A wave is modeled with the function y = 1/2 sin (3Θ) , where Θ is in radians. Describe the graph of this function, including its period, amplitude, and points of intersection with the x-axis.

A wave is modeled with the function y 12 sin 3Θ where Θ is in radians Describe the graph of this function including its period amplitude and points of intersect class=

Respuesta :

[tex]y=Asin\left(B\theta\right)[/tex]

In the function above:

A is the amplitude

2π/B is the period

You equal to 0 the function and solve θ to find the points of intersection with the x-axis.

For the given function:

[tex]y=\frac{1}{2}sin\left(3\theta\right)[/tex]

Amplitude:

[tex]A=\frac{1}{2}[/tex]

Period:

[tex]P=\frac{2\pi}{3}[/tex]

Points of intersection with the x-axis:

[tex]\begin{gathered} \frac{1}{2}sin\left(3\theta\right)=0 \\ \\ Multiply\text{ both sides of the equatio by 2:} \\ 2*\frac{1}{2}sin\left(3\theta\right)=2*0 \\ \\ sin\left(3\theta\right)=0 \end{gathered}[/tex]

Using the unit circle you get that the angles with sine equal to 0 are: 0 and π.

[tex]\begin{gathered} 3\theta=0 \\ 3\theta=\pi \end{gathered}[/tex]

Solve θ:

[tex]\begin{gathered} \theta=0 \\ \\ \theta=\frac{\pi}{3} \end{gathered}[/tex]

Add the period to each solution multiplied by k of θ to find all the intersections:

[tex]\begin{gathered} \theta=0+\frac{2k\pi}{3} \\ \\ \theta=\frac{\pi}{3}+\frac{2k\pi}{3} \end{gathered}[/tex]

Combine the solutions:

[tex]\theta=\frac{k\pi}{3}[/tex]

Then, the given function has Amplitude 1/2; period 2π/3, and points of intersection kπ/3 (k is a whole number)

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