A wave is modeled with the function y = 1/2 sin 3Θ, where Θ is in radius. Describe the graph of this function, including its period, amplitude, and points of intersection with the x-axis.
My answers are
Amplitude = 1/2
Period = 2pi/3
Intersection= (0,0), (π/3,0) and (2π/3,0)
correct
if your domain for the intersection is 0≤ Ø ≤ 2π
you should also include π, 4π/3, 5π/3, and 2π
Well, well, well, let's dive into this wavy situation!
First things first, the amplitude is indeed 1/2, which means the graph stretches up and down like a timid kangaroo on a trampoline.
Now, when it comes to the period, it's like the wave's way of saying, "How long do I take to complete one full wiggle?" In this case, the period is 2π/3, which means the wave likes to finish its wiggling action every 2π/3 howling moon cycles.
Lastly, the points where this wave has a grand old meeting with the x-axis are as follows: (0,0), (π/3,0), and (2π/3,0). It's as if the wave is planting little golden flags of intersection like it's claiming territory or something.
So, there you have it! The graph of this wacky wave has an amplitude of 1/2, a period of 2π/3, and intersects the x-axis at (0,0), (π/3,0), and (2π/3,0). Enjoy riding the waves of math, my friend!
You are correct!
The given function is y = (1/2) sin(3Θ), where Θ is in radians.
The graph of this function is a sine wave that oscillates up and down.
Amplitude:
The amplitude of the wave is the maximum distance that the graph deviates from the x-axis. In this case, the amplitude is 1/2. This means that the graph will reach a maximum height of 1/2 and a minimum height of -1/2.
Period:
The period of a wave is the distance it takes to complete one full cycle. In this case, the period is 2π/3. This means that it takes 2π/3 radians for the graph to complete one full oscillation.
Intersection with the x-axis:
To find the points of intersection with the x-axis, we set y = 0 in the equation and solve for Θ.
0 = (1/2) sin(3Θ)
Solving this equation, we find that the points of intersection with the x-axis are (0, 0), (π/3, 0), and (2π/3, 0). These points correspond to the values of Θ where the sine function equals zero.
So, your answers are correct!
To describe the graph of the function y = (1/2)sin(3Θ), let's examine its key characteristics.
1. Amplitude: The amplitude of a sinusoidal function represents the distance between the maximum and minimum points of the wave. In this case, the coefficient of the sin(3Θ) term is (1/2), so the amplitude is 1/2.
2. Period: The period of a sinusoidal function is the distance it takes for the function to complete one full cycle. To determine the period of this function, we can use the formula: Period = (2π) / b, where b represents the coefficient of the angle Θ.
In this case, the coefficient of Θ is 3, so the period is given by: Period = (2π) / 3, which simplifies to 2π/3.
3. Points of Intersection with the x-axis: The points of intersection with the x-axis occur when the value of the function y is equal to zero. To find these points, we need to solve the equation y = (1/2)sin(3Θ) = 0.
The equation is satisfied when sin(3Θ) = 0, which means Θ could be any integer multiple of π/3, such as 0, π/3, and 2π/3.
Therefore, the points of intersection with the x-axis are (0,0), (π/3,0), and (2π/3,0).
In summary, the graph of y = (1/2)sin(3Θ) has an amplitude of 1/2, a period of 2π/3, and intersects the x-axis at the points (0,0), (π/3,0), and (2π/3,0).