91Ó°ÊÓ

The amplitude of a trigonometric function like a sine function refers to its height from the centerline to its peak or trough. In the equation you provided, the term outside the sine function, here "2" in \( y = 2 \sin(x - \frac{\pi}{2}) \), determines the amplitude. Thus, the amplitude is 2. This means:

• The wave reaches a maximum height of 2 units above its midline.
• It also dips 2 units below the midline.

Typically, the amplitude tells us how "tall" the graphs look compared to the standard sine or cosine curve with an amplitude of 1. If the amplitude was negative, it would imply a reflection over the x-axis, but since it is positive, the graph maintains the traditional sinusoidal peaks and valleys.
Amplitudes do not affect the period or phase shift, helping to focus purely on vertical stretching or compressing alone.

The period of a trigonometric function describes how long it takes for the function to complete one full cycle of its pattern before repeating. For the standard sine function \( y = \sin(x) \), this period is \( 2\pi \).
In equations of the form \( y = a \sin(bx - c) \), the coefficient \( b \) affects the period, calculated by the formula \( \frac{2\pi}{b} \). In this case, our equation has \( b = 1 \), so the period is exactly \( 2\pi \), meaning:

• The wave completes one full repetition every \( 2\pi \) units along the x-axis.
• If \( b \) were greater than 1, the period would decrease, compacting the wave horizontally.
• If \( b \) were less than 1, the wave would spread out, increasing the period.

In summary, a period of \( 2\pi \) for our function will display a repetitive pattern with consistent highs and lows, primarily affected by the phase shift, rather than the amplitude.

The phase shift refers to the horizontal shift of the graph of a trigonometric function. It essentially tells us where the wave starts along the x-axis. In the equation \( y = 2 \sin(x - \frac{\pi}{2}) \), the phase shift is calculated from the formula \( \frac{c}{b} \), where \( c \) is subtracted inside the function. For this equation:

• \( c = \frac{\pi}{2} \)
• \( b = 1 \)

Therefore, the phase shift is \( \frac{\pi}{2} \) to the right. Here's what this means:

• The entire sine wave is moved \( \frac{\pi}{2} \) units to the right.
• This causes the typical starting point of the sine wave from \((0, 0)\) to begin instead at \(x = \frac{\pi}{2}\).
• Such shifts can also be translated to the left if the value of \( c \) were subtractive.

The phase shift doesn't alter the wave's height or the length of one period. It simply repositions where features like peaks or troughs appear on the x-axis, allowing easier alignment with real-world or specific data patterns.