Use the sinusoidal regression capability of your calculator to find a sinusoidal model for this data. Round you coefficients to the nearest tenth. 3 6 7 10 12 39 67 78 80 72 80 72 Y 51 y = [ ? ) sin(( )x+ D + []
The Correct Answer and Explanation is :
To find a sinusoidal model for the data provided, we’ll use a general form of the sinusoidal function, which is:
[ y = A \sin(Bx + C) + D ]
Here, ( A ) represents the amplitude, ( B ) the frequency, ( C ) the phase shift, and ( D ) the vertical shift. The provided data seems to only include ( y )-values (51 and another series following). If these are ( y )-values, we’ll assume ( x )-values correspond to each index incrementally (1, 2, 3, etc.). Let’s estimate the model parameters based on the data.
- Amplitude (A): This is half the difference between the maximum and minimum values. Observing the peaks (around 80) and troughs (around 6), we estimate:
[ A \approx \frac{80 – 6}{2} = 37 ] - Vertical Shift (D): This is the average of the maximum and minimum values.
[ D \approx \frac{80 + 6}{2} = 43 ] - Frequency (B): We need to determine how often the data completes one full cycle (peak to peak or trough to trough). It seems one cycle is approximately completed between indexes 3 and 12. The period (( T )) can be estimated as ( T \approx 9 ) (from index 3 to index 12). Thus, ( B ) can be calculated as ( B = \frac{2\pi}{T} \approx \frac{2\pi}{9} ).
- Phase Shift (C): This parameter shifts the wave left or right. Without more specific information, we will assume ( C \approx 0 ) for simplicity.
Combining these, the sinusoidal model can be approximated as:
[ y \approx 37 \sin\left(\frac{2\pi}{9}x\right) + 43 ]
This model suggests that the amplitude of oscillation around the vertical shift is 37 units, with a period of about 9 units along the x-axis, and a mean value around 43. Adjustments may be needed for a more accurate fit, particularly in adjusting the phase shift or recalculating parameters with precise data points and fitting techniques such as least squares.
I’ll generate an image to visualize this sinusoidal model against hypothetical ( x )-values from 1 to 12 to illustrate how the curve might look.
Here’s a graph illustrating the sinusoidal model based on the parameters we estimated. This visualization helps in understanding how the function behaves across the given range. If the fit isn’t perfect, further adjustments to parameters like the phase shift or more precise measurements might be needed.
