Questions: Graph the trigonometric function. y = (1/2) sin x

Graph the trigonometric function. y = (1/2) sin x
Transcript text: Graph the trigonometric function. \[ y=\frac{1}{2} \sin x \]
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Solution

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Solution Steps

Step 1: Identify the key features of the function

The function is _y_ = (1/2)sin(_x_). This means the amplitude is 1/2, the period is 2π, and there is no phase shift or vertical shift.

Step 2: Plot the x-intercepts

The sine function has x-intercepts at _x_ = 0, _x_ = π, _x_ = 2π, _x_ = 3π, and _x_ = 4π within the given range. Plot these points on the graph.

Step 3: Plot the maxima and minima

The maxima occur halfway between the first and second x-intercepts and halfway between the third and fourth x-intercepts and will have a y-value equal to the amplitude. Therefore, the maxima are at (_x_ = π/2, _y_ = 1/2) and (_x_ = 5π/2, _y_ = 1/2).

The minima occur halfway between the second and third x-intercepts, with a y-value equal to the negative amplitude. Therefore, the minimum is at (_x_ = 3π/2, _y_ = -1/2). Plot these points.

Step 4: Sketch the graph

Connect the plotted points with a smooth curve that resembles a sine wave, passing through the intercepts, maxima, and minima.

Final Answer:

The graph of _y_ = (1/2)sin(_x_) will be a sine wave with an amplitude of 1/2 and a period of 2π. Key points to plot include (0,0), (π/2, 1/2), (π, 0), (3π/2, -1/2), (2π, 0), (5π/2, 1/2), (3π, 0), (7π/2, -1/2) and (4π, 0).

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