Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
Let's break down the solution for the function [tex]\( y = 3.5 \sin \left(\frac{2 \pi}{365}(x-90)\right) + 12 \)[/tex]:
### Step-by-Step Analysis:
1. Understanding the Function:
- The given function is [tex]\( y = 3.5 \sin \left(\frac{2 \pi}{365}(x-90)\right) + 12 \)[/tex].
- This is a sinusoidal function, which means it represents a wave-like pattern.
2. Amplitude, Period, Phase Shift, and Vertical Shift:
- Amplitude: The coefficient of the sine function is 3.5, so the amplitude is 3.5. This means the function oscillates 3.5 units above and below the central value.
- Period: The period of the sine function is given by the coefficient of [tex]\( x \)[/tex] inside the sine function. Here, the period is [tex]\( P = \frac{2\pi}{\frac{2\pi}{365}} = 365 \)[/tex]. This means the function completes one full cycle every 365 units.
- Phase Shift: The term [tex]\( (x-90) \)[/tex] indicates a phase shift. The function is shifted to the right by 90 units.
- Vertical Shift: The +12 outside the sine function indicates that the entire function is shifted up by 12 units.
3. Range of Values:
- The sine function varies between -1 and 1.
- Therefore, [tex]\( 3.5 \sin \left(\frac{2 \pi}{365}(x-90)\right) \)[/tex] varies between -3.5 and 3.5.
- Adding 12 shifts this range to between [tex]\( 12 - 3.5 = 8.5 \)[/tex] and [tex]\( 12 + 3.5 = 15.5 \)[/tex].
4. Values for Specific X:
- To analyze the function at various points, consider [tex]\( x \)[/tex] values between 0 and 365, repeating every 365 units.
### Values of [tex]\( y \)[/tex] for a range of [tex]\( x \)[/tex] :
Here are selected values for the function [tex]\( y = 3.5 \sin \left(\frac{2 \pi}{365}(x-90)\right) + 12 \)[/tex] calculated between [tex]\( x = 0 \)[/tex] and [tex]\( x = 365 \)[/tex]:
[tex]\[ \begin{array}{c|c} x & y \\ \hline 0 & 8.50081024 \\ 90 & 12.0 \\ 182.5 & 15.50023694 \\ 273.75 & 12.0 \\ 365 & 8.50081024 \\ \end{array} \][/tex]
### Brief Conclusion:
- At [tex]\( x = 0 \)[/tex], the function starts at approximately [tex]\( y = 8.50081024 \)[/tex].
- At [tex]\( x = 90 \)[/tex], the function reaches its central value [tex]\( y = 12.0 \)[/tex].
- At [tex]\( x = 182.5 \)[/tex], the function reaches its maximum value [tex]\( y = 15.50023694 \)[/tex].
- At [tex]\( x = 273.75 \)[/tex], the function returns to the central value [tex]\( y = 12.0 \)[/tex].
- At [tex]\( x = 365 \)[/tex], the function completes one full cycle and returns to [tex]\( y = 8.50081024 \)[/tex].
This detailed analysis shows the behavior of the sinusoidal function over one period, illustrating its periodic, oscillating nature.
### Step-by-Step Analysis:
1. Understanding the Function:
- The given function is [tex]\( y = 3.5 \sin \left(\frac{2 \pi}{365}(x-90)\right) + 12 \)[/tex].
- This is a sinusoidal function, which means it represents a wave-like pattern.
2. Amplitude, Period, Phase Shift, and Vertical Shift:
- Amplitude: The coefficient of the sine function is 3.5, so the amplitude is 3.5. This means the function oscillates 3.5 units above and below the central value.
- Period: The period of the sine function is given by the coefficient of [tex]\( x \)[/tex] inside the sine function. Here, the period is [tex]\( P = \frac{2\pi}{\frac{2\pi}{365}} = 365 \)[/tex]. This means the function completes one full cycle every 365 units.
- Phase Shift: The term [tex]\( (x-90) \)[/tex] indicates a phase shift. The function is shifted to the right by 90 units.
- Vertical Shift: The +12 outside the sine function indicates that the entire function is shifted up by 12 units.
3. Range of Values:
- The sine function varies between -1 and 1.
- Therefore, [tex]\( 3.5 \sin \left(\frac{2 \pi}{365}(x-90)\right) \)[/tex] varies between -3.5 and 3.5.
- Adding 12 shifts this range to between [tex]\( 12 - 3.5 = 8.5 \)[/tex] and [tex]\( 12 + 3.5 = 15.5 \)[/tex].
4. Values for Specific X:
- To analyze the function at various points, consider [tex]\( x \)[/tex] values between 0 and 365, repeating every 365 units.
### Values of [tex]\( y \)[/tex] for a range of [tex]\( x \)[/tex] :
Here are selected values for the function [tex]\( y = 3.5 \sin \left(\frac{2 \pi}{365}(x-90)\right) + 12 \)[/tex] calculated between [tex]\( x = 0 \)[/tex] and [tex]\( x = 365 \)[/tex]:
[tex]\[ \begin{array}{c|c} x & y \\ \hline 0 & 8.50081024 \\ 90 & 12.0 \\ 182.5 & 15.50023694 \\ 273.75 & 12.0 \\ 365 & 8.50081024 \\ \end{array} \][/tex]
### Brief Conclusion:
- At [tex]\( x = 0 \)[/tex], the function starts at approximately [tex]\( y = 8.50081024 \)[/tex].
- At [tex]\( x = 90 \)[/tex], the function reaches its central value [tex]\( y = 12.0 \)[/tex].
- At [tex]\( x = 182.5 \)[/tex], the function reaches its maximum value [tex]\( y = 15.50023694 \)[/tex].
- At [tex]\( x = 273.75 \)[/tex], the function returns to the central value [tex]\( y = 12.0 \)[/tex].
- At [tex]\( x = 365 \)[/tex], the function completes one full cycle and returns to [tex]\( y = 8.50081024 \)[/tex].
This detailed analysis shows the behavior of the sinusoidal function over one period, illustrating its periodic, oscillating nature.
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
