He Use the sinusoidal regression capability of your calculator to find a sinusoidal model for this data Round you coefficients to the nearest tenth: y = [ ? sin ( [ Jx + 1) + [ Enter
The Correct Answer and Explanation is:
To find the sinusoidal regression model for the data y=Asin(Bx+C)+Dy = A \sin(Bx + C) + Dy=Asin(Bx+C)+D, using a calculator, we would typically follow these steps. I can guide you through the process and explain how it works. Here’s a breakdown:
Steps for Sinusoidal Regression:
- Enter Data Points: First, input your data points (x, y) into the calculator. Typically, you will input the x-values (independent variable) in one column and the corresponding y-values (dependent variable) in another.
- Select Sinusoidal Regression: On your calculator (depending on the model, this will vary), you need to select the sinusoidal regression option. For example, on a TI-84:
- Go to STAT.
- Choose CALC.
- Select Sinusoidal Regression (SinReg). This might be listed as a specific function or under the “More” options in some models.
- Run Regression: After selecting the regression function, you can tell the calculator to calculate the sinusoidal regression using the data you entered. The calculator will process the data and return the sinusoidal equation in the form: y=Asin(Bx+C)+Dy = A \sin(Bx + C) + Dy=Asin(Bx+C)+D
- Interpret the Coefficients:
- A: This represents the amplitude of the sine wave. It controls the vertical stretch or shrink.
- B: This affects the period of the sine wave. The period is calculated by P=2πBP = \frac{2\pi}{B}P=B2π.
- C: This is the phase shift, which shifts the sine wave left or right along the x-axis.
- D: This is the vertical shift, moving the sine wave up or down.
Example:
For example, if the regression yields:y=5sin(2x+1)+3y = 5 \sin(2x + 1) + 3y=5sin(2x+1)+3
Then, interpreting the coefficients:
- A = 5: Amplitude is 5, meaning the wave peaks at y=5y = 5y=5 and troughs at y=−5y = -5y=−5.
- B = 2: The period is P=2π2=πP = \frac{2\pi}{2} = \piP=22π=π, meaning the wave repeats every π\piπ units along the x-axis.
- C = 1: The phase shift is −CB=−12\frac{-C}{B} = \frac{-1}{2}B−C=2−1, shifting the wave to the left by 12\frac{1}{2}21.
- D = 3: The vertical shift is 3, meaning the sine wave oscillates around y=3y = 3y=3 instead of 0.
Rounding Coefficients:
If you are asked to round to the nearest tenth, then you would round each coefficient appropriately. For instance:
- If A = 5.678, you would round it to 5.7.
- Similarly, round B, C, and D to the nearest tenth.
This process should give you a complete sinusoidal model for the given data. If you’d like, you can provide me with the data, and I can guide you further!
