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Let's go through the detailed steps to understand how to graph the function [tex]\(y = 3 \sin \left(\frac{1}{3} x\right) + 4\)[/tex].
### Step-by-Step Solution
1. Understanding the Structure of the Sine Function:
- The standard sine function is [tex]\(y = \sin(x)\)[/tex].
- The given function is [tex]\(y = 3 \sin \left(\frac{1}{3} x\right) + 4\)[/tex].
2. Amplitude:
- The amplitude of the sine function is given by the coefficient of the sine term.
- Here, the coefficient is [tex]\(3\)[/tex], so the amplitude is [tex]\(3\)[/tex].
3. Period:
- The period of the sine function is affected by the coefficient inside the sine function argument, [tex]\(\frac{1}{3}\)[/tex] in this case.
- The period formula for [tex]\(y = \sin(bx)\)[/tex] is [tex]\(\frac{2\pi}{|b|}\)[/tex].
- Here, [tex]\(b = \frac{1}{3}\)[/tex], so the period is [tex]\(\frac{2\pi}{\frac{1}{3}} = 6\pi\)[/tex].
4. Vertical Shift:
- The vertical shift is given by the constant term added outside the sine function.
- Here, it is [tex]\(+4\)[/tex], so the entire graph of the sine function is shifted 4 units upwards.
5. Graph Characteristics:
- The midline [tex]\(y = 4\)[/tex] is the equilibrium line around which the sine wave oscillates.
- The maximum value of the function is [tex]\(4 + 3 = 7\)[/tex] (top of the amplitude range).
- The minimum value of the function is [tex]\(4 - 3 = 1\)[/tex] (bottom of the amplitude range).
6. Key Points:
To graph the function, we need to identify key points within one period [tex]\(6\pi\)[/tex]:
- Start: [tex]\(x = 0\)[/tex]
- [tex]\( y = 3 \sin \left(\frac{1}{3} \cdot 0 \right) + 4 = 4\)[/tex]
- Quarter Period: [tex]\(x = \frac{6\pi}{4} = \frac{3\pi}{2}\)[/tex]
- [tex]\( y = 3 \sin \left(\frac{1}{3} \cdot \frac{3\pi}{2} \right) + 4 = 3 \sin \left(\frac{\pi}{2}\right) + 4 = 3(1) + 4 = 7 \)[/tex]
- Half Period: [tex]\(x = \frac{6\pi}{2} = 3\pi\)[/tex]
- [tex]\( y = 3 \sin \left(\frac{1}{3} \cdot 3\pi \right) + 4 = 3 \sin (\pi) + 4 = 3(0) + 4 = 4\)[/tex]
- Three Quarter Period: [tex]\(x = \frac{9\pi}{2}\)[/tex]
- [tex]\( y = 3 \sin \left(\frac{1}{3} \cdot \frac{9\pi}{2} \right) + 4 = 3 \sin \left(\frac{3\pi}{2}\right) + 4 = 3(-1) + 4 = 1\)[/tex]
- Full Period: [tex]\(x = 6\pi\)[/tex]
- [tex]\( y = 3 \sin \left(\frac{1}{3} \cdot 6\pi \right) + 4 = 3 \sin (2\pi) + 4 = 3(0) + 4 = 4\)[/tex]
Hence, the key points are:
- [tex]\( (0, 4) \)[/tex]
- [tex]\( \left(\frac{3\pi}{2}, 7\right) \)[/tex]
- [tex]\( (3\pi, 4) \)[/tex]
- [tex]\( \left(\frac{9\pi}{2}, 1\right) \)[/tex]
- [tex]\( (6\pi, 4) \)[/tex]
### Summary:
- Amplitude = 3
- Period = [tex]\(6\pi\)[/tex]
- Midline = [tex]\(y = 4\)[/tex]
- Vertical Shift = 4 units up
Using these characteristics, you should adjust your graphing widget to create the sine wave oscillating above and below the equation [tex]\(y = 4\)[/tex], with a full period of [tex]\(6\pi\)[/tex] and key points identified above. This provides the overall shape of the graph: starting at [tex]\(y = 4\)[/tex], peaking at [tex]\(y = 7\)[/tex], crossing back to [tex]\(y = 4\)[/tex], dipping to [tex]\(y = 1\)[/tex], and finishing one period back at [tex]\(y = 4\)[/tex].
