Question: Determine the amplitude, midline, period, and an equation involving the sine function for the graph…

Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown below.

[Graph of a sine function with x-axis from -7 to 7 and y-axis from -8 to 8. The graph shows one complete cycle of the sine wave from x = -7 to x = 7, with peaks at y = 4 and troughs at y = -4.]

Enter the exact answers.

Amplitude: A = [Number] Midline: y = [Number]

[Input box for Period: P =]

[Input box for equation involving the sine function]

Solution

To determine the amplitude, midline, period, and the equation of the sine function based on the provided graph, we can follow these steps: Step 1: Identify the Amplitude The amplitude of a sine function is half the distance between the maximum and minimum values of the function. \[ \text{Amplitude} = \frac{\text{Maximum} - \text{Minimum}}{2} = \frac{8 - (-8)}{2} = \frac{16}{2} = 8 & \] Step 2: Determine the Midline The midline is the average of the maximum and minimum values of the function. \[ \text{Midline} = \frac{\text{Maximum} + \text{Minimum}}{2} = \frac{8 + (-8)}{2} = \frac{0}{2} = 0 & \] Step 3: Calculate the Period The period of the sine function is the length of one complete cycle. From the graph description, the x-axis ranges from -7 to 7, covering one full cycle. \[ \text{Period} = 14 \text{ units} & \] Step 4: Formulate the Sine Function Equation The general form of a sine function is: \[ y = A \sin\left(\frac{2\pi}{P}x\right) + D & \] Where: - \( A \) is the amplitude - \( P \) is the period - \( D \) is the midline Substituting the known values: \[ y = 8 \sin\left(\frac{2\pi}{14}x\right) + 0 \] This simplifies to: \[ y = 8 \sin\left(\frac{\pi}{7}x\right) \]