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Magazine Chapter 4: Problem 33
In Exercises 25-40, graph the given sinusoidal functions over one period. $$ y=-3 \cos \left(\frac{1}{2} x\right) $$
Short Answer
Expert verified
The graph of the function completes a full negative cosine wave from \( x = 0 \) to \( x = 4\pi \) with an amplitude of 3.
Step by step solution
01
Identify Components of the Function
The given function is \( y = -3 \cos \left( \frac{1}{2} x \right) \). Here, the coefficient of \( x \), which is \( \frac{1}{2} \), affects the period of the cosine function. The coefficient \( -3 \) affects the amplitude. The formula for the cosine function is generally \( y = a \cos(bx) \), where \( |a| \) is the amplitude, and the period is given by \( \frac{2\pi}{b} \).
02
Determine the Amplitude
The amplitude of the function is the absolute value of the coefficient in front of the cosine. Here, it is \( |-3| = 3 \). This means the graph will oscillate 3 units above and below the midline, which is the x-axis in this case.
03
Determine the Period
The period of the cosine function is calculated using \( \frac{2\pi}{b} \). Here, \( b = \frac{1}{2} \), so the period is \( \frac{2\pi}{\frac{1}{2}} = 4\pi \). This means one complete cycle of the cosine wave occurs from \( x = 0 \) to \( x = 4\pi \).
04
Determine Key Points for the Graph
The key points in one period of a cosine function occur at fractions of the period. The cosine function starts at its maximum, reaches zero at \( \frac{1}{4} \) of the period, its minimum at \( \frac{1}{2} \) of the period, zero again at \( \frac{3}{4} \) of the period, and returns to its maximum at the period. With a negation, the points will be reflected across the x-axis.
05
Plot the Key Points
For \( y = -3\cos\left(\frac{1}{2}x\right) \): - At \( x = 0 \), \( y = -3 \). - At \( x = \pi \), \( y = 0 \). - At \( x = 2\pi \), \( y = 3 \). - At \( x = 3\pi \), \( y = 0 \). - At \( x = 4\pi \), \( y = -3 \).Plot and connect these points with a smooth curve.
06
Sketch the Graph
Draw the x-axis and y-axis. Mark the key points found in Step 5. Sketch the sinusoidal curve through these points, dipping and rising smoothly, creating one complete wave from \( x = 0 \) to \( x = 4\pi \). Ensure the wave is inverted due to the negative amplitude, starting at a minimum value of -3.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Amplitude of Cosine Function
The amplitude of a cosine function gives you an idea of how high and low the wave reaches vertically from its midline. In the function \( y = -3 \cos \left( \frac{1}{2} x \right) \), the amplitude is represented by the coefficient of the cosine term, which is \(-3\).
This means the amplitude is actually the absolute value \(|-3| = 3\).
This tells us that the wave oscillates 3 units above and 3 units below the midline, which for this function is the x-axis.
This means the amplitude is actually the absolute value \(|-3| = 3\).
This tells us that the wave oscillates 3 units above and 3 units below the midline, which for this function is the x-axis.
- The amplitude of 3 indicates that the highest value the function reaches is 3, and the lowest value is -3.
- The negative sign in front of the coefficient indicates the waveform will be inverted.
Period of Cosine Function
The period of a cosine function determines how long it takes to complete one full cycle of its wave.
In the function \( y = -3 \cos \left( \frac{1}{2} x \right) \), the coefficient \(\frac{1}{2}\) inside the cosine affects the period.
The formula to calculate the period of a basic cosine function \( y = a \cos (bx) \) is \( \frac{2\pi}{b} \).
Substituting \( b = \frac{1}{2} \) gives us the period \( \frac{2\pi}{\frac{1}{2}} = 4\pi \).
This means that the curve completes one full cycle from \( x = 0 \) to \( x = 4\pi \).
In the function \( y = -3 \cos \left( \frac{1}{2} x \right) \), the coefficient \(\frac{1}{2}\) inside the cosine affects the period.
The formula to calculate the period of a basic cosine function \( y = a \cos (bx) \) is \( \frac{2\pi}{b} \).
Substituting \( b = \frac{1}{2} \) gives us the period \( \frac{2\pi}{\frac{1}{2}} = 4\pi \).
This means that the curve completes one full cycle from \( x = 0 \) to \( x = 4\pi \).
- A period of \(4\pi\) means the wave will stretch out wider compared to the standard cosine curve, which has a period of \(2\pi\).
Key Points in Graphing
Graphing a sinusoidal function like a cosine involves plotting the key points over one period of the cycle.
Key points for the cosine function occur at fractional parts of the period.
For \( y = -3 \cos \left( \frac{1}{2} x \right) \), these points include:
Key points for the cosine function occur at fractional parts of the period.
For \( y = -3 \cos \left( \frac{1}{2} x \right) \), these points include:
- Maximum value at the start (inverted to minimum due to negation): \((0, -3)\)
- Zero crossings occur at \( \frac{1}{4} \) and \( \frac{3}{4} \) of the period: \((\pi, 0)\) and \((3\pi, 0)\)
- Minimum value (inverted to maximum) at half period: \((2\pi, 3)\)
- Ends at the starting inverted minimum \((4\pi, -3)\)
Negative Amplitude Reflection
When a cosine function has a negative amplitude, it reflects the graph across the x-axis.
For \( y = -3 \cos \left( \frac{1}{2} x \right) \), this reflection means:
Remember, the mathematical transformation affects only the graph's orientation, not its location or period.
For \( y = -3 \cos \left( \frac{1}{2} x \right) \), this reflection means:
- The peaks and troughs are inverted; what normally would be a peak (+3) now becomes a trough (-3).
- Similarly, positions that normally hit the trough (-3) will now reflect to a peak (+3).
Remember, the mathematical transformation affects only the graph's orientation, not its location or period.
