.0 8), (7, -6)6. Find an equation of the rine thar passes through the point (3, g) and is (a)

parallel to and (b) perpendicular to the line - 4x -t- jy- : _ S.

'T+97. Evaluatef(r) : ffareach value: (a) /(7) (b) f(-5) @t f(x _ 9)

ln Exercises I and 9, determine the domain of the function.

8. ,f(") : a100 - ? e.f(*):l-x+61 +2

ffi ln Exercises'ro-12, (a) find the zeros of the function, (b) use a graphing utirity tograph the function, (c) approximate the intervals over which the function is increas-ing, decreasing, or constant, and (d) determine whether the function is even, odd, orneither.

70. f(x): Zx6 + 5xa - x2 11. f(x): qx_/T- IZ. f(x): lr + 5l

13. Skerch the graph of f (x) : [t*.* ', x < -3

l4x,-7. x>-3'ln Exercises I 4-1 6, sketch a graph of the function.

7a. hG) : *x3 - 7 ts. h(x) : -!GT + 8 M. hlx): jlx + rl _ :ln Exercises 17-2$,letf(x) = gyz - 7 and g(x) = _x2 _ 4x * 5. Find the indicatedvalue.

17. (f + g)(2) ls. ("f - cX-3) le. (/exO)

ln Exercises 2'l-23,find the inverse function, if possible.

2I. f(x): x3 + 8 22. f(r): lxz - 3l + 6

20. (s./X* 1)

23. f(x) : JX-J X

8


d*# y*c* $earrc?

describe angles

use radian and degree measure

ilto use angles to model and solve real-life problems

4""rto identify a unit circle and its relationship to real numbers

to evaluate trigonometric functions using the unit circle

to use the domain and period to evaluate sine and cosine functions

to use a calculator to evaluate trigonometric functions

r+,ii

to evaluate trigonometric functions of acute angles

to use the fundamental trigonometric identities


to use trigonometric functions to model and sotve real-life problems

tr "!+

ow to evaluate trigonometric functions of any angle

to use reference angles to evaluate trigonometric functions

to evaluate trigonometric functions of real numbers

Keview Exercrses1-4

5-20

21,22

eY; 4"5to use amplitude and period to sketch the graphs of sine and

:cosine functions,How to sketch translations of graphs of sine and cosine functions

How to use sine and cosine functions to model real-life data

** 4.6

23-26

27-30

31-34

3s-38

39-42

43-46

47-52

53,54

s5-68

69*74

75-82

83-86

87-90

91,92

93-96

97-100

101,142

1 03-1 08

109-120

121-128

129,130

131

132

How to sketch the graphs oftangent and cotangent functionsHow to sketch the graphs of secant and cosecant functions

How to sketch the graphs of damped trigonometric functions

i,*{t d+"7

How to evaluate the inverse sine functionHow to evaluate the other inverse trigonometric functions

How to evaluate the compositions of trigonometric functions

.j:lrSec?{etr 4,$

. I How to solve real-life problems involving right triangles

','U How to solve real-life problems involving directional bearings

, I How to solve real-fife problems involving harmonic motion


d*# 3r*u $eanr:?

to describe angles

use radian and degree measure

to use angles to model and solve real-life problems

4,€to identify a unit circle and its relationship to real numbers

to evaluate trigonometric functions using the unit circle

to use the domain and period to evaluate sine and cosine functions

to use a calculator to evaluate trigonomeftic functions

t, ."5

to evaluate trigonometric functions of acute angles

to use the fundamental trigonometric identities


to use trigonometric functions to model and solve real-life problems

:+- *to evaluate trigonometric functions of any angle

to use reference angles to evaluate trigonometric functions

to evaluate trigonometric functions of real numbers

c* d*"5

to use amplitude and period to sketch the graphs of sine and.:Eosine functions

,,How to sketch translations of graphs of sine and cosine functions

How to use sine and cosine functions to model real-life data

*lc 4-6How to sketch the graphs oftangent and cotangent functions

How to sketch the graphs of secant and cosecant functions

How to sketch the graphs of damped trigonometric functions

Revieus Exercises1-4

5-20

21,22

4.7

23-26

27-30

31-34

3s-38

39-42

43-46

47-52

53,54

55-68

69-74

75-82

83.-86

87-90

91,92

93-96

97-100

101,102

1 03-1 08

109-120

121-128

129,130

131

132

How to evaluate the inverse sine functionHow to evaluate the other inverse trigonometric functions

;:,:,[ How to evaluate the compositions of trigonometric functions

Sel"ton +"gI How to solve real-life problems involving right triangles

,, E How to solve real-life problems involving directional bearings

' tr How to solve real-life problems involving harmonic motion

SELF TEST

Take this test as you would take a test in class. After you are done, check your workagainst the answers given in the back ofthe book.