### Step-by-Step Solution
1. Understanding the Structure of the Sine Function:
- The standard sine function is [tex]\(y = \sin(x)\)[/tex].
- The given function is [tex]\(y = 3 \sin \left(\frac{1}{3} x\right) + 4\)[/tex].
2. Amplitude:
- The amplitude of the sine function is given by the coefficient of the sine term.
- Here, the coefficient is [tex]\(3\)[/tex], so the amplitude is [tex]\(3\)[/tex].
3. Period:
- The period of the sine function is affected by the coefficient inside the sine function argument, [tex]\(\frac{1}{3}\)[/tex] in this case.
- The period formula for [tex]\(y = \sin(bx)\)[/tex] is [tex]\(\frac{2\pi}{|b|}\)[/tex].
- Here, [tex]\(b = \frac{1}{3}\)[/tex], so the period is [tex]\(\frac{2\pi}{\frac{1}{3}} = 6\pi\)[/tex].
4. Vertical Shift:
- The vertical shift is given by the constant term added outside the sine function.
- Here, it is [tex]\(+4\)[/tex], so the entire graph of the sine function is shifted 4 units upwards.
5. Graph Characteristics:
- The midline [tex]\(y = 4\)[/tex] is the equilibrium line around which the sine wave oscillates.
- The maximum value of the function is [tex]\(4 + 3 = 7\)[/tex] (top of the amplitude range).
- The minimum value of the function is [tex]\(4 - 3 = 1\)[/tex] (bottom of the amplitude range).
6. Key Points:
To graph the function, we need to identify key points within one period [tex]\(6\pi\)[/tex]:
- Start: [tex]\(x = 0\)[/tex]
- [tex]\( y = 3 \sin \left(\frac{1}{3} \cdot 0 \right) + 4 = 4\)[/tex]
- Quarter Period: [tex]\(x = \frac{6\pi}{4} = \frac{3\pi}{2}\)[/tex]
- [tex]\( y = 3 \sin \left(\frac{1}{3} \cdot \frac{3\pi}{2} \right) + 4 = 3 \sin \left(\frac{\pi}{2}\right) + 4 = 3(1) + 4 = 7 \)[/tex]
- Half Period: [tex]\(x = \frac{6\pi}{2} = 3\pi\)[/tex]
- [tex]\( y = 3 \sin \left(\frac{1}{3} \cdot 3\pi \right) + 4 = 3 \sin (\pi) + 4 = 3(0) + 4 = 4\)[/tex]
- Three Quarter Period: [tex]\(x = \frac{9\pi}{2}\)[/tex]
- [tex]\( y = 3 \sin \left(\frac{1}{3} \cdot \frac{9\pi}{2} \right) + 4 = 3 \sin \left(\frac{3\pi}{2}\right) + 4 = 3(-1) + 4 = 1\)[/tex]
- Full Period: [tex]\(x = 6\pi\)[/tex]
- [tex]\( y = 3 \sin \left(\frac{1}{3} \cdot 6\pi \right) + 4 = 3 \sin (2\pi) + 4 = 3(0) + 4 = 4\)[/tex]
Hence, the key points are:
- [tex]\( (0, 4) \)[/tex]
- [tex]\( \left(\frac{3\pi}{2}, 7\right) \)[/tex]
- [tex]\( (3\pi, 4) \)[/tex]
- [tex]\( \left(\frac{9\pi}{2}, 1\right) \)[/tex]
- [tex]\( (6\pi, 4) \)[/tex]
### Summary:
- Amplitude = 3
- Period = [tex]\(6\pi\)[/tex]
- Midline = [tex]\(y = 4\)[/tex]
- Vertical Shift = 4 units up
Using these characteristics, you should adjust your graphing widget to create the sine wave oscillating above and below the equation [tex]\(y = 4\)[/tex], with a full period of [tex]\(6\pi\)[/tex] and key points identified above. This provides the overall shape of the graph: starting at [tex]\(y = 4\)[/tex], peaking at [tex]\(y = 7\)[/tex], crossing back to [tex]\(y = 4\)[/tex], dipping to [tex]\(y = 1\)[/tex], and finishing one period back at [tex]\(y = 4\)[/tex].
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