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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(xd) + c   and   y = a cos k(xd) + 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