given the transformed function sinusoidal function y=-1.5[1/3(x+30)]+2
a) state the period, maximum, minimum, amplitude, phase shift, equation of axis
b) what are the vertical and horizontal transformations of the function?
a) The given function is in the form of y = A*sin(B(x-C))+D, where A is the amplitude, B is the period, C is the phase shift, and D is the vertical shift.
Comparing the given function y = -1.5[1/3(x+30)]+2 to this standard form, we can determine the values:
Amplitude (A) = -1.5
Period (B) = 2π/|1/3| = 6π
Phase Shift (C) = -30 (since x+30 = 0 at the vertical midline)
Vertical Shift (D) = 2
To find the maximum and minimum value of the function, we need to consider the amplitude. The maximum value is the vertical shift (D) + amplitude (A), and the minimum value is the vertical shift (D) - amplitude (A):
Maximum = 2 + (-1.5) = 0.5
Minimum = 2 - (-1.5) = 3.5
Equation of Axis: This is the horizontal line that goes through the midline of the graph. In this case, it is the vertical shift (D):
Equation of Axis: y = 2
b) Vertical Transformation: The vertical transformation is determined by the amplitude.
In this case, the function is multiplied by -1.5, which means that it is reflected vertically and stretched by a factor of 1.5. The negative sign indicates a reflection over the x-axis, and the factor of 1.5 indicates a vertical stretch.
Horizontal Transformation: There is a horizontal shift or phase shift of -30 units to the left. This means the graph is shifted 30 units to the left from the standard sine graph.
So, the vertical transformations are reflection and vertical stretch, and the horizontal transformation is a shift to the left.