The required sine function is [tex]\bold{y=2~sin(\frac{3\pi}{4} x)+3}[/tex]
"It is the horizontal center line about which the function oscillates above and below. "
"It is the distance between the highest point and midline."
"It is the length of one complete cycle of the curve."
"It is the distance between midline and x-axis."
"The generalized equation for a sine graph is as follows:
[tex]y = A~sin(B(x + C)) + D[/tex]
where, amplitude = A
period = 2π/B
phase shift = C
vertical shift = D"
For given function,
A sine function that has a midline of 3, an amplitude of 2 and a period of 3pi/4
A = 2
T = [tex]\frac{3\pi}{4}[/tex]
In this case, the midline is 3
⇒ D = 3
Using the generalized equation of sine function,
[tex]\Rightarrow y = A~sin(B(x + C)) + D\\\\\Rightarrow y = 2~sin(\frac{3\pi}{4} (x + 0)) + 3\\\\\Rightarrow y=2~sin(\frac{3\pi}{4} x)+3[/tex]
Therefore, the required sine function is [tex]\bold{y=2~sin(\frac{3\pi}{4} x)+3}[/tex]
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