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The amplitude of a trigonometric function, such as a sine or cosine wave, tells us how tall the waves are. It represents the maximum displacement from the center or the "midline" of the wave. This is a crucial feature because it reflects the wave's strength or intensity.

In the general sinusoidal function, given by \( y = A \sin\left(\frac{2\pi}{B}(x - C)\right) + D \), the amplitude is represented by \( A \). To determine the amplitude, look directly at the coefficient of the sine function. In our specific example:

• The given function is \( y = \frac{1}{2} \sin(\pi x - \pi) + \frac{1}{2} \).
• Here, \( A = \frac{1}{2} \).

This indicates that the sine wave reaches 0.5 units above and below its midline. Therefore, the wave peaks at its highest point are at 0.5 units above the midline, and the lowest points dip 0.5 units below it.

When you look at the graph, the amplitude directly affects the height of the wave rather than its width or position.

The period of a trigonometric function tells us how long it takes for the wave to repeat itself. In other words, it's the distance along the x-axis that covers one complete cycle of the wave.

For the general function \( y = A \sin\left(\frac{2\pi}{B}(x - C)\right) + D \), the period can be calculated using the formula \( \frac{2\pi}{B} \). In our example function, we have the expression inside the sine:

• The initial expression is \( \pi x - \pi \), which can be rewritten as \( \pi(x - 1) \).
• Here, you're solving for \( B \) by equating \( \frac{2\pi}{B} = \pi \).
• Solving gives \( B = 2 \).

Hence, the period of the sine wave is 2 units. This means that from any point on the graph, if you move 2 units along the x-axis, the wave pattern will start repeating itself.

Understanding the period is essential because it helps in predicting the wave's characteristics over a certain interval.

The phase shift of a trigonometric function shows how the graph of the function is translated horizontally along the x-axis. It's essentially where the function starts its wave cycle.

In the general form \( y = A \sin\left(\frac{2\pi}{B}(x - C)\right) + D \), the term \( C \) determines the phase shift. The function shifts to the right if \( C > 0 \) and to the left if \( C < 0 \). Let's break it down for our example:

• The function component inside the sine is \( x - 1 \) after rewriting \( \pi x - \pi = \pi(x - 1) \).
• This indicates a phase shift of \( C = 1 \).

This phase shift means the entire wave graph moves 1 unit to the right on the x-axis. This avoids starting the wave from the origin, causing it to begin at \( x = 1 \) instead.

Recognizing and calculating phase shift is key to accurately sketching and interpreting trigonometric wave graphs.