Question

Amplitude and period Identify the amplitude and period of the following functions. $$p(t)=2.5 \sin \left(\frac{1}{2}(t-3)\right)$$

Short answer

Answer: The amplitude of the function $$p(t)$$ is 2.5, and the period is $$4\pi$$.

Step-by-step solution

Identify the amplitude

Let's first find the amplitude of the function. In the given function, the coefficient of the sine term is 2.5, which can be represented as $$A=2.5$$. The amplitude is the absolute value of the coefficient, which is already positive, so the amplitude is 2.5.

Identify the period

Now, let's find the period of the function. The function has a sine term in the format $$\sin (B(t - C))$$. Here, the coefficient of t inside the sine term is $$B = \frac{1}{2}$$. The period of the sine function can be found by dividing \(2\pi\) by B, so the period of this function is $$T=\frac{2\pi}{B} = \frac{2\pi}{\frac{1}{2}} = 4\pi$$. To summarize:

Amplitude and Period of $$p(t)$$

The amplitude of the function $$p(t)$$ is 2.5, and the period is $$4\pi$$.

Key concepts

Amplitude of Sine Function

Understanding the amplitude of a sine function is essential for grasping how it behaves. The amplitude of a sine (or cosine) function represents the maximum value that the function can reach. It defines the height of the sine wave from the central axis (the equilibrium position) to its peak.

In the function \( p(t)=2.5 \sin \left(\frac{1}{2}(t-3)\right) \), the coefficient before the sine term is 2.5. This coefficient tells us how much the sine wave is stretched or compressed vertically. In this case, the sine wave reaches 2.5 units above and below the central axis, indicating the amplitude is 2.5. A higher amplitude means the wave extends further from the central axis, while a lower amplitude means it is closer.

Period of Sine Function

The period of a sine function refers to the length of one complete cycle of the wave. In other words, it is the distance on the horizontal axis before the function starts repeating itself. For a standard sine function \( y=\sin(x) \), the period is \(2\pi\), since sine waves repeat every \(2\pi\) radians.

When you have \( p(t)=2.5 \sin \left(\frac{1}{2}(t-3)\right) \), the function inside the sine, \( \frac{1}{2}(t-3) \), affects the period. The coefficient \( \frac{1}{2} \) of \( t \) compresses the wave horizontally, making it twice as long to complete one cycle. Therefore, to get the period, you divide \(2\pi\) by \( \frac{1}{2} \), resulting in a period of \( 4\pi \). This means it takes \( 4\pi \) units along the \( t \)-axis for one complete wave cycle.

Sine Function Transformation

A sine function can undergo various transformations that alter its graph. These transformations include vertical shifts, horizontal shifts, reflections, and changes in amplitude and period which we've already discussed.

Let's consider our function \( p(t)=2.5 \sin \left(\frac{1}{2}(t-3)\right) \). The presence of \( -3 \) within the sine function's argument translates to a horizontal shift. The graph of the sine function will shift 3 units to the right. This kind of horizontal movement does not affect the amplitude or the period, but it does change where the sine wave starts along the \( t \)-axis.

Transformations of sine functions provide us with a powerful toolkit to model oscillating phenomena, like sound waves, light waves, or the motion of a pendulum. By adjusting the amplitude, period, and shifts, a sine wave can be tailored to fit a vast array of real-world applications.