GRAPHS OF TRIGONOMETRIC FUNCTIONS a sin(bx c) d If y , then a cos(bx c) d Domain is ( , ) and Range is [ | a | d , | a | d ] 2 is called the period of function |b| |a| is called Amplitude c is called Phase Shift b d is called vertical translation If y a sec(bx c) d , then c Domain is all real numbers except x (2n 1) 2b b Range is ( , | a | d ] [| a | d , ) 2 is called the period of function |b|

No Amplitude for the functions sec, csc, tan and cot c is called Phase Shift b d is called vertical translation c x (2n 1) are the asymptotes of the function 2b b If y a csc(bx c) d , then n c Domain is all real numbers except x b b Range is ( , | a | d ] [| a | d , ) n c x are the asymptotes of the function b b 2 is called the period of function |b| c is called Phase Shift b d is called vertical translation

If y a tan(bx c) d , then Domain is all real numbers except x (2n 1) c 2b b Range is ( , ), Period is |b| c x (2n 1) are the asymptotes of the function 2b b If y a cot(bx c) d , then Domain is all real numbers except x Range is ( , ), Period is n c 2b b |b|

n c x are the asymptotes of the function 2b b EXAM QUESTION 2x The period of f ( x) 3cos is 3 3 4 A)2 B) C )3 D)4 E ) 4 3 EXAM QUESTION Let f ( x) a sin(bx), where b 0. If the periond of f is 12 and f (3) 4, then f (25) ? A)2 B)6

C )4 D)0 E )8 EXAM QUESTION Let f ( x) a tan(bx), where a 0, b 0. 3 If the periond of f is 3, then f ( ) is 4 a a A)equal to B)undefined C ) equal to b b D )equal to a E )equal to b EXAM QUESTION Let f ( x) a cos(bx). If the periond of f is 8 and f (4) 3, then f (12) ? A)3

B )4 C )12 D )8 E )0 EXAM QUESTION The range of f ( x) 1 4sec x is A)( , 3] [5, ) B )( , 3) (5, ) C )( , 1) (1, ) D)( , 1] [1, ) E )( 1,4) (5, ) EXAM QUESTION Let n be any integer, then the equation of the vertical asymptote of the function x f ( x) 2csc is 2

A) x 2n B ) x 2n 1 C ) x 4n D) x (2n 1) E ) x 2n EXAM QUESTION If f ( x) 2cot 2 x, then the number of the vertical asymptotes over the interval 3 , 4 4 A)2 is equal to B )1 C )3 D)4 E )0

EXAM QUESTION 2x If f ( x) 3cot , then the number of 3 the vertical asymptotes over the interval 3 15 4 , 4 is equal to A)3 B )2 C )4 D)5 E )6 EXAM QUESTION x If f ( x) 3cot , then the number of 2 the vertical asymptotes over the interval , is equal to 2 A)1

B )2 C )0 D)3 E )4 EXAM QUESTION 3x If f ( x) 2cot , then the number of 2 the vertical asymptotes over the interval ,3 is equal to 6 A)5 B)9 C )3 D)2 E )4

EXAM QUESTION x If f ( x) 3 2cot , then the number of 3 the vertical asymptotes over the interval 4,4 A)6 is equal to B)3 C )2 D)1 E )4 EXAM QUESTION The number of the x intercepts of the graph f ( x) 2cot 2 x on the interval ( , ) is

A)4 B )3 C )2 D)1 E )5 y a cos bx a 0 y a x- intercepts 4 2b 3 2b 2

2b 2 2b 2b 0 a 2b 3 2b 4 2b x

y a cos bx a0 y a 4 2b 4 2b 3 2b 2 2b

2b 0 x- intercepts a 2b 2 2b 3 2b x y 3 cos 2 x

a 3 b 2 y 3 2 3 4 2 4 0 3

4 3 4 x y 2 cos 3 x a 2 b 3 y 2 2 3 2 3

2 3 6 0 2 6 3

2 x y a sin bx a 0 y a 3 2b 0 2b a 2 2b

4 2b 5 2b x y a sin bx a 0 y a sin bx a 0 y a 5 2b 2b 0 a

2 2b 3 2b 4 2b x Draw one full period of y=2sin(3x/2) Period = 2/|b| = b|b| = = 2/3 Amplitude = |b| = a|b| = = 2 2 Phase shift = -c/b = /6 /2 /6 2/3

5/6 /3 -2 This is the graph of 2sin(3x). Now click to see the phase shift and to get 2sin(3x/2) Graph one full period of sin(x /2) 1/2 a =1, b =1,c = /2 and d = 1/2 Section 5.7 Question 43 Amplitude = |a| =1 Period = 2/b = 2 Phase shift = c/b = /2 Vertical translation units down Vertical translation: 1/2 units down 1 1/2

y = sin(x /2) 2 1 y = sin(x) 3/2 Phase shift /2 units right y = sin(x /2) 1/2 Graph one full period of 2sin(3x /2) +1 y a =2, b =3,c = /2 and d=1 3 Amplitude = |a| =2

2 Period = 2/b = 2/3 1 /2 Phase shift = c/b = / 6 /6 Vertical translation: 1 1 unit up 2 2/3 /3

This is the graph of 2sin(3x). Now click to see the phase shift , vertical translation and to get 2sin(3x/2)+1 x Graph one full period of sin(x+ /6) y Section 5.7 Question 18 a =1, b =1,c = /6 Amplitude = |a| =1 Period = 2/b = 2 1 Phase shift = c/b = /6 3/2 /6 /2

