given the transformed function sinusoidal function y=-1.5sin[1/3(x+30)]+2
a) state the period max, min, amplitude, phase shift, equation of axis
b) what are the vertical and horizontal transformations of the function?
a)
- The period of the function is 3 times the period of the standard sine function, which is 2π. So, the period of the given function is 3(2π) = 6π.
- The maximum value of the function can be found by adding the amplitude to the vertical shift. In this case, the amplitude is 1.5 and the vertical shift is 2, so the maximum value is 1.5 + 2 = 3.5.
- The minimum value of the function can be found by subtracting the amplitude from the vertical shift. In this case, the amplitude is 1.5 and the vertical shift is 2, so the minimum value is 2 - 1.5 = 0.5.
- The amplitude of the function is the absolute value of the coefficient in front of the sine function, which is 1.5.
- The phase shift of the function can be found by setting the argument of the sine function equal to 0 and solving for x. In this case, the argument is 1/3(x + 30) and setting it equal to 0 gives x = -30. So, the phase shift is -30.
- The equation of the axis is the vertical shift, which is 2.
b)
- The vertical transformation of the function is a reflection about the x-axis, which is indicated by the negative sign in front of the sine function. In this case, the function is flipped upside down.
- The horizontal transformation of the function is a translation to the left, which is indicated by the constant inside the argument of the sine function. In this case, the function is shifted 30 units to the left.