Graph the function shown below in the interval from 0 to 2π.
y=sin2(x-π/6)-1/2
To graph the function y = sin^2(x-π/6) - 1/2 in the interval from 0 to 2π, we will first find the key points and then plot the graph.
1. Key Points:
a) The period of the sin^2 function is π, so one full cycle will complete from 0 to 2π. Therefore, we will focus on finding the key points within this interval.
b) The general form of the sin^2 function is y = A sin^2(Bx-C) + D, where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.
c) Amplitude (A): The amplitude of sin^2(x) is 1, but here, it is multiplied by a constant -1/2, so the amplitude will be -1/2.
d) Frequency (B): The frequency of sin^2(x) is 1, so there is no change in the frequency.
e) Phase Shift (C): The phase shift of sin^2(x) is 0, but here, it is shifted to the right by π/6, so the phase shift will be π/6 to the right.
f) Vertical Shift (D): The whole sin^2 function is lowered by 1/2, so the vertical shift will be -1/2 below the x-axis.
2. Key Points Calculation:
a) Amplitude (A): -1/2
b) Frequency (B): 1 (unchanged)
c) Phase Shift (C): π/6 (to the right)
d) Vertical Shift (D): -1/2 (below the x-axis)
3. Graph:
a) Start by drawing the x and y-axis.
b) Find the key points:
- Since the amplitude is -1/2, the highest point on the graph will be -1/2, and the lowest point will be -3/2.
- Since one full cycle completes in π, we will plot key points at intervals of π/2.
- Start from 0 and plot key points at π/6, π/3, π/2, 2π/3, π, 7π/6, 4π/3, 3π/2, 5π/3, and 11π/6. These are our important points.
c) Connect the key points smoothly to create the graph of y = sin^2(x-π/6) - 1/2.
d) The final graph will be a wave-like shape with amplitude -1/2, shifted π/6 to the right, and -1/2 units below the x-axis.
Note: The graph cannot be drawn in this text-based environment. Please refer to a graphing tool or software to plot the graph accurately.