2 1 This is the graph of sin(x). Now click to see the phase shift and to get sin(x+/6) x Graph one full period of cos(2x/3) y Section 5.7 Question 20 a =1, b = 2,c = /3 Amplitude = |a| =1 Period = 2/b = 1 Phase shift = c/b = /6

/2 /6 /4 3/4 7/6 1 This is the graph of cos(2x). Now click to see the phase shift and to get cos(2x-/3) x Graph one full period of y=(1/2)sin(x/3) y a =1/2, b = /3 Amplitude = |a| =1/2 1/2 Period = 2/b = 6

9/2 3/2 1/2 3 6 x 3 y 3 sin x 2 3 a 3 b 2 2b 3 y 3

0 5 3 3 2 3 3 4 3 x Graph one full period of y=2sinx and y= sinx In [0 , ] , 0 sinx 2sinx 2

In [ , 2] , 1 2sinx sinx 0 2 1 2 3 2 2 y a sec(bx) a

5 2b 4 2b 3 2b 2 2b 2b 0 a

2b 2 2b 3 2b 4 2b 5 2b y a csc(bx) a 5 2b 4 2b

3 2b 2 2b 2b 0 a 2b 2 2b 3 2b

4 2b 5 2b y 2 4b 4b a a 4b

2 4b y a tan bx a 0 x y 2 4b a 4b 4b

2 4b a y a tan bx a 0 x y y a cot bx a 0 a a 4b 2 4b 3 4b

4 4b x y y a cot bx a 0 a 4b a 2 4b 3 4b 4 4b

x Draw one full period of y = 2tan(x/2) y a = 2 and b = 1/2 , 4b = 2 Asymptotes: x = 2/4b = 2/2 = Lets draw asymptotes 2 2 Mark 2 and 2 on the y-axis and /4b = /2 on the x-axis Now we can draw the graph Section 5.6 Question 29 2

2 x Graph one full period of (3/2)csc(3x) a =3/2, b = 3 Period = 2/b = 2/3 3/2 /2 /6 3/2 Section 5.6 Question 34 /3 2/3

Graph one full period of (1/3)tanx a =1/3, b =14b = 4 Period = /|b| = /4 /2 Section 5.6 Question 22 1/3 1/3 /4 /2 Graph one full period of 2cscx y a =2, b = 1 Period = 2/b = 2 2 /2

2 Section 5.6 Question 28 3/2 2 x Graph one full period of -3sec(2x/3) y a = 3 , b = 2/3 Period = 2/b = 3 3 3/4 3

Section 5.6 Question 36 3/2 9/4 3 x y Draw one full period of y = 3tan(3x) a = 3 and b = 3 , 4b = 12 Asymptotes: x = 2/4b = 2/12 = /6 Period = /b = /2 Lets draw asymptotes 3 12

Mark 3 and 3 on the y-axis and /4b = /12 on the x-axis Now we can draw the graph 6 12 3 6 x Draw one full period of y = (1/2)cot(2x) y

a = 1/2 and b = 2 , 4b = 8 Asymptotes: x = /b = /2 and x = 0(y-axis) Period = /b = /2 Lets draw asymptotes Mark 1/2 and 1/2 on the y-axis and /8, 2/8, 3/8 and 4/8 on the x-axis Now we can draw the graph 1/2 0 1/2 3 8 8 2 8

4 8 x Graph one full period of y=3/2sin(x /4+3 /4) y a =3/2, b = 1/4,c = 3/4 Amplitude = |a| = 3/2 3/2 Period = 2/b = 8 Phase shift = c/b = 3 6 3 2 8

4 3/2 This is the graph of y=3/2sin(x/4). Now click to see the phase shift and to get y=3/2sin(x/4+3/4) x Graph one full period of y= sec(x /2 )+1 a = 1 , b = 1, c = /2, d = 1 y sec(x) Period = 2/|b| = 2 Phase shift = c/b = /2 Vertical translation : 2 1 (d =) 1 unit up Click to shift /2 unit to right

Click to shift 1 unit up cos(x) |b| = /2 1 |b| = |b| = 3/2 |b| = 2 x Graph one full period of y= csc(x/3-/12)+4 y

a = 1, b = 1/3, c = /3, d = 4 Period = 2/|b| = 6 5 Phase shift = -c/b = /4 Vertical translation: 4 unit up 3 sin(x/3) 1 /4 | 3/2 | 3 9/2 | 1

csc(x/3) | 6 x Sketch the graph of y = |b| = (1/2)sin(3x)|b| = 1/2sin(3x) 0 -2/3 1/2sin(3x) 0 1/2 0 -/3 1/2sin(3x) 0

/3 -1/2 Section 5.5 Question 48 2/3 1/2sin(3x) 0 Sketch the graph of y = cos2(x) 1 1/2 -3/2 - -/2 -/4

/4 Section 5.5 Question 65 /2 3/2 Sketch the graph of y = sin|b| = x|b| = sin(x) if x 0 y = sin|b| = x| = sin(x) 1 -2 - 0

-1 Section 5.5 Question 68 if x 0 2