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Chapter 4: Problem 83

Write an equation for the function that is described by the given characteristics. A sine curve with a period of \(\pi,\) an amplitude of 2 a right phase shift of \(\pi / 2,\) and a vertical translation up 1 unit

Short Answer

Expert verified
The equation of the function is \( y = 2\sin(2(x - \frac{\pi}{2})) + 1 \).

Step by step solution

01

Understand the characteristics

Given characteristics are: Amplitude = 2, Period = \( \pi \), Phase shift = \( \frac{\pi}{2} \) to the right, Vertical translation = Up 1 unit. It's important to remember that the generic sine function format is \( y = A\sin(B(x-h)) + k \) where A is the amplitude, B affects the period, h is the phase shift, and k is the vertical translation.
02

Use the Period formula

To find B, we use the formula for the period of a sine function which is \( \frac{2\pi}{|B|} = Period \). Substituting given Period = \( \pi \), we get \( B = \frac{2\pi}{\pi} = 2 \).
03

Find the phase shift and the vertical shift

The given phase shift is to right by \( \frac{\pi}{2} \), hence the value of h is \( \frac{\pi}{2} \). The vertical shift is 1 unit up, so k=1.
04

Substitute A,B,h, and k into the general equation

Now we have all the necessary information to form the equation. Substitute the values A=2, B=2, h= \( \frac{\pi}{2} \), and k=1 into the general sine equation \( y = A\sin(B(x-h)) + k \). This gives us the equation for the function: \( y = 2\sin(2(x - \frac{\pi}{2})) + 1 \)

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