Questions: Sketch at least one cycle of the graph of the function. Determine the period, phase shift, and range of the function. Label the five key points of the graph y=3 sin [2(x-(π/6))]+2 The period is (Type an exact answer in terms of π).

Solution Steps

The period of a sine function \( y = a \sin(bx + c) + d \) is given by \( \frac{2\pi}{|b|} \).

For the given function \( y = 3 \sin \left[2\left(x-\frac{\pi}{6}\right)\right]+2 \), we have \( b = 2 \).

Thus, the period is: \[ \text{Period} = \frac{2\pi}{|2|} = \pi \]

The phase shift of a sine function \( y = a \sin(bx + c) + d \) is given by \( -\frac{c}{b} \).

For the given function \( y = 3 \sin \left[2\left(x-\frac{\pi}{6}\right)\right]+2 \), we can rewrite the argument of the sine function as \( 2x - \frac{\pi}{3} \).

Thus, the phase shift is: \[ \text{Phase shift} = -\frac{-\frac{\pi}{3}}{2} = \frac{\pi}{6} \]

The range of a sine function \( y = a \sin(bx + c) + d \) is given by \( [d - |a|, d + |a|] \).

For the given function \( y = 3 \sin \left[2\left(x-\frac{\pi}{6}\right)\right]+2 \), we have \( a = 3 \) and \( d = 2 \).

Thus, the range is: \[ \text{Range} = [2 - 3, 2 + 3] = [-1, 5] \]

Final Answer