1. Graphs of the Circular Functions
MAC 1114
Module 4
Graphs of the Circular Functions
Rev.S08

2. Learning Objectives
Upon completing this module, you should be able to:
Recognize periodic functions.
Determine the amplitude and period, when given the equation of a periodic function.
Find the phase shift and vertical shift, when given the equation of a periodic function.
Graph sine and cosine functions.
Graph cosecant and secant functions.
Graph tangent and cotangent functions.
Interpret a trigonometric model.
Click link to download other modules.
Rev.S08

3. Graphs of the Circular Functions
There are three major topics in this module:
- Graphs of the Sine and Cosine Functions
- Translations of the Graphs of the Sine and Cosine Functions
- Graphs of the Other Circular Functions
Click link to download other modules.
Rev.S08

4. Introduction to Periodic Function
A periodic function is a function f such that
f(x) = f(x + np),
for every real number x in the domain of f, every integer n, and some positive real number p. The smallest possible positive value of p is the period of the function.
Click link to download other modules.
Rev.S08

5. Example of a Periodic Function
Click link to download other modules.
Rev.S08

6. Example of Another Periodic Function
Click link to download other modules.
Rev.S08

7. What is the Amplitude of a Periodic Function?
The amplitude of a periodic function is half the difference between the maximum and minimum values.
The graph of y = a sin x or y = a cos x, with a ≠ 0, will have the same shape as the graph of y = sin x or y = cos x, respectively, except the range will be [−|a|, |a|]. The amplitude is |a|.
Click link to download other modules.
Rev.S08

8. How to Graph y = 3 sin(x) ? −3 3 3sin x −1 1 sin x π 3π/2 π/2 x−3 3
3sin x −1 1
sin x π
3π/2
π/2 x
Note the difference between sin x and 3sin x. What is the difference?
Click link to download other modules.
Rev.S08

9. How to Graph y = sin(2x)?
The period is 2π/2 = π. The graph will complete one period over the interval [0, π].
The endpoints are 0 and π, the three middle points are:
Plot points and join in a smooth curve.
Click link to download other modules.
Rev.S08

10. How to Graph y = sin(2x)? (Cont.)
Note the difference between sin x and sin 2x. What is the difference?
Click link to download other modules.
Rev.S08

11. Period of a Periodic Function
Based on the previous example, we can generalize the following:
For b > 0, the graph of y = sin bx will resemble that of y = sin x, but with period 2π/b.
The graph of y = cos bx will resemble that of y = cos x, with period 2π/b.
Click link to download other modules.
Rev.S08

12. How to Graph y = cos (2x/3) over one period?
The period is 3π.
Divide the interval into four equal parts.
Obtain key points for one period. 1 −1
cos 2x/3 2π
3π/2 π
π/2
2x/3 3π
9π/4
3π/4 x
Click link to download other modules.
Rev.S08

13. How to Graph y = cos(2x/3) over one period? (Cont.)
The amplitude is 1.
Join the points and connect with a smooth curve.
Click link to download other modules.
Rev.S08

14. Guidelines for Sketching Graphs of Sine and Cosine Functions
To graph y = a sin bx or y = a cos bx, with b > 0, follow these steps.
Step 1 Find the period, 2π/b. Start with 0 on the x-axis, and lay off a distance of 2π/b.
Step 2 Divide the interval into four equal parts.
Step 3 Evaluate the function for each of the five x-values resulting from Step The points will be maximum points, minimum points, and x-intercepts.
Click link to download other modules.
Rev.S08

15. Guidelines for Sketching Graphs of Sine and Cosine Functions Continued
Step 4 Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude |a|.
Step 5 Draw the graph over additional periods, to the right and to the left, as needed.
Click link to download other modules.
Rev.S08

16. How to Graph y = −2 sin(4x)?
Step 1 Period = 2π/4 = π/2. The function will be graphed over the interval [0, π/2] .
Step 2 Divide the interval into four equal parts.
Step 3 Make a table of values 2 −2
−2 sin 4x −1 1
sin 4x 2π
3π/2 π
π/2 4x
3π/8
π/4
π/8 x
Click link to download other modules.
Rev.S08

