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To identify which graph represents the equation [tex]\( y = 3 \sin \left(\frac{1}{3} x\right) \)[/tex], let's analyze the characteristics of the graph step-by-step.
### Step 1: Understand the Basic Shape
The function involved is a sine function, [tex]\( \sin(x) \)[/tex], which generally oscillates between -1 and 1 with a period of [tex]\( 2\pi \)[/tex]. Given that our function is modified, we need to see how these modifications affect it.
### Step 2: Amplitude
The amplitude of the function [tex]\( \sin(x) \)[/tex] is the maximum distance from the midline (x-axis) to the peak of the sine wave, which is 1 for the basic sine function. In [tex]\( y = 3 \sin \left(\frac{1}{3} x\right) \)[/tex], the amplitude is multiplied by 3. Therefore, the new amplitude is:
[tex]\[ 3 \times 1 = 3 \][/tex]
This means the function will oscillate between -3 and 3.
### Step 3: Period
The period of the basic [tex]\( \sin(x) \)[/tex] function is [tex]\( 2\pi \)[/tex]. The period of the sine function is affected by the coefficient inside the argument of the sine function. For the function [tex]\( y = \sin(bx) \)[/tex], the period [tex]\( P \)[/tex] is given by:
[tex]\[ P = \frac{2\pi}{|b|} \][/tex]
Here, the inside argument of our sine function is [tex]\( \frac{1}{3} x \)[/tex]. So,
[tex]\[ P = \frac{2\pi}{\left|\frac{1}{3}\right|} = 2\pi \times 3 = 6\pi \][/tex]
Thus, the period of the function [tex]\( y = 3 \sin \left(\frac{1}{3} x\right) \)[/tex] is [tex]\( 6\pi \)[/tex].
### Step 4: Phase Shift and Vertical Shift
There are no horizontal shifts (phase shifts) or vertical shifts in the equation [tex]\( y = 3 \sin \left(\frac{1}{3} x\right) \)[/tex]. The function has not been translated left, right, up, or down. The midline remains the x-axis, [tex]\( y = 0 \)[/tex].
### Step 5: Putting It All Together
To summarize, the complete analysis indicates that the graph of [tex]\( y = 3 \sin \left(\frac{1}{3} x\right) \)[/tex] has:
- An amplitude of 3
- A period of [tex]\( 6\pi \)[/tex]
- No phase shift
- No vertical shift
### Conclusion
Considering the above characteristics, the correct graph will show a sine wave oscillating between -3 and 3, completing one full cycle (from peak to peak or trough to trough) every [tex]\( 6\pi \)[/tex] units along the x-axis.
Thus, the graph you are looking for will reflect these properties: a sine wave pattern that reaches its maximum value of 3 and minimum value of -3 and completes one period over an interval of [tex]\( 6\pi \)[/tex]. Look for the graph that exhibits these features precisely.
If you are given options, you can now identify the correct one based on this detailed explanation.
### Step 1: Understand the Basic Shape
The function involved is a sine function, [tex]\( \sin(x) \)[/tex], which generally oscillates between -1 and 1 with a period of [tex]\( 2\pi \)[/tex]. Given that our function is modified, we need to see how these modifications affect it.
### Step 2: Amplitude
The amplitude of the function [tex]\( \sin(x) \)[/tex] is the maximum distance from the midline (x-axis) to the peak of the sine wave, which is 1 for the basic sine function. In [tex]\( y = 3 \sin \left(\frac{1}{3} x\right) \)[/tex], the amplitude is multiplied by 3. Therefore, the new amplitude is:
[tex]\[ 3 \times 1 = 3 \][/tex]
This means the function will oscillate between -3 and 3.
### Step 3: Period
The period of the basic [tex]\( \sin(x) \)[/tex] function is [tex]\( 2\pi \)[/tex]. The period of the sine function is affected by the coefficient inside the argument of the sine function. For the function [tex]\( y = \sin(bx) \)[/tex], the period [tex]\( P \)[/tex] is given by:
[tex]\[ P = \frac{2\pi}{|b|} \][/tex]
Here, the inside argument of our sine function is [tex]\( \frac{1}{3} x \)[/tex]. So,
[tex]\[ P = \frac{2\pi}{\left|\frac{1}{3}\right|} = 2\pi \times 3 = 6\pi \][/tex]
Thus, the period of the function [tex]\( y = 3 \sin \left(\frac{1}{3} x\right) \)[/tex] is [tex]\( 6\pi \)[/tex].
### Step 4: Phase Shift and Vertical Shift
There are no horizontal shifts (phase shifts) or vertical shifts in the equation [tex]\( y = 3 \sin \left(\frac{1}{3} x\right) \)[/tex]. The function has not been translated left, right, up, or down. The midline remains the x-axis, [tex]\( y = 0 \)[/tex].
### Step 5: Putting It All Together
To summarize, the complete analysis indicates that the graph of [tex]\( y = 3 \sin \left(\frac{1}{3} x\right) \)[/tex] has:
- An amplitude of 3
- A period of [tex]\( 6\pi \)[/tex]
- No phase shift
- No vertical shift
### Conclusion
Considering the above characteristics, the correct graph will show a sine wave oscillating between -3 and 3, completing one full cycle (from peak to peak or trough to trough) every [tex]\( 6\pi \)[/tex] units along the x-axis.
Thus, the graph you are looking for will reflect these properties: a sine wave pattern that reaches its maximum value of 3 and minimum value of -3 and completes one period over an interval of [tex]\( 6\pi \)[/tex]. Look for the graph that exhibits these features precisely.
If you are given options, you can now identify the correct one based on this detailed explanation.
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