1. A New Twist on Trigonometric Graphs
Mark Turner
Cuesta College

2. ISSUE
How can we simplify the process of graphing the six trigonometric functions for our students?

3. 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

4. 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.

5. GeoGebra Applets Internet Link Unit Circle: Sine and Cosine
Unit Circle: Tangent
Unit Circle: Secant
Unit Circle: Coscant

6. THE BASIC CYCLES
y = sin(x)
y = cos(x)

7. THE BASIC CYCLES
y = csc(x)
y = sec(x)

8. THE BASIC CYCLES
y = tan(x)
y = cot(x)

9. What are the advantages of using these definitions?

10. 1. CONSISTENCY All six basic cycles begin at zero
By knowing the period, students know the cycle interval

11. 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

12. What are some possible disadvantages of using these definitions?

13. 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

14. Step-by-Step Process

15. 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.

16. Solve the compound inequalityor
for the variable x.

17. After solving:
the left value is the horizontal shift, and indicates where a cycle begins
the right value indicates where a cycle ends
the difference of the two values is the period

18. Find the period and divide it by four
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).

19. Lightly draw vertical lines where the cycle begins and ends
Lightly draw vertical lines where the cycle begins and ends. These lines form the left and right sides of the frame.

20. Subdivide the frame (lengthwise) into four equal sections
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.

21. Plot the anchor points and sketch any asymptotes (don’t forget to check for a reflection and adjust the anchor points accordingly).
Within the frame, sketch the graph of a complete cycle through the anchor points.

22. 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.

23. EXAMPLES

24. Comments or Questions?