Lesson 3 Using the Sine Function to Model Periodic Graphs

Transcription

1 Lesson 3 Using the Sine Function to Model Periodic Graphs Objectives After completing this lesson you should 1. Know that the sine function is one of a family of functions which is used to model periodic behavior. 2. Understand how changing the values of A, B, C, and D in y = A sin(b(x + c)) + D changes the appearance of the sine curve. 3. Know that the sine function can be used to model sound waves. Warm-Up Questions 1. What kinds of graphs do you remember studying in high school? 2. Without doing any calculating and without using the graphing calculator, make your best prediction of what the graph of y=3x+2 would look like. 3. Follow your teachers instructions to produce a graph of y=3x+2 on your graphing calculator. Was your prediction correct? 4. Without doing any calculating and without using the graphing calculator, make your best prediction of what the graph of y=x 2 would look like. 5. Follow your teachers instructions to produce a graph of y=x 2 on your graphing calculator. Was your prediction correct? 6. Have you ever studied the function y=sin x? If so, what do you remember about it? (We will study this function in today s lesson.) Activity Equipment: One TI graphing calculator with link cable for each pair or group and for the instructor. One TI presenter for projecting graphs.

2 Part I 1. Set your graphing calculator window to Xmin = Xmax = 6.15 Xscl = 1 Ymin = -4 Ymax = 4 Yscl = 1 2. Store the following values in your calculator: 1 STO> A 1 STO> B 0 STO> C 0 STO> D 2. Press the Y= button and enter this formula: A sin( B (X + C)) + D. 3. Press the Graph button and describe the resulting graph. a. What is the formula? 1 sin (1 (X + 0)) + 0 b. Where is the graph centered (vertically)? c. Approximately what is its period? d. What is its amplitude (i.e., how high does it go above its vertical center)? 4. Now we will change A, B, C, and D one at a time to see how changes to each affect the graph. a. Press Clear and then change D to 2 by pressing 2 STO>D. Press Graph. b. What is the new formula? c. Describe the change in the graph. d. Predict what will happen if you change D to -3. Then make the change and see if your prediction was correct. e. Change D back to 0. ( Clear then 0 STO> D ) 5. Experiment with different values of A. Use whole numbers and fractions. Keep a record of each value of A that you try. In each case, write down what the formula is and sketch the resulting graph. Write a summary of the effect of A on the graph. Change A back to 1 before going on to the next step. 6. Experiment with different values of B. Use whole numbers and fractions. Keep a record of each value of B that you try. In each case, write down what the formula is and sketch the resulting graph. Write a

3 summary of the effect of B on the graph. Change B back to 1 before going on to the next step. 7. Experiment with different values of C. Use both positive and negative numbers. (If you don t know how to enter a negative number, your teacher will show you.) Keep a record of each value of C that you try. In each case, write down what the formula is and sketch the resulting graph. Write a summary of the effect of C on the graph. Change C back to 0 before going on to the next step. 8. Without using the graphing calculator, try to make a sketch of the graph of this function Y = 2 sin (1(X-1)) + 3. Test your prediction by graphing the function with the calculator. Set C and D back to 0 and A and B back to 1 before going on to Part 2. Part 2 In Part 2 you will use what you have learned about the sine function to write a formula for the periodic graph that you generated by blowing across the bottle in Lesson Why is the graph of the sound data not on the same screen as the graph from Part 1? (Hint: You can use press Zoom and select ZoomStat to see the graph of the sound data. Look at the window dimensions and compare them to the window dimensions that you used in Part 1.) Can you modify the window settings so that both graphs are seen on the same screen? 2. Press Zoom and ZoomStat to return to an appropriate screen for the sound data. On approximately what number are the data vertically centered? Store this number as D. Now can you see both graphs on the same screen? 3. Find a value of B which makes the formula graph have approximately the same frequency as the sound data. (Hint: Multiply the frequency of the sound data by 2 π or 6.28 and try this value as B.) 4. Change the value stored in A to adjust the amplitude of the formula graph to match the amplitude of the sound data as closely as possible. 5. Determine the amount that the formula graph needs to be shifted horizontally to match the sound data and enter this value as C. (Hint: The Trace button may help with this step.)

4 6. Make any other small adjustments to A, B, C, and/or D that you think will improve the fit of your graph and write your new formula below (with numbers rather than letters). Homework Without using a graphing calculator, sketch the graphs of each of these functions. The graph of Y = 1 sin (1 (X + 0 )) + 0 is included for reference. 1. Y = 3 sin (1 (X + 0)) + 0

5 2. Y = 1 sin (1 (X + 0 )) Y = 1 sin (1 (X + 2 )) + 0

6 4. Y = 1 sin (2 (X + 0 )) Y = 2 sin (1 (X - 1 )) + 1