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.
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.
