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Name: ________________________________ Date: _______________ Hr: _______

Trigonometric Models WS #1

Period, Amplitude, Frequency

The three most important characteristics of a periodic graph are its period, its amplitude, and its frequency. See the picture and explanation below:

[pic]

1. Find the value for the period, the amplitude, and the frequency of the velocity vs. time graph shown above.

(a) Period: The amount of time (along the x-axis) it takes to complete ONE cycle. On the above graph, there are about __________seconds in one cycle.

(b) Frequency: The amount of cycles in one unit of time. On the above graph, there are about __________ cycles in one second. Another way to get this value is to INVERT the period. If the period from part (a) is inverted, there are ____________ cycles per second. Another name for the unit of cycles per second is a ______________.

(c) Amplitude: The distance (along the y-axis) between the MIDDLE of the wave and the TOP of the wave. Notice this is the same as HALF the distance between the high point and low point on the graph. On the above graph, the amplitude is about ________________ in units of __________.

(d) Would the graph shown above be better modeled with a sine function or a cosine function? Explain why (think about what you learned in section 6-3).

2. Use your graphing calculator in RADIAN mode to complete each of the following.

(a) Press the Y= key and enter Y1 = sin(x). Make sure all other functions are empty.

(b) Press the ZOOM key and choose 7:ZTrig. Sketch the resulting graph on the axes provided. Then fill in the following information:

Period: _________________

Frequency: ______________

Amplitude: ______________

c) Press the WINDOW key to see the settings for this ZTrig window. Fill in the values for each of the following AS INCREMENTS of π!! One of them have been started for you. (Don’t worry about Xres).

Xmin = _6.2 = 2π_ Ymin = __________

Xmax = __________ Ymax = __________

Xscl = ___________ Yscl = ___________

(d) NOW label each tick mark on your GRAPH from part (b). Along the x-axis it should be labeled with increments of π!!

3. Sketching y = A sin(x). Graph each of the following on your graphing calculator and sketch the graph on the axes provided. Next to each graph fill in the PERIOD and AMPLITUDE!!

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

[pic] [pic]

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

[pic] [pic]

4. Sketching y = A cos(x). Graph each of the following on your graphing calculator and sketch the graph on the axes provided. Next to each graph fill in the PERIOD and AMPLITUDE!!

(a) y = cos(x) (b) y = -2cos(x)

[pic] [pic]

(c) y = 4cos(x) (d) y = -1.5cos(x)

[pic] [pic]

5. Look at your results for questions 3 and 4. What effect did changing the A value have on the graphs of y = A sin(x) and y = A cos(x)? Explain. How does the A value relate to the amplitude of the graph? Explain.

6. Sketching y = sin(Bx). Graph each of the following on your graphing calculator and sketch the graph on the axes provided. Next to each graph fill in the PERIOD and AMPLITUDE!!

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

[pic] [pic]

(c) y = sin(0.5x) (d) y = sin(-x)

7. Sketching y = cos(Bx). Graph each of the following on your graphing calculator and sketch the graph on the axes provided. Next to each graph fill in the PERIOD and AMPLITUDE!!

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

[pic] [pic]

(c) y = cos(0.25x) (d) y = cos(-x)

[pic] [pic]

8. Look at your results for questions 6 and 7. What effect did changing the B value have on the graphs of y = sin(Bx) and y = cos(Bx)? Explain. How does the B value relate to the period of the graph? Explain.

|Summary: When you changed A and B in the graphs of y = A sin(Bx) and y = A cos(Bx), hopefully this is what you found: |

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|▪Changing A stretches or shrinks the graph vertically. A>1 stretches the graph, A1 shrinks the graph, B ................
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