Module 4  Parametric Equations, Trigonometric and Inverse Trigonometric Functions  
Introduction  Lesson 1  Lesson 2  Lesson 3  Lesson 4  SelfTest  
Lesson 4.3: Sine Regression  
In the previous lesson you started with an equation and found the corresponding graph by looking at the transformations of the basic graph. A more realistic situation might be to start with a graph of realworld data and then find an equation to fit the data. This can be done on your TI83 using regression. If the data appear to be sinusoidal, then you can use sine regression. In this lesson you will download data sets produced by a turning fork to your calculator and explore finding the equation that fits the data. Tuning Fork Data A vibrating object that disturbs the surrounding air and causes the air molecules to vibrate produces sound. These vibrations cause a periodic change in air pressure that travels through the air much like ripples on a body of water. The air pressure waves are more commonly called sound waves. Although most sounds are a combination of many different sounds, the sound from a turning fork is a single tone that can be described mathematically using a transformed sine function. A Texas Instruments CBL™ and microphone probe were used to measure a sound wave from a tuning fork. The information in the data will allow you to determine the specific note that was sounded. Downloading the Data to Your Computer
Transferring the Data to the TI83 Click here to get information about how to obtain the needed cable and to review the procedure to transfer the lists from your computer to your calculator.
Plotting the Data
Sine Regression Because the graph appears to be a transformed sine function, perform sine regression and store the regression equation.
After a minute or two a sine equation that fits the data will be displayed on the Home screen and stored in Y1. The Forms Notice that the regression equation is slightly different from the form y = Asin[B(x + C] + D described in Lesson 4.2. A, B and D are the same as the values in the regression equation, but the value of C in the transformed equation is found by dividing the value of C in the regression equation by the value of B. B must be factored out to determine the horizontal shift of the transformed graph. The transformed regression equation has the characteristics listed below.
Period and Frequency The period of this sine wave is , which indicates that it takes approximately 0.004 seconds per cycle. The frequency, which is the reciprocal of the period, is cycles per second. The Tuning Fork's Note A frequency of 256 cycles per second is the frequency of the note middle C, so the tuning fork must be a "C". 4.3.1 The period is the distance from one peak to the next peak. If another tuning fork were used with a higher frequency than the tuning fork in this lesson, how would its graph compare with the graph above? Write your response then click here for the answer. 

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