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Chapter 5: Problem 22

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$y=3 \cos \frac{\pi}{2} x$$

Short Answer

Expert verified
Amplitude: 3, Period: 4, Phase Shift: 0; Sketch completes at (4,3).

Step by step solution

01

Identify the Amplitude

The amplitude of a cosine function in the form \( y = a \cos(bx + c) + d \) is the absolute value of \( a \). In this equation, \( a = 3 \). Therefore, the amplitude is \( |3| = 3 \).
02

Determine the Period

The period of a cosine function is given by \( \frac{2\pi}{b} \). In the equation \( y = 3 \cos \frac{\pi}{2} x \), \( b = \frac{\pi}{2} \). Thus, the period is \( \frac{2\pi}{\frac{\pi}{2}} = 4 \).
03

Calculate the Phase Shift

The phase shift of a cosine function \( y = a \cos(bx + c) + d \) is \( -\frac{c}{b} \). Here, \( c = 0 \), so the phase shift is \( -\frac{0}{\frac{\pi}{2}} = 0 \). There is no horizontal shift.
04

Sketch the Graph

The graph of \( y = 3 \cos \frac{\pi}{2} x \) will have an amplitude of 3, a period of 4, and no phase shift. The graph starts at (0,3), reaches the minimum at (2,-3), and completes the cycle at (4,3). Plot these points and sketch a smooth wave between them.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude
The amplitude of a trigonometric function, like the cosine function, determines the height of its wave, or how far the wave reaches above and below its central axis. For a standard cosine function written as \( y = a \cos(bx + c) + d \), the amplitude is given by the absolute value of \( a \), the coefficient in front of the cosine.
  • Amplitude tells us about the maximum and minimum values that the function can reach.
  • In our example, the equation is \( y = 3 \cos \frac{\pi}{2} x \).
  • The amplitude here is calculated as \( |3| = 3 \), meaning the wave peaks at 3 and troughs at -3.
Understanding amplitude gives insight into the "loudness" or intensity of the wave's oscillation. The wave doesn't stretch or compress horizontally鈥攁mplitude solely affects vertical stretching.
Period of a Function
The period of a trigonometric function indicates the length of one complete cycle of the wave. For the cosine function \( y = a \cos(bx + c) + d \), the period is determined by the formula \( \frac{2\pi}{b} \).
  • This measurement tells us how quickly the function repeats itself.
  • For \( y = 3 \cos \frac{\pi}{2} x \), we have \( b = \frac{\pi}{2} \).
  • By plugging in \( b \), we find the period to be \( \frac{2\pi}{\frac{\pi}{2}} = 4 \).
This means every four units along the x-axis, the function starts its wave cycle over again. The length of the period can visually impact how "compressed" or "stretched out" the waves appear on a graph.
Phase Shift
The phase shift in trigonometric functions is the horizontal displacement of the wave from its usual starting point. It's determined using the formula \( -\frac{c}{b} \) for a function in the form \( y = a \cos(bx + c) + d \).
  • Phase shift dictates where the wave cycle begins along the x-axis.
  • In the case of \( y = 3 \cos \frac{\pi}{2} x \), \( c = 0 \) which results in \( -\frac{0}{\frac{\pi}{2}} = 0 \).
  • So, there is no horizontal shift; the wave starts at the y-axis.
Without a phase shift, the cosine wave starts exactly where expected, making it simpler to draw and understand. On a graph, this means the peak, typically starting at (0, amplitude), aligns with the y-axis, reflecting the function's natural behavior without additional shifts.

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