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1. 

A sine function has the following key features:

Frequency = 18/π

Amplitude = 6

Midline: y = 3

y-intercept: (0,3)

The function is not a reflection of its parent function over the x-axis.

2. 

A sine function has the following key features:

Period = 4

Amplitude = 3

Midline: y=−1

y-intercept: (0,-1)

The function is not a reflection of its parent function over the x-axis. 

3. 

A sine function has the following key features:

Period = 4π

Amplitude = 3

Midline: y = 2

y-intercept: (0,2)

The function is a reflection of its parent function over the x-axis.

 Jan 29, 2021
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I think I will do one of these, and you should be able to complete the rest of these because the process does not change significantly with each problem anyway.

 

\(f(x)=A\left(\sin\left[B(x-C)\right]\right)+D\\\) is the general form of a sine curve where \(|A|\) is the amplitude, \(\frac{2\pi}{|B|}\) is the period (also known as frequency), C = horizontal shift, and the equation of the midline is \(y=D\).

 

Let's put all of this information to use to do all these problems.

 

The amplitude is specified to be 6, so let's solve for the amplitude.

 

\(|A|=6\\ A=\pm6\\ A=6\)

 

We only need one sine curve that satisfies the conditions, so I will just favor the positive answers, if I can help it.

 

The period (or frequency is \(\frac{18}{\pi}\)), so let's figure out what \(B\) should be.

\(\frac{2\pi}{|B|}=\frac{18}{\pi}\\ |B|=\frac{\pi^2}{9}\\ B=\pm\frac{\pi^2}{9}\\ B=\frac{\pi^2}{9}\)

 

Once again, I will just get rid of the negative answer as we do not need it.

 

The midline occurs at y = 3, so \(D=3\).

 

Just by chance, the sine function with its current parameters already intersects the y-intercept at (0, 3), so there is no need to do a horizontal shift of any kind.

 

Therefore, the equation of this particular sine curve is \(f(x)=6\sin(\frac{\pi^2}{9}x)+3\)

 

To be honest, I am a tad puzzled with "the function is not a reflection of its parent function over the x-axis" requirement. I am not sure what this means, but I am pretty sure that this curve satisfies this cryptic condition.

 Jan 29, 2021

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