1. Consider the angle of magnitude 5rf 4 radians.

(a) Sketch the angle in standard position.

(b) Determine two coterminal angles (one positive and one negative).(c) Convert the angle to degree measure.

2. A truck is moving at arate of 90 kilometers per hour, and the diameter of itswheels is 1 meter. Find the angular speed of the wheels in radians per minute.

3. Find the exact values of the six trigonometric functions of the angle 0 shownin the figure.

4. Given that tan e : ), nna the other five trigonometric functions of g.

5. Determine the reference angle 0'of the angle 0 : 290'and sketch 0 and 0,in standard position.

6. Determine the quadrant in which 0lies if sec g < 0 and tan 0 > O.

7. Findtwovaluesof gindegrees (0 < g < 360")if cos d : *J1/2.(Donotuse a calculator.)

8. use a calculator to approximate two values of g in radians (o < e < 2r) ifcsc d : i.030. Round the result to two decimal places.

ln Exercises 9 and 10, find the remaining five trigonometric functions of 0 satisfyingthe conditions.

9.cosg:!,tang<0 10. sec 0: -+. sin d > 0

ln Exercises 1 1 and 12, graph the function through two full periods without the aidof a graphing utility.

/^\11.s(x) :-2sin{*-+}

\ 4/ 12. f(a) : )tunzo

In Exercises 13 and 14, use a graphing utility to graph the function. lf the function isperiodic, find its period.

13. ), : sin2nx * 2 cos nx 14. y:6e-012t cos(o.zsr), o < t<3215. Find a, b, and c for the function/(r) : a sin(bx + c) such rhar rhe graph of

/ matches the figure.

16. Find the exact value of tan(arccos l) witrrout rhe aid of a calcularor.

17. Graph the function/(r) : Zarc.in(jr).18. A plane is 80 miles south and 95 miles east of an airport. what bearing

should be taken to fly directly to the airport?

19. write the equation for the simple harmonic motion of a ball on a spring thatstarts at its lowest point of 6 inches below equilibrium, bounces to its maxi-mum height of 6 inches above equilibrium, and returns to its lowest point ina total of 2 seconds.


Wtzat did you learn?

Scction 5,1 Reuiew Exercisesn How to recognize and write the fundamental trigonometric identities 1-6

tr How to use the fundamental trigonometric identities to evaluate 7-23trigonometric tunctions, simplify trigonometric expressions, and rewritetrigonometric expressions

F- -!--- F A5ecElon 5"cI How to plan a strategy for verifying trigonometric identities 24-31

x How to verify trigonometric identities 24-31

>ecnon 5,5I How to use standard algebraic techniques to solve trigonometric 32-37

equations

n How to solve trigonometric equations of quadratic type 38-41

I How to solve trigonometric equations involving multiple angles 42-45

n How to use inverse trigonometric functions to solve trigonometric 46-49equations

bectron 5.4I How to use sum and difference formulas to evaluate trigonometric 50-63

functions

I How to use sum and difference formulas to verify identities and solve 64-67trigonometric equations

SELF TEST

Take this test as you would take a test in class. After you are done, check your workagainst the answers given in the back of the book.

1. If tan I : ] and cos 0 < 0, use the fundamental identities to evaluate theother five trigonometric functions of 6.

2. Use the fundamental identities to simplify csc2 B(1 - cos2 B).

3. Factor and simplify sec4 -{ - tana xsec'x + tan'x

4. Add and simplify TY . 99.' sin 0 cos 0'

5. Determine the values of 0, 0 < 0 < 2r, for which tan 0 : - JT;A 0 - |is true.

H 6. Use a graphing utiiity to graph the functions y1 : cosx * sin xlanx and)2 : sec x. Make a conjecture about y, and yr. Verify the result analytically.

ln Exercises 7-12,verify the identity.

7. sin9sec0:tan9 8. sec2xtanzx *sec2;r:sec4x

^ csca*secd ( n\9.

-:cota

i tana 10. cos{x+ :l : -sinxsin a * cos a -"' -""\" Zl11. sin(nrr + g) : (- 1),' sin 0, n is an integer.

12. (sin .r * cos x)z : 1 t sin 2x

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PRECALCULUS CHAPTER 2 STUDY GUIDE f dEd yoti learn?