Problem 23 - Precalculus: Functions and Graphs

Question

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=2 \cos \frac{\pi}{2} x\)

Step-by-Step Solution

Answer

Amplitude: 2, Period: 4, Phase Shift: 0, Cosine wave starting at maximum.

Step 1: Identify the Amplitude

The amplitude of a cosine function is the absolute value of the coefficient in front of the cosine function. In the equation \( y = 2 \cos \frac{\pi}{2} x \), the amplitude is \( 2 \).

Step 2: Determine the Period

The period of a cosine function is given by the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). In this case, \( b = \frac{\pi}{2} \). So, the period is \( \frac{2\pi}{\frac{\pi}{2}} = 4 \).

Step 3: Calculate the Phase Shift

The phase shift for a basic cosine function \( y = a \cos(bx - c) \) is given by \( \frac{c}{b} \). In this equation, there is no phase shift because there is no \( c \) value (since \( c = 0 \)). Therefore, the phase shift is \( 0 \).

Step 4: Sketch the Graph

To sketch the graph, use the amplitude, period, and phase shift calculated. The cosine wave starts at its maximum at \( x = 0 \), has a period of \( 4 \), and returns to its starting point. Plot one cycle from \( 0 \) to \( 4 \), marking points at maximum (\( y = 2 \)), minimum (\( y = -2 \)), and equilateral crossing at \( x = 2 \). The graph is completed by extending it for additional cycles as needed.

Key Concepts

Understanding Amplitude

In the world of trigonometric functions, particularly the cosine function, the amplitude represents the peak value of the wave. It's essentially the height of the wave from its central axis. For any equation of the form

1. The amplitude comes from the coefficient of the cosine function. In simpler terms, it's the number that multiplies the cosine.
2. In our example equation, \( y = 2 \cos \frac{\pi}{2} x \), the amplitude is \( 2 \). This tells us that the wave reaches a peak at \( y = 2 \) and a trough at \( y = -2 \).
3. Amplitude is always a positive number, as it's derived from taking the absolute value of the coefficient.

Recognizing the amplitude is crucial in graphing trigonometric functions because it dictates the vertical stretch or compression of the wave. Thus, it defines how intense or subdued the oscillations appear on the graph.

Period of Trigonometric Functions

The period of a trigonometric function indicates how long it takes for the wave to complete one full cycle. In mathematical terms, it's how wide the graph stretches before repeating itself.

The standard formula to find the period for a cosine function is \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) in the argument of the cosine.

• For our cosine function, \( y = 2 \cos \frac{\pi}{2} x \), the value of \( b \) is \( \frac{\pi}{2} \).
• Using the formula, we find the period to be \( \frac{2\pi}{\frac{\pi}{2}} = 4 \). This means the pattern of the wave repeats every 4 units along the x-axis.

Knowing the period is essential when sketching graphs, as it helps identify space between these repeating cycles. It impacts the frequency and makes it clear how the wave behaves over time.

What is Phase Shift?

Phase shift describes the horizontal movement of a trigonometric function along the x-axis. If a function shifts left or right, it affects where the wave begins its cycle compared to the basic cosine wave.

For any function \( y = a \cos(bx - c) \), the phase shift can be calculated using \( \frac{c}{b} \). However, in cases where \( c = 0 \), as seen in our equation \( y = 2 \cos \frac{\pi}{2} x \), the phase shift is \( 0 \). This means:

• The wave starts exactly at the typical starting point without any horizontal movements.
• It's important to remember that a positive c would indicate a shift to the right, while a negative c would shift the wave to the left.

Understanding phase shift is vital for plotting and interpreting the graph accurately since it influences the initial position of the cosine function and helps in aligning subsequent waves correctly.