Questions: Write the equation of a sine function that has the following characteristics. Amplitude: 3 Period: 7π Phase shift: 1/9

Transcript text: Write the equation of a sine function that has the following characteristics. Amplitude: 3 Period: $7 \pi$ Phase shift: $\frac{1}{9}$

Solution

Solution Steps

Step 1: Identify the Given Parameters

We are given the following characteristics for the sine function:

Amplitude: $3$
Period: $7\pi$
Phase shift: $\frac{1}{9}$

Step 2: Calculate the Frequency

The frequency $B$ of a sine function is calculated using the formula: \[ B = \frac{2\pi}{\text{Period}} \] Substituting the given period: \[ B = \frac{2\pi}{7\pi} = \frac{2}{7} \]

Step 3: Construct the Sine Function

The general form of a sine function is: \[ y = A \cdot \sin(B \cdot (x - C)) \] where $A$ is the amplitude, $B$ is the frequency, and $C$ is the phase shift. Substituting the given values: \[ y = 3 \cdot \sin\left(\frac{2}{7} \cdot \left(x - \frac{1}{9}\right)\right) \]

Step 4: Simplify the Expression

Simplifying the expression inside the sine function: \[ y = 3 \cdot \sin\left(\frac{2}{7}x - \frac{2}{63}\right) \]