Module 4 - Parametric Equations, Trigonometric and Inverse Trigonometric Functions

Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Lesson 4 | Self-Test

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 real-world data and then find an equation to fit the data. This can be done on your TI-83 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.

• The data are at the two links L1 and L2. Right-click on each link and choose to save the link as a file on your computer. L1, L2
• Save the files on your local hard disk on the desktop or in a folder that you can access later.

Transferring the Data to the TI-83

• Send the lists L1 and L2 from your computer to your TI-83.
• Display the Stat List editor to confirm that the lists were stored.

Plotting the Data

• Define a scatter plot with L1 in the X-list and L2 in the Y-list.
• Select dots as the Mark type.
• Select ZoomStat from the ZOOM menu and display the graph.

Sine Regression

Because the graph appears to be a transformed sine function, perform sine regression and store the regression equation.

• Open the STAT CALC menu by pressing .
• Highlight C:SinReg by pressing repeatedly or by pressing once.
• Paste the SinReg command to the Home screen by pressing
• Complete the sine regression command "SinReg L1,L2,Y1".
Recall that L1 and L2 are above and , and that Y1 is found by pressing .
• Execute the command by pressing .

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.

 A 0.202 Compress vertically by a factor of 0.202. B 1609 The period of the transformed graph is Horizontal shift right 0.00160 units. D 0.014 Vertical shift up by 0.0141 units.

• Press to see the graph of the regression equation along with the scatter plot.
• The graph and scatter plot in the ZoomStat window

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.

< Back | Next >