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Unformatted text preview: Nov Z2 Trigonometry  Functions and Graphs Lesson #10:
Reciprocal and Absolute Value Tr igonometr iC Functions Review of Reciprocal Functions Recall the properties of reciprocal functions by completing the following: 1 .
1.  Whenﬂx) = 0, the graph ofy = ﬂ—x) may have a yer‘hCQﬂ (25%93i'b‘l76  Whenﬂx) is positive,ﬂ+lr) is (3,159 E‘Slfl'iilé
 When ﬁx) is negative, m is 622ch Qgﬁa’l‘Vc l l I
2. When x= ,—= .l. . When x=—1,—=’ .
,1“ ﬁx) . f” m .
 The for a recuprocal transformation can be found where the lines
1 intersect the graphs of ﬁx) and . . l .
3. ' Whenﬂx) increases over an Interval, — (1M (CGSQS over the same interval. ﬁch)
 When ﬁx) decreases over an interval, m ilk KEGSQS over the same interval.
1 _ l
4.  When ﬁx) approaches zero, m approaches 1 so and the graph oi ﬂ—x) approaches
a 3M1 :i [C jag asymptote.
1 l
c When ﬁx) approaches 1 00, m approaches zero and the graph of @ approaches
a is; N l 2 Qﬂ‘l‘aﬂasymptote.
@  Remember: sin‘l x does NOT mean Sinx. sin—1x represents the inverse of the function sin x. The reciprocal of sin x is csc x. ' The above properties can be used as a general aid to sketch the reciprocal trigonometric
functions. Sketching the Graph of a Reciprocal Trigonomem'c Function Use the following general procedure to sketch the graph of a reciprocal trigonometric function. t Sketch the Mark the Where , arg§>gV§€~=g
> ﬂats 369%» Copyright © by Absolute Vaiue Publications. This book is NOT covered by the Cancopy agreement. 302 Trigonometry — Functions and Graphs Lesson #10: Reciprocal and Absolute Value Trig Functions Class Ex. #1 The graph ofy = sin x, —2:r 5 x 5 23: is shown. (z e ’w
L...»
,>..a.\ a) Graph: the reciprocal
ofy = sin x, using the
properties on the previous
page. b) State the equations of the
asymptotes. 1: NY
nél C) List the invariant points. m
m «(a Si
[1+ .6: a... >,. Class Ex. #2 Use a graphing calculator to: a) Graph y csc .r. and y = csc 2x. b) State an appropriate graphing calculator window where x is in radians. e) Complete the table for .I‘Eiﬁ. Function Domain Range Period Equation of
Asymptotes ‘
é; y:cscx 12,515an 5.4—! 0V9>x 21p xeryr‘r ‘né
(\
= .‘ 2x _ : ‘ Rear 34%» «r x as
‘2. Z :1) Complete the following statements based on your observations in a), b), c). i) The graph of y = csc 2;: is a transformation of the graph of y = csc x by: aw compaggfm byafactor of 22 ' ii) Compared to the asymptotes of y = csc x, the asymptotes of the graph of y = csc 2x are 111‘)“; e as frequent. Copyright @ by Absolute Value Publications. This book is NOT covered by the Canoopy agreement. Class Ex. #3 Trigonometry — Functions and Graphs Lesson #10: Reciprocal and Absolute Voiue Trig Functions WarmUp #2 Review of A bsolute Value Functions Recall the properties of absolute value functions by completing the following: 0 When ﬁx) a 0, (i.e. the graph of y =f(x) is above giggins), the graph of y = f(.r)
is idﬂ n’l'l‘CQZ to the graph ofy =ﬂx). 303  Whenﬂx) s 0, (i.e. the graph of y =_f[.r) is W), the graph ofy = f[.r)
is a of the graph of y =f[.r) in the .r—axis. The graph ofy = cos x, —2;rrs x s 23:13 shown. a) Sketch the graph
ofy: cosx b) State the domain and range
of y =  cos x . D43
Q: Oéj Assignment at l e (3 1. The graph ofy = cos x, —211' s x s Znis shown. < l \ a) Graph y = secx, the reciprocal
of y = cos x. b) State where sec x is undefined. c) List the invariant points. [1) Complete the table for xEEli. _—l'—l'—1
—— Copyright @ by Absolute Vaiue Publications. This book is NOT covered by the Canoopy agreement. 304 Trigonometry  Functions and Graphs Lesson #10: Reciprocal and Absolute Value Trig Functions 3:: 33:, 2. The graph ofy = tan x, 77 s x 5 71s shown.
