Question

Write an equation of a sine function that has the given characteristics. Amplitude: 2 Period: \(\pi\) Phase shift: \(\frac{1}{2}\)

Short answer

The equation is \[ y = 2 \, \text{sin}(2(x - \frac{1}{2})) \]

Step-by-step solution

Understand the General Form

The general form of a sine function is given by \[ y = A \, \text{sin}(B(x - C)) + D \]where: - A is the amplitude,- B affects the period,- C is the phase shift,- D is the vertical shift.

Identify Amplitude

The given amplitude is 2. Therefore, \[ A = 2 \]

Determine the Period

The period of a sine function is calculated by \[ \text{Period} = \frac{2\pi}{B} \]Given the period is \[ \pi \]So, \[ \pi = \frac{2\pi}{B} \]Solving for B,\[ B = 2 \]

Find the Phase Shift

The given phase shift is \[ \frac{1}{2} \].Therefore, \[ C = \frac{1}{2} \]

Assemble the Equation

Substitute the values A, B, and C into the general form:\[ y = 2 \, \text{sin}(2(x - \frac{1}{2})) \]

Key concepts

Amplitude

The amplitude of a sine function represents the height of the wave. It is the distance from the middle of the wave to either its highest or lowest point. In mathematical terms, amplitude is always a positive number and in our given equation, it is represented by the parameter \( A \). For the equation \[ y = A \, \text{sin}(B(x - C)) + D \], the amplitude is given as \( 2 \). This means the wave will reach a maximum value of 2 and a minimum value of -2. The sine wave splashes up and down based on this amplitude, making it a crucial factor that controls the vertical stretch of the sine wave.

Period

The period of a sine function is the length of one complete cycle of the wave. It is calculated using the parameter \( B \) in the equation. The formula to determine the period is \[ \text{Period} = \frac{2\pi}{B} \]. For our sine function, the given period is \( \pi \). By substituting into the period formula, we get \[ \pi = \frac{2\pi}{B} \], which leads us to \( B = 2 \). Therefore, our sine function completes a full cycle from start to end in \( \pi \) units along the x-axis. This periodic behavior is essential in predicting how the wave repeats over time or distance.

Phase Shift

The phase shift of a sine function represents the horizontal movement of the wave along the x-axis. It tells us how far the wave is shifted left or right from its standard position. In the equation \[ y = A \, \text{sin}(B(x - C)) + D \], the phase shift is denoted by \( C \). For this problem, we are given a phase shift of \( \frac{1}{2} \). This shift means that the entire sine wave moves to the right by \( \frac{1}{2} \) units. Such shifts help in aligning the wave to a specific point of interest along the x-axis, aligning the sine function's peaks, and troughs.

Trigonometric Functions

Trigonometric functions, including the sine function, are fundamental in mathematics for modeling periodic phenomena. These functions, like sine, cosine, and tangent, relate angles in a triangle to the ratios of the triangle’s sides. The sine function, specifically, is vital in describing oscillations, waves, and other repeating patterns. It is defined as the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle. The general sine function equation \[ y = A \, \text{sin}(B(x - C)) + D \] incorporates parameters for amplitude, period, phase shift, and vertical shifts. Understanding these parameters allows for precise control and application in various scientific and engineering contexts.