Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=\sin \left(x-\frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

Amplitude: 1, Period: \(2\pi\), Phase Shift: \(\frac{\pi}{2}\) to the right.

Step 1: Identify the Amplitude

The amplitude of a sine function \(y = a \sin(bx - c)\) is given by the absolute value of \(a\). Since in your equation \(y = \sin \left(x - \frac{\pi}{2}\right)\), \(a = 1\). Therefore, the amplitude is 1.

Step 2: Determine the Period

The period of the sine function \(y = a \sin(bx - c)\) is calculated as \(\frac{2\pi}{b}\). In this equation, \(b = 1\), thus the period is \(2\pi\).

Step 3: Calculate the Phase Shift

The phase shift is determined by solving \(bx - c = 0\). For \(\sin\left(x - \frac{\pi}{2}\right)\), we solve for \(x - \frac{\pi}{2} = 0\). This gives \(x = \frac{\pi}{2}\). Hence, the phase shift is \(\frac{\pi}{2}\) to the right.

Step 4: Sketch the Graph

To sketch the graph, take into account: amplitude = 1, period = \(2\pi\), and a phase shift of \(\frac{\pi}{2}\) to the right. Begin the sine wave at \(x = \frac{\pi}{2}\), and complete one full cycle by \(x = \frac{\pi}{2} + 2\pi\). Plot key points: maximum, zero crossings, and minimum over the cycle ensuring the peak amplitude reaches 1.

Key Concepts

Amplitude

Amplitude is a fundamental concept when graphing trigonometric functions, particularly sine and cosine. For any sine function in the form \( y = a \sin(bx - c) \), the amplitude is the absolute value of the coefficient \( a \). This tells us how "tall" the wave is, or in other words, how far the wave reaches from its central position, known as the midline.

• If \( a \) is positive, the wave will still form its standard undulating shape with peaks and valleys.
• If \( a \) is negative, the peaks and valleys invert, but the amplitude remains unchanged as it's always a positive value.

For example, in the function \( y = \sin(x - \frac{\pi}{2}) \), \( a = 1 \), which makes the amplitude equal to 1. This means that while graphing, the maximum and minimum points of the sine wave will be 1 and -1 respectively.

Period

The period of a trigonometric function describes how long it takes for the function to complete one full cycle of its pattern. In the context of the sine and cosine functions, the period is determined by the value of \( b \) in the function equation \( y = a \sin(bx - c) \).The formula used to find the period is \( \frac{2\pi}{b} \).

• This formula is derived from the standard period of the sine wave, which is \( 2\pi \) radians (or 360 degrees).
• Adjusting \( b \) changes how quickly or slowly the wave oscillates through its cycle.

For the equation in our example, \( b = 1 \). Therefore, the period of \( y = \sin(x - \frac{\pi}{2}) \) is simply \( 2\pi \). This tells us that the graph will repeat its complete waveform every \( 2\pi \) units along the x-axis.

Phase Shift

Phase shift in a trigonometric function describes a horizontal shift in the graph of the function. For the sine function in the form \( y = a \sin(bx - c) \), the phase shift is calculated as \( \frac{c}{b} \).The phase shift indicates how much the entire graph of the function moves left or right on the Cartesian plane.

• If \( c > 0 \), the graph moves to the right.
• If \( c < 0 \), the graph moves to the left.

For the equation \( y = \sin(x - \frac{\pi}{2}) \), solving \( x - \frac{\pi}{2} = 0 \) reveals a phase shift of \( \frac{\pi}{2} \) to the right.This means that, compared to the regular sine function starting at the origin (0,0), the beginning of the wave is now offset by \( \frac{\pi}{2} \) units to the right on the x-axis.