Describe the Transformations For Each Function

a)f(x)=3sin(1/2x+pi/2)+1

b)-1/3cos(2x-pi)-2

how about if we start by saying

y-1 = 3sin((1/2)(x+pi))
and
y+2 = -1/3 cos(2(x - pi/2))

the shifts and scalings should now be clearer.

a) For the function f(x) = 3sin(1/2x + π/2) + 1:

1. Amplitude: The amplitude of the sine function is 3, which determines the maximum value the function can reach. In this case, it is positive 3.

2. Horizontal compression/stretch: The coefficient of x is 1/2, which represents the horizontal compression or stretch of the function. Since the coefficient is less than 1, the function is horizontally compressed.

3. Phase shift: The phase shift of the function is -π/2 units to the left. This means the graph of the function is shifted horizontally to the left by π/2 units compared to the usual sine function.

4. Vertical shift: The function is shifted vertically upward by 1 unit, indicated by the addition of +1 to the sine function.

b) For the function f(x) = -1/3cos(2x - π) - 2:

1. Amplitude: The amplitude of the cosine function is 1/3, which determines the maximum value the function can reach. In this case, it is positive 1/3.

2. Horizontal compression/stretch: The coefficient of x is 2, which represents the horizontal compression or stretch of the function. Since the coefficient is greater than 1, the function is horizontally stretched.

3. Phase shift: The phase shift of the function is π units to the right. This means the graph of the function is shifted horizontally to the right by π units compared to the usual cosine function.

4. Vertical shift: The function is shifted vertically downward by 2 units, indicated by the addition of -2 to the cosine function.

To describe the transformations for each function, we need to identify the effects of the changes made to the original function, which is the sine and cosine function respectively.

a) f(x) = 3sin(1/2x + π/2) + 1

For this function, the transformations are as follows:
1. Amplitude: The amplitude of the sine function is multiplied by 3, resulting in a vertical stretching or compression. In this case, the amplitude is increased to 3 times the original amplitude.
2. Period: The period of the sine function is determined by the coefficient of x inside the sin function. The coefficient in this case is 1/2, so the period is 2π divided by 1/2, which simplifies to 4π. This means that the graph of the function completes a full cycle every 4π units.
3. Phase Shift (Horizontal Translation): The argument (1/2x + π/2) inside the sine function controls the horizontal shift. The original function sine(x) has no phase shift (π/2x will be π/2 when x=1), but the addition of π/2 in this case shifts the graph to the left by π/2 units.
4. Vertical Shift: The constant term (+1) outside the sine function shifts the graph vertically. In this case, it is shifted upwards by 1 unit.

b) -1/3cos(2x - π) - 2

For this function, the transformations are as follows:
1. Amplitude: The amplitude of the cosine function is given by the absolute value of the coefficient (-1/3), which means the amplitude is 1/3. This represents a vertical stretching or compression compared to the basic cosine function.
2. Period: The period of the cosine function is determined by the coefficient of x inside the cos function. The coefficient in this case is 2, so the period is 2π divided by 2, which simplifies to just π. This means that the graph of the function completes a full cycle every π units.
3. Phase Shift (Horizontal Translation): The argument (2x - π) inside the cosine function controls the horizontal shift. The original function cosine(x) has no phase shift, but the subtraction of π in this case shifts the graph to the right by π units.
4. Vertical Shift: The constant term (-2) outside the cosine function shifts the graph vertically. In this case, it is shifted downwards by 2 units.