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To graph the function [tex]\( y = 4 \sin \left( \frac{1}{4} x \right) \)[/tex], follow these steps:
### Step-by-Step Solution
1. Identify the Basic Sine Function Characteristics:
- The standard sine function is [tex]\( y = \sin(x) \)[/tex], which has:
- An amplitude of 1.
- A period of [tex]\( 2\pi \)[/tex] (since it completes one cycle from [tex]\( 0 \)[/tex] to [tex]\( 2\pi \)[/tex]).
2. Adjust for Amplitude:
- The given function [tex]\( y = 4 \sin \left( \frac{1}{4} x \right) \)[/tex] has an amplitude of 4. This means the peak values of the sine wave will be scaled by a factor of 4.
- Therefore, the maximum value is [tex]\( 4 \)[/tex] and the minimum value is [tex]\( -4 \)[/tex].
3. Adjust for Period:
- The coefficient [tex]\( \frac{1}{4} \)[/tex] inside the sine function affects the period.
- The period of a sine function [tex]\( y = \sin(Bx) \)[/tex] is given by [tex]\( \frac{2\pi}{|B|} \)[/tex].
- For [tex]\( y = 4 \sin \left( \frac{1}{4} x \right) \)[/tex], [tex]\( B = \frac{1}{4} \)[/tex].
- Therefore, the period is [tex]\( \frac{2\pi}{\frac{1}{4}} = 8\pi \)[/tex].
4. Generate Points for Plotting:
- We need to plot points over one complete cycle of the sine function from [tex]\( -4\pi \)[/tex] to [tex]\( 4\pi \)[/tex] (centered around zero for symmetry).
- The chosen [tex]\( x \)[/tex]-values ranging from [tex]\( -2\pi \)[/tex] to [tex]\( 2\pi \)[/tex], seem to generate a standard cycle of the sine wave.
5. Plot the Points:
- Calculate [tex]\( y \)[/tex]-values using [tex]\( y = 4 \sin \left( \frac{1}{4} x \right) \)[/tex].
- Let's use some key points:
- When [tex]\( x = -2\pi \)[/tex], [tex]\( y = 4 \sin \left( \frac{-2\pi}{4} \right) = 4 \sin \left( -\frac{\pi}{2} \right) = -4 \)[/tex].
- When [tex]\( x = -\pi \)[/tex], [tex]\( y = 4 \sin \left( \frac{-\pi}{4} \right) = 4 \sin \left( -\frac{\pi}{4} \right) = -4\sqrt{2}/2 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 4 \sin(0) = 0 \)[/tex].
- When [tex]\( x = \pi \)[/tex], [tex]\( y = 4 \sin \left( \frac{\pi}{4} \right) = 4\sqrt{2}/2 \)[/tex].
- When [tex]\( x = 2\pi \)[/tex], [tex]\( y = 4 \sin \left( \frac{2\pi}{4} \right) = 4 \sin \left( \frac{\pi}{2} \right) = 4 \)[/tex].
6. Graph Incrementally:
- Begin plotting from [tex]\( -2\pi \)[/tex] to [tex]\( 2\pi \)[/tex] ensuring to get the key points right.
- The points:
- [tex]\( (-2\pi, -4) \)[/tex]
- [tex]\( (-\pi, -4\sqrt{2}/2) \)[/tex]
- [tex]\( (0, 0) \)[/tex]
- [tex]\( (\pi, 4\sqrt{2}/2) \)[/tex]
- [tex]\( (2\pi, 4) \)[/tex]
- Connect these points smoothly following the sine curve behavior.
### Graph Representation:
The graph of the function will show a sine wave with:
- An amplitude of 4.
- A period of [tex]\( 8\pi \)[/tex].
- Peaks at [tex]\( 4 \)[/tex] and troughs at [tex]\( -4 \)[/tex].
To graph:
- Click and drag the highlighted points to adjust the graph in accordance with the periodicity and amplitude as described.
- Shift + Mouse Wheel to zoom in or out to better view the graph.
- Shift + Mouse Drag to pan across the graph.
Remember that the sine function is continuous and periodic, so the wave will repeat every [tex]\( 8\pi \)[/tex] units along the [tex]\( x \)[/tex]-axis.
