Question
x 0 π2 π 3π2 2π y 0 34 0 −34 0 Write the equation of the periodic function based on the table. (1 point) Responses y=−34sinx y equals negative Start Fraction 3 over 4 End Fraction sine x y=34cosx y equals Start Fraction 3 over 4 End Fraction cosine x y=32sinx y equals Start Fraction 3 over 2 End Fraction sine x y=34sinx
Answers
GPT-4o mini
To derive the equation of the periodic function based on the provided values in the table, let’s first understand the pattern of the points:
- At \( x = 0 \), \( y = 0 \)
- At \( x = \frac{\pi}{2} \), \( y = 34 \)
- At \( x = \pi \), \( y = 0 \)
- At \( x = \frac{3\pi}{2} \), \( y = -34 \)
- At \( x = 2\pi \), \( y = 0 \)
These values suggest that this is a sine function that oscillates between 34 and -34.
1. **Amplitude**: The maximum value is 34 and the minimum value is -34. The amplitude \( A \) is given by:
\[
A = \frac{\text{max} - \text{min}}{2} = \frac{34 - (-34)}{2} = \frac{68}{2} = 34
\]
2. **Period**: Since the function returns to its starting point after \( 2\pi \), the period \( T \) is \( 2\pi \). Thus, it fits the standard sine function with no horizontal stretching/compression.
3. **Starting Point**: The sine function starts at zero, which matches \( y = 34 \sin(x) \) as \( \sin(0) = 0 \).
From this analysis, we can conclude that:
\[
y = 34 \sin(x)
\]
Since the sine function swings between its maximum and minimum values and aligns perfectly with the provided data points, the correct equation of the periodic function based on the table is:
\[
\boxed{y = 34 \sin x}
\]
- At \( x = 0 \), \( y = 0 \)
- At \( x = \frac{\pi}{2} \), \( y = 34 \)
- At \( x = \pi \), \( y = 0 \)
- At \( x = \frac{3\pi}{2} \), \( y = -34 \)
- At \( x = 2\pi \), \( y = 0 \)
These values suggest that this is a sine function that oscillates between 34 and -34.
1. **Amplitude**: The maximum value is 34 and the minimum value is -34. The amplitude \( A \) is given by:
\[
A = \frac{\text{max} - \text{min}}{2} = \frac{34 - (-34)}{2} = \frac{68}{2} = 34
\]
2. **Period**: Since the function returns to its starting point after \( 2\pi \), the period \( T \) is \( 2\pi \). Thus, it fits the standard sine function with no horizontal stretching/compression.
3. **Starting Point**: The sine function starts at zero, which matches \( y = 34 \sin(x) \) as \( \sin(0) = 0 \).
From this analysis, we can conclude that:
\[
y = 34 \sin(x)
\]
Since the sine function swings between its maximum and minimum values and aligns perfectly with the provided data points, the correct equation of the periodic function based on the table is:
\[
\boxed{y = 34 \sin x}
\]