6.3 Graphing Trig FunctionsLast section we analyzed graphs, now we will graph them.

Graph: y = sin θ - 1First, look at y = sin θ

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-1

Since the – 1 is on the outside that means we are shifting DOWN ONE unit

Graph: y = cos θ + 2First, look at y = cos θ

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Since the + 2 is on the outside that means we are shifting UP TWO units

Graph: y = 4sin 2θFirst, look at y = sin θ

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Amplitue = 4

Period = 360/2 = 180

Phase Shift = 0°

I will change the period first

Then change the amplitude

Graph: y = -2cos (θ + 90°)First, look at y = cos θ

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Amplitue = 2

Period = 360/1 = 360

Phase Shift = Left 90°I will change the amplitude

first

Then change the phase shift

Graph: y = 2tan( θ +45)First, look at y = 2tan x

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Since 2 in front changes the “amplitude”?? Then each output is doubled

Asymptotes are still 90° + 180k°

We’re not done, go to next slide

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Graph: y = 2tan( θ +45)

Continued Now let’s shift 45° tothe right

Graph: y = sin ( + 90°) See if you can graph this without graphing each step.

Amplitude = 1

Period = 360/½ = 720

Phase Shift = 180° Left

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θ2

(π,0) (2π,-1) (3π,0)

(4π,1)(5π,0)

(0,1)

Θ 0 90 180 270 360 450 540 630 720

y 1 0.7 0 -0.7 -1 -0.7 0 0.7 1

See if you can graph this without graphing each step.

Amplitude = 1

Period = 180/½ = 360

Phase Shift = 0°

Θ 0 90 180 270 360 450 540 630 720

y 0 1 UD -1 0 1 UD -1 0

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y = tan 12 x( )Graph:

Graph: y = 3cos (θ - 90°)First, look at y = cos θ

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Amplitue = 3

Period = 360/1 = 360°

Phase Shift = 90°

I will change the period first

Then change the amplitude

FIX THIS!!!

Graph: y = cot (θ – 90°)Cot 0 = Does Not Exist

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-1

Amplitue = none

Period = 180/1 = 180°

Phase Shift = 90° Right

I will change the period first

Then change the amplitude

FIX THIS!!!

Graph: y = sin x + cos x

Best approach - table

θ cos θ

sin θ

sum

0° 1 0 1

45° .71 .71 1.4

90° 0 1 1

135° -.71 .71 0

180° -1 0 -1

225° -.71 -.71 -1.4

270° 0 -1 -1

315° .71 -.71 0

360° 1 0 1

Period = 360

Graph: y = cos 2x – cos x

Best approach - table

θ cos 2θ

cos θ -

0° 1 1 0

45° 0 .71 -.71

90° -1 1 -1

135° 0 -.71 .71

180° 1 -1 2

225° 0 -.71 .71

270° -1 0 -1

315° 0 .71 -.71

360° 1 1 0

Period = ???

Graph: y = tan ( - )

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x

2

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π8 Amplitude = 1

Period = 180/½ = 360

Phase Shift = π/4 right

Graph: y = 2sin x + 3cos x

Best approach - table

θ 3cos θ

2sin θ sum

0° 0 3 3

45° 1.4 2.1 3.5

90° 2 0 2

135°

1.4 -2.1 -.7

180°

0 -3 -3

225°

-1.4 -2.1 -3.5

270°

-2 0 -2

315°

-1.4 2.1 .7

360°

0 3 3

Period = 360???