Answer :
To determine the truthfulness of each statement given the problem, we can go through them one by one, ensuring we understand each concept.
### Statement 1: The amplitude is 6.
- Amplitude is the maximum displacement from the equilibrium position.
- According to the problem, the weight bobs 3 inches above and 3 inches below the equilibrium position.
- Therefore, the amplitude is 3 inches, not 6 inches.
- This statement is false.
### Statement 2: The graph is stretched vertically by a factor of 3.
- Stretching vertically by a factor means that the amplitude of the motion is multiplied by that factor.
- Since the amplitude is given as 3 inches, the vertical stretch factor refers to this amplitude.
- This statement is true.
### Statement 3: The frequency is \( \frac{1}{8} \).
- Frequency is the reciprocal of the period. It represents how many cycles occur per unit of time.
- Given the period \( T = 8 \) seconds, the frequency \( f \) is \( \frac{1}{T} = \frac{1}{8} \) cycles per second (Hz).
- This statement is true.
### Statement 4: The graph is compressed horizontally by a factor of \( \frac{\pi}{4} \).
- Horizontal compression or stretching relates to changes in the period, typically involving phase shifts or frequency multipliers.
- In the given context, there is no indication of a horizontal compression by a factor of \( \frac{\pi}{4} \).
- This statement is false.
### Statement 5: The frequency is 8.
- Again, the frequency \( f \) is the reciprocal of the period \( T = 8 \) seconds, i.e., \( f = \frac{1}{8} \) Hz.
- Therefore, the frequency is not 8 Hz.
- This statement is false.
After checking all the statements based on the information provided in the problem:
- The amplitude is 6. False
- The graph is stretched vertically by a factor of 3. True
- The frequency is \( \frac{1}{8} \). True
- The graph is compressed horizontally by a factor of \( \frac{\pi}{4} \). False
- The frequency is 8. False
The correct assessments match with the following results:
- (False, True, True, False, False)
### Statement 1: The amplitude is 6.
- Amplitude is the maximum displacement from the equilibrium position.
- According to the problem, the weight bobs 3 inches above and 3 inches below the equilibrium position.
- Therefore, the amplitude is 3 inches, not 6 inches.
- This statement is false.
### Statement 2: The graph is stretched vertically by a factor of 3.
- Stretching vertically by a factor means that the amplitude of the motion is multiplied by that factor.
- Since the amplitude is given as 3 inches, the vertical stretch factor refers to this amplitude.
- This statement is true.
### Statement 3: The frequency is \( \frac{1}{8} \).
- Frequency is the reciprocal of the period. It represents how many cycles occur per unit of time.
- Given the period \( T = 8 \) seconds, the frequency \( f \) is \( \frac{1}{T} = \frac{1}{8} \) cycles per second (Hz).
- This statement is true.
### Statement 4: The graph is compressed horizontally by a factor of \( \frac{\pi}{4} \).
- Horizontal compression or stretching relates to changes in the period, typically involving phase shifts or frequency multipliers.
- In the given context, there is no indication of a horizontal compression by a factor of \( \frac{\pi}{4} \).
- This statement is false.
### Statement 5: The frequency is 8.
- Again, the frequency \( f \) is the reciprocal of the period \( T = 8 \) seconds, i.e., \( f = \frac{1}{8} \) Hz.
- Therefore, the frequency is not 8 Hz.
- This statement is false.
After checking all the statements based on the information provided in the problem:
- The amplitude is 6. False
- The graph is stretched vertically by a factor of 3. True
- The frequency is \( \frac{1}{8} \). True
- The graph is compressed horizontally by a factor of \( \frac{\pi}{4} \). False
- The frequency is 8. False
The correct assessments match with the following results:
- (False, True, True, False, False)