17. How to Graph y = −2 sin(4x)? (Cont.)
Plot the points and join them with a sinusoidal curve with amplitude 2.
Click link to download other modules.
Rev.S08

18. What is a Phase Shift?
In trigonometric functions, a horizontal translation is called a phase shift.
In the equation
the graph is shifted π/2 units to the right.
Click link to download other modules.
Rev.S08

19. How to Graph y = sin (x − π/3) by Using Horizontal Translation or Phase Shift?
Find the interval for one period.
Divide the interval into four equal parts.
Click link to download other modules.
Rev.S08

20. How to Graph y = sin (x − π/3) by Using Horizontal Translation or Phase Shift? (Cont.)−1 1
sin (x − π/3) 2π
3π/2 π
π/2
x − π/3
7π/3
11π/6
4π/3
5π/6
π/3 x
Click link to download other modules.
Rev.S08

21. How to Graph y = 3 cos(x + π/4) by Using Horizontal Translation or Phase Shift?
Find the interval.
Divide into four equal parts.
Click link to download other modules.
Rev.S08

22. How to Graph y = 3 cos(x + π/4) by Using Horizontal Translation or Phase Shift?−3
3 cos (x + π/4) 1 −1
cos(x + π/4) 2π
3π/2 π
π/2
x + π/4
7π/4
5π/4
3π/4
π/4
−π/4 x
Click link to download other modules.
Rev.S08

23. How to Graph y = 2 − 2 sin 3x by Using Vertical Translation or Vertical Shift?
The graph is translated 2 units up from the graph y = −2 sin 3x. 2 4
2 − 2 sin 3x −2
−2 sin 3x 2π
3π/2 π
π/2 3x
2π/3
π/3
π/6 x
Click link to download other modules.
Rev.S08

24. How to Graph y = 2 − 2 sin 3x by Using Vertical Translation or Vertical Shift? (Cont.)
Plot the points and connect.
Click link to download other modules.
Rev.S08

25. Further Guidelines for Sketching Graphs of Sine and Cosine Functions
Method 1: Follow these steps.
Step 1 Find an interval whose length is one period 2π/b by solving the three part inequality 0 ≤ b(x − d) ≤ 2π.
Step 2 Divide the interval into four equal parts.
Step 3 Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and points that intersect the line y = c (middle points of the wave.)
Click link to download other modules.
Rev.S08

26. Further Guidelines for Sketching Graphs of Sine and Cosine Functions (Cont.)
Step 4 Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude |a|.
Step 5 Draw the graph over additional periods, to the right and to the left, as needed.
Method 2: First graph the basic circular function. The amplitude of the function is |a|, and the period is 2π/b. Then use translations to graph the desired function. The vertical translation is c units up if c > 0 and |c| units down if c < The horizontal translation (phase shift) is d units to the right if d > 0 and |d| units to the left if d < 0.
Click link to download other modules.
Rev.S08

27. How to Graph y = −1 + 2 sin (4x + π)?
Write the expression in the form c + a sin b(x − d) by rewriting 4x + π as
Step 1
Step 2: Divide the interval.
Step 3 Table
Click link to download other modules.
Rev.S08

28. How to Graph y = −1 + 2 sin (4x + π)?(Cont.)−3 1
−1 + 2sin(4x + π) 2 −2
2 sin 4(x + π/4)
sin 4(x + π/4) 2π
3π/2 π
π/2
4(x + π/4)
3π/8
π/4
π/8
x + π/4
−π/8
−π/4 x
Click link to download other modules.
Rev.S08

29. How to Graph y = −1 + 2 sin (4x + π)? (Cont.)
Steps 4 and 5
Plot the points found in the table and join then with a sinusoidal curve.
Click link to download other modules.
Rev.S08

