Problem 27 - Precalculus: Functions and Graphs

Question

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=5 \sin \left(3 x-\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

Amplitude: 5, Period: \(\frac{2\pi}{3}\), Phase shift: \(\frac{\pi}{6}\) right.

Step 1: Identify the amplitude

The general form of the sine function is given by \( y = a \sin(bx + c) \). In this equation, \( a \) represents the amplitude. Comparing this to the given equation \( y = 5 \sin(3x - \frac{\pi}{2}) \), we can see that \( a = 5 \). Therefore, the amplitude is 5.

Step 2: Find the period

The period of a sine function is calculated using the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) in the function. In the equation \( y = 5 \sin(3x - \frac{\pi}{2}) \), \( b = 3 \). Thus, the period is \( \frac{2\pi}{3} \).

Step 3: Determine the phase shift

The phase shift is calculated by transforming the formula \( bx + c = 0 \) into the form \( x = -\frac{c}{b} \). In our equation, \( bx + c \) is \( 3x - \frac{\pi}{2} \). Setting this equal to zero, we solve \( 3x - \frac{\pi}{2} = 0 \) to find \( x = \frac{\pi}{6} \). So, the phase shift is \( \frac{\pi}{6} \) to the right.

Step 4: Sketch the graph

The graph of \( y = 5 \sin(3x - \frac{\pi}{2}) \) can be plotted by starting at the phase shift \( x = \frac{\pi}{6} \) and plotting the sine wave with an amplitude of 5 and a period of \( \frac{2\pi}{3} \). The sine wave completes one full cycle every \( \frac{2\pi}{3} \) units along the x-axis, with each peak and valley reaching 5 units and -5 units respectively from the midline \( y=0 \).

Key Concepts

Amplitude

The amplitude of a sine function is a measure of how far its peaks and valleys deviate from the central axis, usually the x-axis. Essentially, it tells us how "tall" or "deep" the wave is.
For the sine function expressed as \( y = a \sin(bx + c) \), the amplitude is the absolute value of \( a \). It indicates the maximum and minimum values the sine wave will reach from its central line or the equilibrium position.

• For example, if \( a = 5 \), then the function climbs up to 5 units above the x-axis and dips down to 5 units below it. That's a total vertical stretch or compression of 10 units, centered at the x-axis.
• The amplitude is always non-negative because it represents a distance.

In our function \( y = 5 \sin(3x - \frac{\pi}{2}) \), the amplitude is 5. This means the graph will oscillate between 5 and -5 around the horizontal axis. Understanding amplitude is crucial for sketching sine waves, as it dictates how high and low they go.

Period of a Sine Function

The period of a sine function refers to the distance along the x-axis for which the function completes one entire cycle.
In mathematical terms, it's how long it takes for the sine wave to repeat its pattern.

• The general formula to find the period of a sine function, \( y = a \sin(bx + c) \), is \( \frac{2\pi}{b} \).
• Here, \( b \) is the frequency coefficient, which affects how "squeezed" or "stretched" the wave is along the x-axis.

For the equation \( y = 5 \sin(3x-\frac{\pi}{2}) \), \( b = 3 \). Therefore, the period is \( \frac{2\pi}{3} \).
This means the wave completes one full oscillation every \( \frac{2\pi}{3} \) units along the x-axis. Knowing the period helps you determine where the peaks, troughs, and intercepts of the sine wave will occur, showing the regularity with which these features appear.

Phase Shift in Trigonometry

In trigonometry, the phase shift refers to the horizontal translation of the sine or cosine wave along the x-axis.
It's determined by the values of \( c \) and \( b \) in the sine function \( y = a \sin(bx + c) \).

• We use the formula \( x = -\frac{c}{b} \) to find the phase shift, which tells us how much the graph shifts left or right.
• If the result is positive, the shift is to the right, whereas a negative result indicates a shift to the left.

In the function \( y = 5 \sin(3x - \frac{\pi}{2}) \), the phase shift is obtained by solving \( 3x - \frac{\pi}{2} = 0 \), giving us \( x = \frac{\pi}{6} \). This tells us that the sine wave shifts \( \frac{\pi}{6} \) units to the right from its usual starting position. Understanding phase shift is key for accurately plotting the starting point and the progression of the sine wave along the x-axis.