Problem 27 - Algebra and Trigonometry

Question

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=\cos \left(x-\frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Standard Form

The standard form for the cosine function is: \[ y = a \cdot \cos(b(x - c)) + d \]Here, our function is \( y = \cos \left(x - \frac{\pi}{2}\right) \). We can see that this matches the form with \( a = 1 \), \( b = 1 \), \( c = \frac{\pi}{2} \), and \( d = 0 \). These values will be used to determine other characteristics.

Step 2: Calculate the Amplitude

The amplitude of a cosine function \( y = a \cdot \cos(b(x - c)) + d \) is given by the absolute value of \( a \). Thus, for our function, the amplitude is: \[ |1| = 1 \]

Step 3: Determine the Period

The period of a cosine function \( y = \cos(b(x - c)) \) is given by \( \frac{2\pi}{|b|} \). For our function, where \( b = 1 \), the period is: \[ \frac{2\pi}{1} = 2\pi \]

Step 4: Identify the Phase Shift

The phase shift is determined by the value of \( c \) in the standard form. It is given by \( c = \frac{\pi}{2} \). Since the form is \( x - c \), and \( c \) is positive, the phase shift is: \[ \frac{\pi}{2} \] to the right.

Step 5: Graph One Complete Period

To graph one complete period of \( y = \cos(x - \frac{\pi}{2}) \): 1. Start at \( x = \frac{\pi}{2} \), where the graph begins due to the phase shift.2. Plot the standard cosine points over one period \( 2\pi \): - Original cosine starts at (0, 1). Adjusted for phase shift, start at \( (\frac{\pi}{2}, 1) \). - Next key points are: \((\frac{3\pi}{2}, 0)\), \((\frac{5\pi}{2}, -1)\), \((\frac{7\pi}{2}, 0)\), and ending the period back at \((\frac{9\pi}{2}, 1)\).3. Draw a smooth wave passing through these points, completing one full cycle of the cosine wave with the specified phase shift.

Key Concepts

Amplitude

The amplitude of a function refers to the maximum vertical distance from the midpoint of the wave to its peak or trough. It is a crucial aspect of trigonometric functions like the cosine and sine functions. For any cosine function of the form:

• \( y = a \cdot \cos(b(x - c)) + d \)

the amplitude is determined by the absolute value of the coefficient \( a \). This value represents how "tall" or "short" the wave appears. If \( |a| \) is greater than 1, the wave becomes taller; if less than 1, it is compressed.

In the given function \( y = \cos(x - \frac{\pi}{2}) \), \( a = 1 \). Thus, the amplitude is \( |1| = 1 \). This means the wave reaches a maximum height of 1 unit above, and a minimum of 1 unit below, the horizontal axis. Understanding this concept helps in visually interpreting the graph and predicting how it will look.

Period

In trigonometry, the period of a function is the length, horizontally, over which the wave pattern repeats itself. It's like knowing how long a loop of the wave will take before it starts again. For the standard cosine function:

• \( y = \cos(b(x - c)) \)

the period is calculated using the formula:\[ \frac{2\pi}{|b|} \] Here, \( b \) determines how many cycles occur in a typical duration. If \( |b| \) is large, the function oscillates more rapidly, and if small, it stretches over a longer span.

For our specific problem with \( y = \cos(x - \frac{\pi}{2}) \), the \( b \) value is 1, which simplifies the period to:\[ \frac{2\pi}{1} = 2\pi \]This means that every \( 2\pi \) units, or one complete cycle on the x-axis, the wave pattern repeats itself. Knowing the period helps in accurately sketching one complete cycle of the wave on a graph and understanding how the function behaves over different intervals.

Phase Shift

Phase shift refers to the horizontal displacement of a trigonometric function. It shows how much the basic wave has been shifted along the x-axis from its usual starting position. In a cosine function of the form:

• \( y = a \cdot \cos(b(x - c)) + d \)

the phase shift is determined by the value of \( c \), and the function is shifted to the right if \( c \) is positive and to the left if it is negative.

In our scenario with the function \( y = \cos(x - \frac{\pi}{2}) \), \( c \) is \( \frac{\pi}{2} \). This indicates a shift of:

• \( \frac{\pi}{2} \) units to the right

Phase shifts are important when graphing because they show how the function aligns with other shifts in a wave pattern, aiding in understanding intersections or overlaps with other mathematical functions or data plotted on the same graph.