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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GeoGebra Applets
Internet Link
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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

(though this does not appear to be an issue with
the sine function)

• Most textbooks do not use these definitions

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Step-by-StepProcess
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Find the vertical translation and draw a
horizontal line (lightly) at this value. Pretend
this is now the x-axis.

• 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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• Solve the compound inequality
• or
• for the variable x.

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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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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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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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• 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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Within the frame, sketch the graph of a complete
cycle through the anchor points.

• Plot the anchor points and sketch any asymptotes
  (dont forget to check for a reflection and
  adjust the anchor points accordingly).

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• Extend the graph by duplicating the cycle as
  necessary

• 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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CommentsorQuestions?