A New Twist on Trigonometric Graphs
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Transcript of A New Twist on Trigonometric Graphs
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A New Twist onTrigonometric Graphs
Mark Turner
Cuesta College
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ISSUE
How can we simplify the process of graphing the six
trigonometric functions for our students?
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My Solution
• Use consistent definitions of the six basic cycles
• Construct a simple frame for one cycle
• Plot a few anchor points
• Sketch the graph and extend as necessary
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First, we will use some interactive java applications to derive the basic cycles for the
circular functions.
I created these applets using
which is free and very user friendly.
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THE BASIC CYCLES
y = sin(x) y = cos(x)
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THE BASIC CYCLES
y = csc(x) y = sec(x)
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THE BASIC CYCLES
y = tan(x) y = cot(x)
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What are the advantages of using
these definitions?
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1. CONSISTENCY
•All six basic cycles begin at zero
•By knowing the period, students know the cycle interval
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2. LESS CONFUSION
•Because the graphs all look different, it is not as easy to confuse one for the other
•The phase shift is easier to identify
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What are some possible disadvantages of using
these definitions?
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•Visualizing the inverse tangent graph
•Most textbooks do not use these definitions
(though this does not appear to be an issue with the sine function)
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Step-by-StepProcess
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1. Find the vertical translation and draw a horizontal line (lightly) at this value. Pretend this is now the x-axis.
2. Find the “amplitude” A. Lightly draw horizontal lines A units above and below the translated x-axis. This gives us the upper and lower sides of the frame.
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3. Solve the compound inequality
or for the variable x.
0 argument 2
0 argument
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After solving:
• the left value is the phase shift, and indicates where a cycle begins
• the right value indicates where a cycle ends
• the difference of the two values is the period
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4. Find the period and divide it by four. Set the scale on your x-axis so that this value is equal to some whole number multiple of squares (like 2).
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5. Lightly draw vertical lines where the cycle begins and ends. These lines form the left and right sides of the frame.
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6. Subdivide the frame (lengthwise) into four equal sections. To label the three intermediary points, start at the left edge of the frame and add one-fourth the period. Repeat two times.
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7. Plot the anchor points and sketch any asymptotes (don’t forget to check for a reflection and adjust the anchor points accordingly).
8. Within the frame, sketch the graph of a complete cycle through the anchor points.
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9. Extend the graph by duplicating the cycle as necessary
10. To verify on a graphing calculator, use the frame to help set up an appropriate window.
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EXAMPLES
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Commentsor
Questions?