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Magazine Chapter 2: Problem 78
The graph of \(y=3 \cos (5(x-\pi))+7\) is a _________ wave.
Short Answer
Expert verified
The graph is a cosine wave.
Step by step solution
01
- Identify the Function Type
The given function is in the form of a cosine function. The general form of a cosine function is: \[ y = A \cos(B(x - C)) + D \]
02
- Determine the Amplitude
In the function \( y = 3 \cos (5(x - \pi)) + 7 \), the amplitude is determined by the coefficient of the cosine function. Therefore, the amplitude is \( A = 3 \).
03
- Identify the Frequency (or B value)
The frequency of the cosine function is represented by the coefficient \( B = 5 \) in \( 5(x - \pi) \). This changes the period of the wave.
04
- Phase Shift Calculation
The phase shift is calculated by \( C / B \). For this function: \[ C = \pi \text{ and } B = 5, \text{ so the phase shift = } \pi / 5 \]. This means the graph shifts \( \pi / 5 \) units to the right.
05
- Vertical Shift
The vertical shift \( D \) is represented by the constant term at the end of the function. Here it is \( D = 7 \), which means the entire graph of the cosine wave is shifted 7 units upwards.
06
- Summarize and Conclude
Combining all the transformations鈥攁mplitude change, frequency modification, phase shift, and vertical shift鈥攖he graph remains a cosine wave with these characteristics.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Amplitude
In the equation of a cosine wave, the amplitude shows how tall or high the peaks of the wave are.
For the given function, the amplitude is 3 because it is the coefficient in front of the cosine function.
This means that the wave鈥檚 highest and lowest points are 3 units away from the centerline (or equilibrium position).
You can think of amplitude as the 'loudness' of the wave in a visual sense.
So, if you see the function is written as: \(y = 3 \cos (5(x-\pi)) + 7\), you know that 3 is the amplitude.
For the given function, the amplitude is 3 because it is the coefficient in front of the cosine function.
This means that the wave鈥檚 highest and lowest points are 3 units away from the centerline (or equilibrium position).
You can think of amplitude as the 'loudness' of the wave in a visual sense.
So, if you see the function is written as: \(y = 3 \cos (5(x-\pi)) + 7\), you know that 3 is the amplitude.
Frequency
The frequency of a cosine wave indicates how often the wave repeats itself over a set distance.
In simpler words, it shows how many cycles occur in a particular interval.
For the given function, the frequency is identified by the value of B, which is 5 in this case.
This influences the period of the wave, which is calculated as \(\frac{2\pi}{B}\).
So, with B being 5, the period will be \(\frac{2\pi}{5}\).
This means that one complete cycle of the wave is completed in a shorter distance, making it stretch or compress horizontally.
In simpler words, it shows how many cycles occur in a particular interval.
For the given function, the frequency is identified by the value of B, which is 5 in this case.
This influences the period of the wave, which is calculated as \(\frac{2\pi}{B}\).
So, with B being 5, the period will be \(\frac{2\pi}{5}\).
This means that one complete cycle of the wave is completed in a shorter distance, making it stretch or compress horizontally.
Phase Shift
The phase shift of a cosine wave tells you how the wave has been moved left or right on the graph.
To calculate the phase shift, use the formula \(\frac{C}{B}\) where C is the value subtracted inside the cosine function, and B is the frequency coefficient.
For the given function \(y = 3 \cos (5(x-\pi)) + 7\), the value of C is \(\pi\) and B is 5.
Thus, the phase shift is \(\frac{\pi}{5}\).
This means the graph of the wave is shifted \(\frac{\pi}{5}\) units to the right.
To calculate the phase shift, use the formula \(\frac{C}{B}\) where C is the value subtracted inside the cosine function, and B is the frequency coefficient.
For the given function \(y = 3 \cos (5(x-\pi)) + 7\), the value of C is \(\pi\) and B is 5.
Thus, the phase shift is \(\frac{\pi}{5}\).
This means the graph of the wave is shifted \(\frac{\pi}{5}\) units to the right.
Vertical Shift
The vertical shift moves the entire cosine wave up or down along the y-axis.
In the equation, this is represented by the constant D added to or subtracted from the cosine function.
For the function \(y = 3 \cos (5(x-\pi)) + 7\), D is 7.
This means the whole wave is lifted 7 units higher than it would be if D were zero.
Vertical shifts are seen as movements of the centerline up or down from the standard y = 0 line.
So here, the midline of the wave is now y = 7.
In the equation, this is represented by the constant D added to or subtracted from the cosine function.
For the function \(y = 3 \cos (5(x-\pi)) + 7\), D is 7.
This means the whole wave is lifted 7 units higher than it would be if D were zero.
Vertical shifts are seen as movements of the centerline up or down from the standard y = 0 line.
So here, the midline of the wave is now y = 7.