### Step-by-Step Solution
1. Identify the Basic Sine Function Characteristics:
- The standard sine function is [tex]\( y = \sin(x) \)[/tex], which has:
- An amplitude of 1.
- A period of [tex]\( 2\pi \)[/tex] (since it completes one cycle from [tex]\( 0 \)[/tex] to [tex]\( 2\pi \)[/tex]).
2. Adjust for Amplitude:
- The given function [tex]\( y = 4 \sin \left( \frac{1}{4} x \right) \)[/tex] has an amplitude of 4. This means the peak values of the sine wave will be scaled by a factor of 4.
- Therefore, the maximum value is [tex]\( 4 \)[/tex] and the minimum value is [tex]\( -4 \)[/tex].
3. Adjust for Period:
- The coefficient [tex]\( \frac{1}{4} \)[/tex] inside the sine function affects the period.
- The period of a sine function [tex]\( y = \sin(Bx) \)[/tex] is given by [tex]\( \frac{2\pi}{|B|} \)[/tex].
- For [tex]\( y = 4 \sin \left( \frac{1}{4} x \right) \)[/tex], [tex]\( B = \frac{1}{4} \)[/tex].
- Therefore, the period is [tex]\( \frac{2\pi}{\frac{1}{4}} = 8\pi \)[/tex].
4. Generate Points for Plotting:
- We need to plot points over one complete cycle of the sine function from [tex]\( -4\pi \)[/tex] to [tex]\( 4\pi \)[/tex] (centered around zero for symmetry).
- The chosen [tex]\( x \)[/tex]-values ranging from [tex]\( -2\pi \)[/tex] to [tex]\( 2\pi \)[/tex], seem to generate a standard cycle of the sine wave.
5. Plot the Points:
- Calculate [tex]\( y \)[/tex]-values using [tex]\( y = 4 \sin \left( \frac{1}{4} x \right) \)[/tex].
- Let's use some key points:
- When [tex]\( x = -2\pi \)[/tex], [tex]\( y = 4 \sin \left( \frac{-2\pi}{4} \right) = 4 \sin \left( -\frac{\pi}{2} \right) = -4 \)[/tex].
- When [tex]\( x = -\pi \)[/tex], [tex]\( y = 4 \sin \left( \frac{-\pi}{4} \right) = 4 \sin \left( -\frac{\pi}{4} \right) = -4\sqrt{2}/2 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 4 \sin(0) = 0 \)[/tex].
- When [tex]\( x = \pi \)[/tex], [tex]\( y = 4 \sin \left( \frac{\pi}{4} \right) = 4\sqrt{2}/2 \)[/tex].
- When [tex]\( x = 2\pi \)[/tex], [tex]\( y = 4 \sin \left( \frac{2\pi}{4} \right) = 4 \sin \left( \frac{\pi}{2} \right) = 4 \)[/tex].
6. Graph Incrementally:
- Begin plotting from [tex]\( -2\pi \)[/tex] to [tex]\( 2\pi \)[/tex] ensuring to get the key points right.
- The points:
- [tex]\( (-2\pi, -4) \)[/tex]
- [tex]\( (-\pi, -4\sqrt{2}/2) \)[/tex]
- [tex]\( (0, 0) \)[/tex]
- [tex]\( (\pi, 4\sqrt{2}/2) \)[/tex]
- [tex]\( (2\pi, 4) \)[/tex]
- Connect these points smoothly following the sine curve behavior.
### Graph Representation:
The graph of the function will show a sine wave with:
- An amplitude of 4.
- A period of [tex]\( 8\pi \)[/tex].
- Peaks at [tex]\( 4 \)[/tex] and troughs at [tex]\( -4 \)[/tex].
To graph:
- Click and drag the highlighted points to adjust the graph in accordance with the periodicity and amplitude as described.
- Shift + Mouse Wheel to zoom in or out to better view the graph.
- Shift + Mouse Drag to pan across the graph.
Remember that the sine function is continuous and periodic, so the wave will repeat every [tex]\( 8\pi \)[/tex] units along the [tex]\( x \)[/tex]-axis.
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