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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.
The graph of y=1/2sin(3Θ) is a sinusoidal function with amplitude of 1/2 and period of 2π/3.
To find the period, we use the formula T=2π/b, where b is the coefficient of Θ in the function. In this case, b=3, so T=2π/3.
To find the amplitude, we note that the coefficient of sin(3Θ) is 1/2. The amplitude of the function is therefore |1/2| = 1/2.
To find the points of intersection with the x-axis, we set y=0 and solve for Θ.
0 = 1/2sin(3Θ)
0 = sin(3Θ)
We know that sin(3Θ) = 0 when 3Θ is equal to multiples of π. Therefore, the points of intersection with the x-axis occur at Θ = 0, Θ = π/3, Θ = 2π/3, Θ = π, Θ = 4π/3, and Θ = 5π/3.
Thus, the graph of y=1/2sin(3Θ) has a period of 2π/3, an amplitude of 1/2, and intersects the x-axis at Θ = 0, Θ = π/3, Θ = 2π/3, Θ = π, Θ = 4π/3, and Θ = 5π/3.
wait im confused what is the answer
The answer is:
The graph of y=1/2sin(3Θ) is a sinusoidal function with a period of 2π/3, an amplitude of 1/2, and intersects the x-axis at Θ = 0, Θ = π/3, Θ = 2π/3, Θ = π, Θ = 4π/3, and Θ = 5π/3.
A sound wave is modeled with the equation y = 1/4 cos 2 pi/3 theta
a. Find the period. Explain your method.
b. Find the amplitude. Explain your method.
c. What is the equation of the midline? What does it represent?
a. The period of the function y = 1/4 cos 2 pi/3 theta can be found using the formula T = 2π/|b|, where b is the coefficient of theta. In this case, b = 2 pi/3, so we have T = 2π/|2 pi/3| = 3 seconds. The period represents the amount of time it takes for the wave to complete one full cycle.
b. The amplitude of the function y = 1/4 cos 2 pi/3 theta is the absolute value of the coefficient of cos theta, which is 1/4. Therefore, the amplitude is 1/4. The amplitude represents the maximum displacement of the wave from its equilibrium position.
c. The equation of the midline is y = 0. The midline represents the equilibrium position of the wave, which is the position the wave would be in if it were undisturbed. Any displacement from this position is represented by the amplitude of the wave.
To describe the graph of the given function y = (1/2)sin(3Θ), let's analyze its period, amplitude, and points of intersection with the x-axis.
1. Period:
The period of a sine function is given by the formula: Period = (2π)/|B|, where B is the coefficient of the angle.
In this case, the coefficient of the angle is 3. Therefore, the period is: Period = (2π)/|3| = (2π)/3.
So, the graph of the function repeats itself every (2π)/3 radians.
2. Amplitude:
The amplitude of a sine function is the absolute value of the coefficient of the sine function. In this case, the coefficient is 1/2.
Thus, the amplitude is |1/2| = 1/2.
This means that the highest point on the graph is 1/2, and the lowest point is -1/2.
3. Points of intersection with the x-axis:
To find the points of intersection with the x-axis, we need to solve the equation y = 0.
When y = 0, we have: (1/2)sin(3Θ) = 0.
This occurs when sin(3Θ) = 0.
A sine function is equal to zero at angles Θ = 0, π, 2π, 3π, etc.
So, the points of intersection with the x-axis are at Θ = 0, Θ = π/3, Θ = 2π/3, Θ = π, Θ = 4π/3, Θ = 5π/3, and so on.
By using the values of the period, amplitude, and points of intersection with the x-axis, we can sketch the graph of the function.
To describe the graph of the function y = 1/2sin(3Θ), we will analyze the role of each element in the equation: the amplitude, period, and points of intersection with the x-axis.
1. Amplitude:
The amplitude of a sinusoidal function determines the vertical distance between the maximum and minimum values of the function. In this case, the coefficient of the sine function is 1/2. Since the amplitude is the absolute value of this coefficient, the amplitude is 1/2. This means that the graph will oscillate between y = 1/2 and y = -1/2.
2. Period:
The period of a sinusoidal function determines the distance between two consecutive repetitions of the graph. To find the period, we need to consider the coefficient of Θ inside the sine function. In this case, the coefficient is 3, which represents the frequency of the wave. The formula to find the period is T = 2π/ω, where ω is the frequency. Therefore, the period is T = 2π/3.
3. Points of intersection with the x-axis:
To find the points where the graph intersects the x-axis, we need to set y = 0 in the equation and solve for Θ. Here is how we can do it:
Set y = 0:
0 = 1/2sin(3Θ)
Now, solve the equation for Θ:
sin(3Θ) = 0
To find the values of Θ for which sin(3Θ) = 0, we need to consider the unit circle and its properties. Since sin(x) = 0 when x is a multiple of π, we can determine that 3Θ is a multiple of π. Therefore,
3Θ = kπ, where k is an integer.
Divide both sides by 3:
Θ = kπ/3
So, the points of intersection with the x-axis occur when Θ takes on the values kπ/3, where k is an integer.
Now, to summarize the graph of the function y = 1/2sin(3Θ):
- The graph oscillates between y = 1/2 and y = -1/2.
- The period of the wave is T = 2π/3.
- The points of intersection with the x-axis occur at Θ = kπ/3, where k is an integer.