Finding the Mathematical Model of a Sinusoid

Finding the Mathematical Model of a Sinusoid

Y = K + A sin(D(X-H))

Using SNDDEMO data recorded as volts vs time.

© 2002 Jerel L. Welker, Lincoln High SchoolPermission is granted to copy and distributewith copyright notation attached.


Step 1:

Find a local maximum and minimum point near the start of the data using the TRACE function.


Step 2:Find a local maximum and minimum point near the end of the data. Record the values.


Find the vertical shift (k) value.

Step 3:

The vertical shift is the midline of the data set. It can be approximated by averaging the local maximum and minimum values.

The midline value (k) is 2.6993 volts.

Graphing the midline value (y = 2.70) on the current data set.


Find the amplitude (a) value.

Step 4:

The amplitude is the maximal change in the data from the midline.

To find the amplitude:

Subtract the midline from the local maximum value.

The amplitude is 0.016 volts.


Find the period and frequency.

Step 5:

•Find two local max/min values.

•Find the difference of the x-coordinates. (.2025 - .06) = .1425 sec•Count the number of cycles from the starting to ending point. 2 cycles•Divide the difference by the number of cycles to determine the period. 0.07125 seconds/cycle or a period of 0.07125•Find the reciprocal of the period for the frequency. 14.04 cps


Find the horizontal stretch/shrink (d) value.

Step 6:

The d value in the model is the number of times the period repeats in 2 radians.

The horizontal stretch/shrink (d) value is 88.1851 radians / second.


Find the horizontal shift (h) value.

Step 7:

A period of sine begins when the graph crossing the midline with increasing value.

The red circle is the beginning of one period of sine. Find the x-value of this point.

Average the x-values of the adjacent minimum and maximum values.

The horizontal shift is .04125 seconds.


Graph the model y = k + a sin(d(x - h)).

Step 8:

Enter the values found into the function grapher:

k = 2.6993 v a = 0.016 v

d = 88.185 rad/sec h = 0.04125 sec


Using the TI-83 SinReg function.Press the STAT key. Use the arrow keys to select CALC and SinReg. Press ENTER.

Enter the list for the y-values (L2) followed by a comma.

Enter the list for the x-values (L1) followed by a comma.

Press VARS followed by Y-VARS and Function...

Enter the function to store the regression. (Y2)


Using the TI-83 SinReg function.The screen should now appear similar to:

Press ENTER and the calculator will attempt to determine the regression equation in the form of y = a*sin(bx+c)+d.

The resulting model is:

The model is stored in the function grapher Y2:


Using the TI-83 SinReg function.The graph will appear similar to:

The two models are:

V = 0.016v * sin(89.590 rad/sec * T sec + 2.428 sec) + 2.700 v

or

V = 2.699v + 0.016v * sin(88.185 rad/sec (T sec - 0.041 sec)

Finding the Mathematical Model of a Sinusoid

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