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Trig. Definitions using {x, y, r}

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 UNIT 7 : TRIGONOMETRIC FUNCTIONS

 LESSON 5 : TRANSFORMATIONS: y = a sin k(x d) + c and y = a cos k(x d) + c

 

 

 

Transformations of y = sin x and y = cos x :

 

x

(radians)

 

0

 

 

x

(degrees)

 

0

 

30

 

60

 

90

 

120

 

150

 

180

 

210

 

240

 

270

 

300

 

330

 

360

sin x

(exact)

 

0

 

1

 

0

 

-1

 

0

sin x

(approx.)

 

0

 

0.5

 

0.87

 

1

 

0.87

 

0.5

 

0

 

-0.5

 

-0.87

 

-1

 

-0.87

 

-0.5

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x

(radians)

 

0

 

 

x

(degrees)

 

0

 

30

 

60

 

90

 

120

 

150

 

180

 

210

 

240

 

270

 

300

 

330

 

360

cos x

(exact)

 

1

 

0

 

-1

 

0

 

1

cos x

(approx.)

 

1

 

0.5

 

0.87

 

1

 

-0.5

 

-0.87

 

-1

 

-0.87

 

-0.5

 

0

 

0.5

 

0.87

 

1

 

 

 

 

The graph of y = asin kx:

Review carefully lessons 3, 6 of Functions & Transformations. The same point mapping method will be applied to the trigonometric functions.

 

The value of a determines the vertical stretch or compression and the amplitude.

If |a| > 1, there is a vertical stretch factor a

If |a| < 1, there is a vertical compression factor a

Amplitude = |a|

The value of k determines the horizontal stretch or compression and the period.

If |k| > 1, there is a horizontal compression factor 1/k

If |k| < 1, there is a horizontal stretch factor 1/k

 

.

 

x0

0

90

180

270

360

y

0

1

0

-1

0

 

 

 

 

 

 

 

 

The graph of y = sin (x d) + c :

Review carefully lessons 3, 6 of Functions & Transformations. The same point mapping method will be applied to the trigonometric functions.

The value of d determines a horizontal translation (shift) d units to the right or left

y = sin (x 300) yields a shift of 300 right relative to y = sin x.

y = sin (x + 450) yields a shift of 450 left relative to y = sin x.

 

The value of c determines a vertical translation (shift) c units up or down

y = sin x 3 yields a shift of 3 units down relative to y = sin x.

y = sin x + 2 yields a shift of 2 units up relative to y = sin x.

 

 

x0

0

90

180

270

360

y

0

1

0

-1

0

 

 

 

 

 

 

 

 

 

 

 

 

 

Definition: The horizontal translation is called the phase shift.

 

 

 

x0

0

90

180

270

360

y

1

0

-1

0

1

 

 

 

 

 

 

 

 

 

 

 

Combinations of Transformations : y = a sin k(x d) + c and y = a cos k(x d) + c

 

 

x0

0

90

180

270

360

y

0

1

0

-1

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x0

0

90

180

270

360

y

1

0

-1

0

1

 


 

 

 

 

 


 

 

 

 

 

 

 

 

 

Text Box: In summary, to graph y = a sin [k(x  d)] + c from the graph of y = sin(x), follow these ideas:

	If a < 0, we have a reflection in the x-axis
	If k < 0, we have a reflection in the y-axis
	If  | a | < 1, we have a vertical compression , factor | a |
	If  | a | > 1, we have a vertical stretch, factor | a |
	
	If  | k | < 1, we have a horizontal stretch, factor 1/k
	If  | k | > 1, we have a horizontal compression, factor 1/k
	The value of d gives the horizontal translation (phase shift)
	The value of c gives the vertical translation (shift)
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


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