GRAPHING TRIGONOMETRIC FUNCTIONS

GRAPHING TRIGONOMETRIC FUNCTIONS Section.6B Precalculus PreAP/Dual, Revised 7 viet.dang@humbleisd.net 8//8 : AM.6B: Graphing Trig Functions

REVIEW OF GRAPHS 8//8 : AM.6B: Graphing Trig Functions

A. Equation: = A trig function B C + D B. A is the amplitude. a: verticall stretches b a factor of a,. a : Verticall compresses b a factor of /a C. B is the period or frequenc. equation: π for sine and cosine, π for tangent B B. B: phase compresses b a factor of π B 3. B : phasel stretches b a factor of b D. C is the phase shift TRANSFORMATIONS.If there no GCF taken out, divide the coefficient E. D is the vertical shift F. Frequenc is defined as the number of ccles per second 8//8 : AM.6B: Graphing Trig Functions 3

STEPS A. Identif A, B, C, and D from the equation, = A trig B C + D B. Identif the phase shift. : π B or π B (for Tan and Cot onl) C. Use the period to identif the spacing. Anchor Point Equation: D. Start with the phase shift as the middle of the trig table (at the origin) and appl the spacing before and after 8//8 : AM.6B: Graphing Trig Functions

= sin BASIC TABLE POINTS = cos = tan C C C 8//8 : AM.6B: Graphing Trig Functions 5

= csc BASIC TABLE POINTS = sec = cot C C C 8//8 : AM.6B: Graphing Trig Functions 6

EXAMPLE ( ) = AtrigB C + D A =, B =, C =, D = Graph = sin + in one period and identif amplitude, period, vertical shift, phase shift, domain (entire graph), and range = sin() C + = + = 3 sin () + Amplitude Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions 7 Up None = = (, ),

EXAMPLE Graph = sin + in one period and identif amplitude, period, vertical shift, phase shift, domain (entire graph), and range = sin () + π/ π 3π/ π 3 8//8 : AM.6B: Graphing Trig Functions 8 Amplitude Vertical Shift Spacing (A.P.) Domain Range Up None = = (, ),

EXAMPLE ( ) = AtrigB C + D A =, B =, C =, D = Graph = cos θ in one period and identif amplitude, period, vertical shift, phase shift, domain (entire graph), and range = cos(θ) C + = + = 3 cosθ / / / cosθ / 3/ / Amplitude Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions 9 Down None = = (, ) 3,

EXAMPLE Graph = cos θ in one period and identif amplitude, period, vertical shift, phase shift, domain (entire graph), and range cosθ / π/ π 3π/ π 3/ / 3 8//8 : AM.6B: Graphing Trig Functions Amplitude Vertical Shift Spacing (A.P.) Domain Range Down None = = (, ) 3,

YOUR TURN Graph = sin t + from π, π and identif amplitude, period, vertical shift, phase shift, domain, and range = sin t + π/ π 3π/ π 5/ 3/ Amplitude Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions Up None (, ) 3 5,

EXAMPLE 3 = AtrigB ( C) + D A =, B =, C =, D = Given = tan + from π, π and amplitude, period, vertical shift, phase shift, domain, and range Amplitude DNE = tan() = C + = = tan tan + 3 Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions Up None = = ( ) (, ),,

EXAMPLE 3 Given = tan + from π, π and amplitude, period, vertical shift, and phase shift Amplitude DNE tan + π/ π/ π/ 3 π/ Vertical Shift Spacing (A.P.) Domain Range Up None = = (, ) (, ) 8//8 : AM.6B: Graphing Trig Functions 3

EXAMPLE ( ) = AtrigB C + D A =, B =, C =, D = Given = csc + from π, π and amplitude, period, vertical shift, phase shift, domain, and range Amplitude DNE = csc() C 3 = csc / / csc + 3/ / Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions Up None = = + n 3,,

EXAMPLE Given = csc + from π, π and amplitude, period, vertical shift, phase shift, domain, and range Amplitude DNE π/ π/ 3π/ π csc + 3/ / Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions 5 Up None = = + n 3,,

YOUR TURN Given = sec from π, π and identif period, vertical 3 shift, phase shift, domain, and range Amplitude Vertical Shift Spacing (A.P.) Domain Range DNE 8//8 : AM.6B: Graphing Trig Functions 6 Down None + n,, 3 3

EXAMPLE 5 = AtrigB C + D ( ) Given = 5 sin π 6 phase shift, and points to graph in one period +, identif amplitude, period, vertical shift, A B C D = 5, =, =, = 6 Amplitude = 5sin + 6 5 Vertical Shift Up 8//8 : AM.6B: Graphing Trig Functions 7

EXAMPLE 5 Given = 5 sin π +, identif amplitude, period, vertical shift, 6 phase shift, and points to graph in one period = 5sin + 6 B is OUTSIDE of the parenthesis = AtrigB C + D ( ) A = 5, B =, C =, D = 6 Right 6 8//8 : AM.6B: Graphing Trig Functions 8

