A wave is modeled with the function y=1/2sin(3θ), where θ is in radians. Describe the graph of this function, including its period, amplitude, and points of intersection with the x-axis.
period: 2π/3
amplitude: 1/2
y=0 when 3θ is a multiple of π
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To describe the graph of the function y = (1/2)sin(3θ), let's analyze its different components:
1. Amplitude: The coefficient 1/2 before the sine function affects the amplitude of the wave. In this case, the amplitude is 1/2. This means that the highest point of the wave will reach 1/2 units above the centerline, and the lowest point will be 1/2 units below the centerline.
2. Period: The period of a wave refers to the distance required for one complete cycle of the wave. For this function, the coefficient 3 in front of θ affects the period. The period can be calculated as 2π divided by the coefficient. In this case, the period is 2π/3. This means that it takes 2π/3 radians for one complete cycle of the wave.
3. Points of Intersection with the x-axis: To find the points of intersection with the x-axis, we refer to the y-values where y = 0. Setting y = 0 and solving for θ, we get:
0 = (1/2)sin(3θ)
Since sine is zero at specific angles (0, π, 2π, etc.), we can set the argument 3θ equal to zero and its multiples of π:
3θ = 0, π, 2π, 3π, ...
Solving for θ:
θ = 0/3, π/3, 2π/3, π, 4π/3, 5π/3, 2π, ...
So, the graph of the function y = (1/2)sin(3θ) intersects the x-axis at θ = 0, π/3, 2π/3, π, 4π/3, 5π/3, 2π, etc.
Combining all these characteristics, the graph of the function y = (1/2)sin(3θ) is a sinusoidal wave with an amplitude of 1/2, a period of 2π/3, and it intersects the x-axis at the angles θ = 0, π/3, 2π/3, π, 4π/3, 5π/3, 2π, etc.
To describe the graph of the function y = 1/2sin(3θ), let's analyze its key components:
1. Period: The period of a trigonometric function is the distance it takes to complete one full cycle. In this case, the coefficient of θ inside the sine function is 3, so the period (P) of the wave can be found by dividing 2π by 3, giving P = 2π/3. Therefore, the graph will complete one cycle every 2π/3 radians.
2. Amplitude: The amplitude determines the maximum displacement from the mean position. In this case, the coefficient in front of the sine function is 1/2, which means the maximum displacement is 1/2 unit above and below the mean position (y = 0). Therefore, the amplitude (A) is 1/2.
3. Points of intersection with the x-axis: The points where the function intersects the x-axis (y = 0) indicate the solutions for θ in the given equation. To find these points, set y = 0 in the equation:
0 = 1/2sin(3θ)
Now, solve for θ. Since we have a sine function, θ can take any value that satisfies sin(3θ) = 0. The sine function equals zero at x = 0, π, 2π, etc. Therefore, we can set 3θ equal to those values:
3θ = 0, π, 2π, ...
Divide both sides of the equation by 3 to solve for θ:
θ = 0/3, π/3, 2π/3, ...
So the points of intersection with the x-axis occur at θ = 0/3, π/3, 2π/3, etc.
Combining all three aspects, the graph of the function y = 1/2sin(3θ) will have a period of 2π/3, an amplitude of 1/2, and points of intersection with the x-axis at θ = 0/3, π/3, 2π/3, etc.