I  i I I i I I I t I
EEEIEEE Iiﬂﬂﬁii
. iiiimﬂﬁiii‘miiii" b) izate the equactlons 01: :he I  ":
ymptotes o y = co . . . . _
ﬂaﬁﬂﬁﬁ;
.....:EEE§iEEE...E.i....:E:
’lﬁi 3) Graph y = cot x, the reciprocal
of y = tan x. c) List the invariant points. (1) Complete the table for x63? 3. Use a graphing calculator to: 4
b) State an appropriate graphing calculator window in radians. a) Graph )2 = csc x and y = csc [x + it] c) Complete the table for xEEii. Function Domain Range Period Equation of
Asymptotes y=cscx y=csc(x+%) 4. Use a graphng calculator to:
a) Graph y = sec )5 and y = sec 3x b) State an appropriate graphing calculator window in radians. 0) Complete the table for xEEil. Function Domain Range Period Equation of
Asymptotes y = see A: y : sec 3x Copyright © by Absolute Value Publications. This book is NOT covered by the Cancopy agreement. Trigonometry  Functions and Graphs Lesson #10: Reciprocal and Absolute Value Trig Functions 305 5. The graph ofy = 2 sin x, 72315 x s 23 is shown. 3) Graph the reciprocal .
of y = 2 sin x. I _
III E _
I’II‘II I I II ‘III .
b) State the equation of the :n‘h‘. IE. [IEEEIIII ‘ reelprocal functlon. I’IIII‘IIIIIIIIIIII‘IIIIII
lllllﬂlllllﬂlllllﬂllllll
"IIIII‘IIIIII"IIIIIIIIIIII b
WIIIIIIIIIIIII IIIIIIIII'
— ﬂIﬂIHtIIIII'II IEiI ﬁll1' I
. HI
::i2::a:i::t::§= :s.:::!
I
I I III. 1‘ I'iIII .Il‘II'J. E l‘ I". I Iain. M:
x III! I I I! A. IIIIIIIIIIIIIIIIIIIIIIII 6. The graph of the function y = sec 2x is shown. 3) Graph the reciprocal of y = sec 2x. [3) State the equation of the reciprocal function. 7. Sketch the following graphs
3) y = \sinx Copyright © by Absoiute Value Publications. This book is NOT covered by the Cancopy agreement. 306 Trigonometry  Functions and Graphs Lesson #10: Reciprocal and Absolute Value Trig Functions 8. The graph represents a reciprocal trigonometric function after a single transformation.
Determine the equation of each graph. Verify with a graphing calculator. 3) Multiple 9. Which of the following describes the asymptotes for y = sec x?
Choice A. x = mt, nEI 13.x Nla + mt, 1461 C. X: 21131“, nEl D. x=3§+2mr= nEI . . . . 1 .
10. The minimum p051tive value of y 011 the graph of y = csc Ex is 1 A. E B. 1 C. 2 D . impossible to determine Copyright © by Absolute Value Publications. This book is NOT covered by the Cancopy agreement. 303 Trigonometry  Functions and Graphs Lesson #10: Reciprocat and Absotme Vaiiie Trig Functions 4 . b) answers may vary c)
It It y = sec 1' x ;r= E + mt, nEi’, XE?“ y s —1 and y 21, yEﬂi 2:": X = E + mt, HE! . J1: 5‘12
ys—landya],y€3i x=—+h—,itEI 6 3 l
5 . a) see graph below b) y = E csc x 6 . a) 7 . a) see graph below b} see graph below 9. B 10. B 11. C 12. 0.25 13.3.0 Copyright @ by Absolute Value Pubtioations. This book is NOT covered by the Cahcopy agreement. ...
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This note was uploaded on 11/05/2011 for the course MATH 24325456 taught by Professor Jack during the Spring '09 term at Adventista de las Antillas.
 Spring '09
 JACK
 Math, Calculus

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