Problem 62
Question
Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=\frac{5}{2} \cos (6 x+\pi)$$
Step-by-Step Solution
Verified Answer
The amplitude is \(\frac{5}{2}\) and the period is \(\frac{\pi}{3}\). The function graph will have a maximum and minimum of \(\frac{5}{2}\) and \(-\frac{5}{2}\) respectively, repeating every \(\frac{\pi}{3}\) units, and shifted \(\pi\) units to the left.
1Step 1: Identify the Amplitude
The amplitude of a function is the absolute value of the coefficient in front of the cosine. In this case, the amplitude is the absolute value of \(\frac{5}{2}\), which is \(\frac{5}{2}\).
2Step 2: Calculate the Period
The period of a cosine function is normally \(2\pi\). If the \(x\) in the cosine function is multiplied by a factor, then the period is divided by the absolute value of that factor. In this case, the period is \(2\pi\) divided by the absolute value of \(6\), which is \(\frac{\pi}{3}\).
3Step 3: Graphing the Function
Use the amplitude and the period to graph the cosine function. The function starts at the maximum (the amplitude), then goes to the minimum (-amplitude), and then returns to the maximum, all within the span of one period. The phase shift, represented by \(\pi\) in this case, shifts the function \(\pi\) units to the left. So the graph would have to be moved to the left by \(\pi\) units. Draw two complete periods of the function to complete the task.
Key Concepts
AmplitudePeriodPhase Shift
Amplitude
When working with trigonometric functions like cosine, the amplitude is a critical feature that tells us about the 'height' of the graph. In the function given, \(y = \frac{5}{2} \cos(6x + \pi)\), the amplitude is found by taking the absolute value of the coefficient in front of the cosine term. Here, the coefficient is \(\frac{5}{2}\). Thus, the amplitude is \(\frac{5}{2}\), which means the graph will oscillate \(\frac{5}{2}\) units above and below the horizontal axis.
Amplitude can be understood as half the distance between the maximum and minimum values of the graph. This feature is significant because it gives us a sense of the vertical stretch or compression of the wave. In real-world applications, such as sound waves, the amplitude can correspond to loudness. So, understanding amplitude helps in grasping how much variation there is in the signal or graph.
Amplitude can be understood as half the distance between the maximum and minimum values of the graph. This feature is significant because it gives us a sense of the vertical stretch or compression of the wave. In real-world applications, such as sound waves, the amplitude can correspond to loudness. So, understanding amplitude helps in grasping how much variation there is in the signal or graph.
Period
The period of a trigonometric function like cosine or sine indicates how long it takes for the graph to complete one full cycle. Normally, for a basic cosine function \(\cos(x)\), the period is \(2\pi\).
However, if the function is modified inside the cosine, as in \(y = \frac{5}{2} \cos(6x + \pi)\), the period changes. This change is determined by the coefficient of \(x\), in this case, 6. The period is calculated by taking the normal period, \(2\pi\), and dividing by the absolute value of this coefficient. So, the period becomes \(\frac{2\pi}{6} = \frac{\pi}{3}\).
What does this mean for the graph? It means that the cosine wave completes one full cycle, from start through the maximum, minimum, and back to the start, in a span of \(\frac{\pi}{3}\) along the x-axis. Compressing the period to a smaller section like this often results in more repetitions of the wave over the same length of the x-axis, making the function oscillate faster.
However, if the function is modified inside the cosine, as in \(y = \frac{5}{2} \cos(6x + \pi)\), the period changes. This change is determined by the coefficient of \(x\), in this case, 6. The period is calculated by taking the normal period, \(2\pi\), and dividing by the absolute value of this coefficient. So, the period becomes \(\frac{2\pi}{6} = \frac{\pi}{3}\).
What does this mean for the graph? It means that the cosine wave completes one full cycle, from start through the maximum, minimum, and back to the start, in a span of \(\frac{\pi}{3}\) along the x-axis. Compressing the period to a smaller section like this often results in more repetitions of the wave over the same length of the x-axis, making the function oscillate faster.
Phase Shift
Phase shift involves the horizontal movement of a function's graph along the x-axis. It's determined by the phase constant, which is the number added inside the cosine function alongside the variable. In the function \(y = \frac{5}{2} \cos(6x + \pi)\), the \(+\pi\) part indicates a phase shift.
The phase shift can be calculated by dividing the phase constant, \(\pi\), by the coefficient of \(x\), which is 6. Thus, the shift is \(\frac{\pi}{6}\) units. Since the phase constant is positive, the cosine graph moves to the left by \(\frac{\pi}{6}\) units.
This feature is quite important because it determines where along the x-axis the graph's cycle starts. If this were a sound wave, for instance, the phase shift would affect when a particular characteristic of the sound starts relative to a reference point, helping to align or offset it in time compared to other waves. Understanding phase shift ensures you're grasping not only the shape of the wave but also its starting position.
The phase shift can be calculated by dividing the phase constant, \(\pi\), by the coefficient of \(x\), which is 6. Thus, the shift is \(\frac{\pi}{6}\) units. Since the phase constant is positive, the cosine graph moves to the left by \(\frac{\pi}{6}\) units.
This feature is quite important because it determines where along the x-axis the graph's cycle starts. If this were a sound wave, for instance, the phase shift would affect when a particular characteristic of the sound starts relative to a reference point, helping to align or offset it in time compared to other waves. Understanding phase shift ensures you're grasping not only the shape of the wave but also its starting position.
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