30. Let’s Take a Look at Other Circular Functions.
Click link to download other modules.
Rev.S08

31. Cosecant Function Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules.
Rev.S08

32. Secant Function Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules.
Rev.S08

33. Guidelines for Sketching Graphs of Cosecant and Secant Functions
To graph y = csc bx or y = sec bx, with b > 0, follow these steps.
Step 1 Graph the corresponding reciprocal function as a guide, using a dashed curve.
y = cos bx
y = a sec bx
y = a sin bx
y = a csc bx
Use as a Guide
To Graph
Click link to download other modules.
Rev.S08

34. Guidelines for Sketching Graphs of Cosecant and Secant Functions Continued
Step 2 Sketch the vertical asymptotes.
- They will have equations of the form x = k, where k is an x-intercept of the graph of the guide function.
Step 3 Sketch the graph of the desired function
by drawing the typical U-shapes branches
between the adjacent asymptotes.
- The branches will be above the graph of the
guide function when the guide function values
are positive and below the graph of the guide
function when the guide function values are
negative.
Click link to download other modules.
Rev.S08

35. How to Graph y = 2 sec(x/2)?
Step 1: Graph the corresponding reciprocal function
y = 2 cos (x/2).
The function has amplitude 2 and one period of the graph lies along the interval that satisfies the inequality
Divide the interval into four equal parts.
Click link to download other modules.
Rev.S08

36. How to Graph y = 2 sec(x/2)? (Cont.)
Step 2: Sketch the vertical asymptotes. These occur at x- values for which the guide function equals 0, such as x = −3π, x = 3π, x = π, x = 3π.
Step 3: Sketch the graph of y = 2 sec x/2 by drawing the typical U-shaped branches, approaching the asymptotes.
Click link to download other modules.
Rev.S08

37. Tangent Function Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules.
Rev.S08

38. Cotangent Function Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules.
Rev.S08

39. Guidelines for Sketching Graphs of Tangent and Cotangent Functions
To graph y = tan bx or y = cot bx, with b > 0, follow these steps.
Step 1 Determine the period, π/b. To locate two adjacent vertical asymptotes solve the following equations for x:
Click link to download other modules.
Rev.S08

40. Guidelines for Sketching Graphs of Tangent and Cotangent Functions continued
Step 2 Sketch the two vertical asymptotes found in Step 1.
Step 3 Divide the interval formed by the vertical asymptotes into four equal parts.
Step 4 Evaluate the function for the first-quarter point, midpoint, and third-quarter point, using the x-values found in Step 3.
Step 5 Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary.
Click link to download other modules.
Rev.S08

41. How to Graph y = tan(2x)?
Step 1: The period of the function is π/2. The two asymptotes have equations x = −π/4 and x = π/4.
Step 2: Sketch the two vertical asymptotes found x = ± π/4.
Step 3: Divide the interval into four equal parts. This gives the following key x-values.
First quarter: −π/8
Middle value: 0 Third quarter: π/8
Click link to download other modules.
Rev.S08

42. How to Graph y = tan(2x)? (Cont.)
Step 4: Evaluate the function
Step 5: Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary. x
−π/8
π/8 2x
−π/4
π/4
tan 2x −1 1
Click link to download other modules.
Rev.S08

43. How to Graph y = tan(2x)? (Cont.)
Every y value for this function will be 2 units more than the corresponding y in y = tan x, causing the graph to be translated 2 units up compared to y = tan x.
Click link to download other modules.
Rev.S08

44. What have we learned? We have learned to:
Recognize periodic functions.
Determine the amplitude and period, when given the equation of a periodic function.
Find the phase shift and vertical shift, when given the equation of a periodic function.
Graph sine and cosine functions.
Graph cosecant and secant functions.
Graph tangent and cotangent functions.
Interpret a trigonometric model.
Click link to download other modules.
Rev.S08

45. Credit
Some of these slides have been adapted/modified in part/whole from the slides of the following textbook:
Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8th Edition
Click link to download other modules.
Rev.S08