Describe the horizontal transformation of this function: y= 3 sin (3 0 + pi) -2

Answer 1

The general transformations for the sine function, y = sin(θ), are:

y = a sin(b(θ -c)) + d

1. a = vertical dilation (stretch) of the amplitude
2. b = horizontal compression (shrink) of the period
3. c = horizontal shift (left/right)
4. d = vertical shift (up/down)

Your function is:

y = 3 sin(3θ + π) - 2

First, put it into the general from by factoring out the three in the argument of the sine:

y = 3 sin(3(θ + π/3)) - 2

1. a = vertical (amplitude) dilation = 3
2. b = horizontal (period) compression = 2π/period = 3
3. c = horizontal shift = π/3 to the left
4. d = vertical shift = -2

Horizontally the b term compresses the sine by a factor of 3 so that the period, which is normally 2π, is now:

3 = 2π/period

period = 2π/3

The c term shifts the whole function π/3 units horizontally to the left. The a and d terms transform the function vertically but do not affect it horizontally.