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Trigonometric Models WS #3

Shifting Trigonometric Graphs

Review:

1. When graphing y = A sin(Bx) or y = A cos(Bx), the ‘A’ and ‘B’ values tell you something about the amplitude and period of the graph. Describe what…

(a) …the ‘A’ value tells you:

(b) …the ‘B’ value tells you:

(c) Explain how you know whether to use “sine” or “cosine” for the equation when you are given the graph. Draw a picture as part of your explanation.

Shifting the Sine or Cosine Graph: Sine and cosine graphs can also be shifted up and down or left and right. For each of the following, describe how the graph has been shifted:

2. (a) Is this graph better modeled with sine or cosine? Why?

b) How has this graph been shifted when compared to the normal y = sin x (or cos x)? Say what direction it has been shifted and by how many units.

3. (a) Is this graph better modeled with sine or cosine? Why?

b) How has this graph been shifted when compared to the normal y = sin x (or cos x)? Say what direction it has been shifted and by how many units.

4. Suppose the graph shown is compared to the graph of y = sinx. Since a normal sine graph keeps repeating the same values over and over, this picture could be viewed as a sine graph that has been shifted EITHER LEFT OR RIGHT!

(a) If this graph is viewed as a sine graph that got shifted to the RIGHT, approximately how many units did it shift?

(b) If this graph is viewed as a sine graph that got shifted to the LEFT, approximately how many units did it shift?

(c) Suppose instead we view this graph as y = cosx that got shifted left or right. If this graph is viewed as a cosine graph that got shifted to the RIGHT, approximately how many units did it shift?

(d) If this graph is viewed as a cosine graph that got shifted to the LEFT, approximately how many units did it shift?

Use your graphing calculator in RADIAN mode to complete each of the following. Press the ZOOM key and choose 7:ZTrig. Sketch the resulting graph on the axes provided.

5. Sketching y = sin(x) + D. Graph each of the following on your graphing calculator and sketch the graph on the axes provided. Next to each graph say whether it got shifted UP, DOWN, LEFT, or RIGHT. Say how many units it got shifted!!

(a) y = sin(x) (b) y = sin(x) + 2

[pic] [pic]

(c) y = sin(x) – 3 (d) y = sin(x) + 0.5

[pic] [pic]

6. Sketching y = cos(x) + D. Graph each of the following on your graphing calculator and sketch the graph on the axes provided. Use the 7:ZTrig window (under the ZOOM menu) as you did in the previous question. Next to each graph say whether it got shifted UP, DOWN, LEFT, or RIGHT. Say how many units it got shifted!!

(a) y = cos(x) (b) y = cos(x) + 3

[pic] [pic]

(c) y = cos(x) – 1 (d) y = cos(x) + 1.5

[pic] [pic]

7. Look at your results for questions 5 and 6. What effect did changing the D value have on the graphs of y = sin(x) + D and y = cos(x) + D? Explain.

8. Did changing the D value effect the amplitude or period of the graph? How do you know?

9. Sketching y = sin(x + C). In order to see the shifts better when the ‘C’ value is changed, press WINDOW on your graphing calculator and set the values to those shown at right. Graph each of the following on your graphing calculator and sketch the graph on the axes provided. Next to each graph say whether it got shifted UP, DOWN, LEFT, or RIGHT. Say how many units it got shifted!!

(a) y = sin(x) (b) y = sin(x + 1)

[pic] [pic]

(c) y = sin(x – 2) (d) y = sin(x + π)

[pic] [pic]

10. Sketching y = cos(x + C). In order to see the shifts better when the ‘C’ value is changed, keep the WINDOW as shown at in the question above. Graph each of the following on your graphing calculator and sketch the graph on the axes provided. Next to each graph say whether it got shifted UP, DOWN, LEFT, or RIGHT. Say how many units it got shifted!!

(a) y = cos(x) (b) y = cos(x – 3)

[pic] [pic]

(c) y = cos(x + 2) (d) y = cos(x – π)

[pic] [pic]

11. Look at your results for questions 9 and 10. What effect did changing the B value have on the graphs of y = sin(x + C) and y = cos(x + C)? Explain. How does the B value relate to the period of the graph? Explain.