EXAMPLE 5 = AtrigB C + D ( ) A = 5, B =, C =, D = 6 Given = 5 sin π +, identif amplitude, period, vertical shift, 6 phase shift, and points to graph in one period Anchor Points = = 8//8 : AM.6B: Graphing Trig Functions 9

EXAMPLE 5 Given = 5 sin π +, identif amplitude, period, vertical shift, 6 phase shift, and points to graph in one period Values = AtrigB C + D ( ) A = 5, B =, C =, D = 6 = sin Y = 5sin() Y = 5sin() + π/6 C 3 5π/ + = + 6 8π/ π/3 + = + π/ + = + π/ 7π/6 5 5 6 5 5 3 8 8 3 Anchor Point: 8//8 : AM.6B: Graphing Trig Functions

EXAMPLE 6 = AtrigB C + D ( ) Given = 3 cos + π phase shift, and points to graph in one period = cos Y = 3cos(), identif amplitude, period, vertical shift, Y = 3co s π/ 3 π/ 3 π 3π/ 3 Amplitude Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions 3 Down Left, ( ),, 3 3

EXAMPLE 7 Given = sin π + 5, identif amplitude, period, vertical shift, and phase shift and points from π, π = sin + 5 ( ) = AtrigB C + D + + + ( ) 8//8 : AM.6B: Graphing Trig Functions

EXAMPLE 7 Given = sin π + 5, identif amplitude, period, vertical shift, and phase shift and points from π, π = sin + 5 Amplitude: : = ( C ) + D : 5 : Vertical Shift: Down 5 : Left 8//8 : AM.6B: Graphing Trig Functions 3

EXAMPLE 7 Given = sin π + 5, identif amplitude, period, vertical shift, and phase shift and points from π, π Amplitude: : Vertical Shift: Down 5 : Left : : : π ππ Phase : Shift: Left Left /π /π = sin = sin sin = cos (π + = cos (π = = ) cos co(π s( π+ ) + = ) = co 5cos s( (π + ) + + ) ) 5 5 /π /π 5 π/ π/ π/ /π π/ π/ π/ /π π π/ π/ /π π π/ π/ /π 5 3π/ 3π/ 3π/ /π 3π/ 3π/ 3π/ /π 9 π π π /π.6b: Graphing Trig Functions π π π /π 5 8//8 : AM

YOUR TURN Given = sin 3 + π 5, identif amplitude, period, vertical shift, 3 phase shift, and points to graph in one period = sin Y = sin() Y = sin() 5 π/3 5 π/3 3 π 5 5π/3 7 8π/3 5 Amplitude Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions 5 3 Down 5 Left 3 3, ( ),, 3 3

EXAMPLE 8 Graph = cos 3 + π in one period and identif amplitude, 3 period, vertical shift, phase shift, domain, and range = cos3+ + 3 Spacing : 3 : : = Left 3 3 = 3 = 6 = cos = cos 3 8//8 : AM.6B: Graphing Trig Functions 6 π/3 C π/6 π/6 π/3 π/3 π/6 π/6 π/3

EXAMPLE 8 Graph = cos 3 + π in one period and identif amplitude, 3 period, vertical shift, phase shift, domain, and range Amplitude ; Reflected Vertical Shift Domain Range 3 None Left 3 (, ) 8//8 : AM.6B: Graphing Trig Functions 7,

EXAMPLE 9 Graph = sin + π from, π and identif amplitude, period, vertical shift, phase shift, domain, and range = Asin B C + D = sin + + + ( ) ( ) : Spacing : = : = Left = = sin = sin + π + π π/ C π/8 π/8 π/ 3 8//8 : AM.6B: Graphing Trig Functions 8 8

EXAMPLE 9 Graph = sin + π from, π and identif amplitude, period, vertical shift, phase shift, domain, and range = sin + π π/ π/8 π/8 π/ 3 3 Amplitude Vertical Shift Anchor Points Domain Range 8//8 : AM.6B: Graphing Trig Functions 9 3 Down Left 8 (, ) 3,

EXAMPLE Given = tan + π from π, π and amplitude, period, vertical shift, phase shift, domain, and range : = = tan + Spacing : = : Left = tan + π/ = tan + π = tan + π π 3π/ π/ C π/ 3 8//8 : AM.6B: Graphing Trig Functions 3

EXAMPLE Given = tan + π from π, π and identif period, vertical shift, phase shift, domain, and range = tan + π π 3π/ π/ π/ 3 Amplitude Vertical Shift Anchor Points Domain Range DNE 8//8 : AM.6B: Graphing Trig Functions 3 Down Left (, ) ; n (,, )

EXAMPLE Given = tan + π from π, π and amplitude, period, vertical shift, phase shift, domain, and range : = = tan + Spacing : = : Left = tan + π/ = tan + π = tan + π π 3π/ π/ C π/ 3 8//8 : AM.6B: Graphing Trig Functions 3

YOUR TURN Given = sec + π + from π, π and identif period, vertical shift, phase shift, domain, and range Amplitude DNE Vertical Shift Anchor Points Domain Up Left ( ), ; n Range (, 3, ) 8//8 : AM.6B: Graphing Trig Functions 33

ASSIGNMENT Worksheet 8//8 : AM.6B: Graphing Trig Functions 3