Question

Write an equation of a sine function that has the given characteristics. Amplitude: 2 Period: \(\pi\) Phase shift: -2

Short answer

The equation is \( y = 2 \sin(2x + 4) \).

Step-by-step solution

Understanding the standard sine function

The standard form of the sine function is given by: \[ y = A \sin(Bx - C) + D \]where:- A is the amplitude- B affects the period- C is the phase shift- D is the vertical shift

Identify and plug in the amplitude

The amplitude of the function is given as 2. Hence, A = 2. Now the equation looks like:\[ y = 2 \sin(Bx - C) + D \]

Determine the value of B to set the period

The period of a sine function is given by the formula: \[ \text{Period} = \frac{2\pi}{B} \]Given the period is \( \pi \), set this equal and solve for B:\[ \pi = \frac{2\pi}{B} \rightarrow B = 2 \]Now the equation is:\[ y = 2 \sin(2x - C) + D \]

Apply the phase shift

The phase shift can be calculated by setting \( \frac{C}{B} \) equal to the given phase shift. Here, the phase shift is -2, and B = 2.\[ \-2 = \frac{C}{2} \rightarrow C = -4 \]The equation now becomes:\[ y = 2 \sin(2x + 4) + D \]

Vertical shift (if any)

There is no vertical shift given in the problem, therefore, D is 0. Hence, the equation remains:\[ y = 2 \sin(2x + 4) \]

Key concepts

Amplitude

Amplitude is a key feature of a sine function. It describes the maximum distance the wave reaches from its central axis, both above and below. For the sine function given, the amplitude is 2, which means the wave peaks at 2 and descends to -2. Mathematically, it's represented as 'A' in the sine function formula. When the amplitude increases, the wave stretches vertically. If the amplitude is decreased, the wave compresses. This control over vertical stretching and compressing makes setting the amplitude an essential step in defining the overall wave shape.

Period

The period of a sine function determines how long it takes for the function to complete one full cycle before repeating. For the given sine function, the period is \(\pi\). To adjust the period, we use the B value in the function. The period is calculated using the formula: [Period = \frac{2\pi}{B}]. Here, we solved for B by setting \(\pi = \frac{2\pi}{B}\)\, resulting in \(B = 2\). A smaller B value makes the wave stretch horizontally, while a larger B value compresses it. Setting the period correctly ensures the wave patterns fit the requirements of the problem.

Phase Shift

Phase shift indicates how the sine function shifts horizontally on the graph. It's calculated by \(\frac{C}{B}\) in the function formula. It's positive, the wave moves left, and if negative, it moves right. For the given function, the phase shift is -2, meaning it shifts right by 2 units. Since we found \(B = 2\), solving \(-2 = \frac{C}{2}\) gives \(C = -4\). This adjusts the position of the sine wave horizontally, starting its cycles at a different point on the x-axis, ensuring timing and placement within a graph align with the given characteristics.

Vertical Shift

The vertical shift moves the entire sine function up or down along the y-axis. It’s represented by D in the formula. For instance, if D = 3, the wave centers around y = 3, instead of y = 0. If D = -3, it centers around y = -3. No vertical shift is given in this problem, so D is 0. Therefore, the wave oscillates around y = 0. This makes vertical shift an essential aspect if you need the wave to fit within certain vertical boundaries or to align with other elements in a graph.