For question 9 and 10, the shifts left and right should have been very straightforward when compared to the actual ‘C’ value. However, they are NOT so straightforward when the ‘B’ value in y = A sin(Bx + C) isn’t equal to 1!! Complete the following to see what happens:

12. Sketching y = sin(Bx + C). For this exercise, press WINDOW on your graphing calculator and set the values to those shown at right. Graph each of the following on your graphing calculator and sketch the graph on the axes provided. Next to each graph say whether it got shifted UP, DOWN, LEFT, or RIGHT when compared to the graph of y = sin(2x). Say how many units it got shifted!!

(a) y = sin(2x) (b) y = sin(2x – 1)

[pic] [pic]

(c) y = sin(2x – 4) (d) y = sin(2x + 2)

[pic] [pic]

13. Sketching y = sin(Bx + C). For this exercise, set the WINDOW to the values shown at right. This time the ‘B’ value will be 3 for each graph. Next to each graph say how it got shifted when compared to the graph of y = sin(3x).

(a) y = sin(3x) (b) y = sin(3x – 1)

[pic] [pic]

(c) y = sin(3x + 3) (d) y = sin(3x – 2)

[pic] [pic]

14. It turns out that the HORIZONTAL shift (left and right) of a sine graph is related to BOTH the ‘B’ and the ‘C’ value. Look over your results for questions 12 and 13 and try to decide how. Write your theory below.

Many real-world situations can be modeled with a sine or cosine graph. In the situations below, the sine or cosine graph will be SHIFTED up or down. Draw a sample graph that might match each description. Identify the period, frequency, amplitude, and vertical shift of each.

14. The mass in the picture at right is pulled to the right from point B to point C and released. The block then oscillates (moves back and forth) between positions A and C. Consider point B to be zero position assume the block is moving towards point C after time 0. The block takes 8 seconds to bounce through ONE CYCLE.

(a) Sketch a graph of the position vs. time of the block as it bounces. Label axes with names, numbers, and units. Notice the block STARTS bouncing (time 0) from position B.

(b) Find the following. Put units on your answer:

Period: ________________

Frequency: ________________

Amplitude: _________________

Vertical Shift: ________________

(c) Would this graph be modeled better with a sine or with a cosine formula? Write a possible formula.

15. The object pictured below left is bouncing such that the highest it reaches is position A and the lowest is position C. The block takes 4 seconds to complete one whole bounce.

(a) Sketch a graph of the height vs. time of the block as it bounces. Label axes with names, numbers, and units. Assume the block STARTS bouncing (time 0) from height A.

(b) Find the following. Put units on your answer:

Period: ________________

Frequency: ________________

Amplitude: _________________

Vertical Shift: ________________

(c) Would this graph be modeled better with a sine or with a cosine formula? Write a possible formula.

16. A bungee jumper leaves the platform and then bounces up and down between position A and position C as shown in the diagram. It takes 8 seconds after she jumps for her to return to position A.

(a) Sketch a graph of the height vs. time of the bungee jumper as she bounces. Label axes with names, numbers, and units. Notice the jumper STARTS her trip (time 0) from position A.

(b) Find the following. Put units on your answer:

Period: ________________

Frequency: ________________

Amplitude: _________________

Vertical Shift: ________________

(c) Would this graph be modeled better with a sine or with a cosine formula? Write a possible formula.

17. A boy swings slowly back and forth on a swing between position A and position C as shown on the picture at right. It takes him 2 seconds to swing back and forth once.

(a) Sketch a graph of the position vs. time of the boy as he swings. Label axes with names, numbers, and units. Suppose the boy STARTS swinging at position B and is heading towards position A on his first swing.

(b) Find the following. Put units on your answer:

Period: ________________

Frequency: ________________

Amplitude: _________________

Vertical Shift: ________________

(c) Would this graph be modeled better with a sine or with a cosine formula? Write a possible formula.

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