To describe the graph of the function y = 12sin(3Θ), we need to analyze its key characteristics:
1. Amplitude: The amplitude of a sine function is the distance from the midline to the peak (or trough) of the function. In this case, the amplitude is 12, since the coefficient of the sine function is 12.
2. Period: The period of a sine function is the distance it takes for the function to complete one full cycle. The period of the function y = sin(3Θ) is given by 2π/b, where b is the coefficient of Θ. In this case, the period is 2π/3, since b = 3.
3. Midline: The midline of a sine function is the horizontal line that the function oscillates around. Since there is no phase shift in this function, the midline is y = 0.
4. Points of Intersection with the x-axis: To find the points of intersection with the x-axis, we set y = 0 and solve for Θ.
0 = 12sin(3Θ)
sin(3Θ) = 0
3Θ = kπ (where k is an integer)
Θ = kπ/3
Therefore, the function intersects the x-axis at Θ = 0, Θ = π/3, Θ = 2π/3, Θ = π, Θ = 4π/3, Θ = 5π/3, etc.
Overall, the graph of y = 12sin(3Θ) has an amplitude of 12, a period of 2π/3, a midline of y = 0, and intersects the x-axis at Θ = kπ/3 for integer values of k. The graph will oscillate between 12 and -12 around the midline as Θ increases.
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A wave is modeled with the function y=12sin(3Θ)
, where Θ is in radians. Describe the graph of this function, including its period, amplitude, and points of intersection with the x-axis.
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