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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.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

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

http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

Rev.S08

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.

http://faculty.valenciacc.edu/ashaw/

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Rev.S08

Example of a Periodic Function

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Rev.S08

Example of Another Periodic Function

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Rev.S08

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|.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

0

π/2

π

3π/2

π

sin x

0

1

0

−1

0

3sin x

0

3

0

−3

0

How to Graph y = 3 sin(x)?Note the difference between sin x and 3sin x. What is the difference?

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

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.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

How to Graph y = sin(2x)?(Cont.)

Note the difference between sin x and sin 2x. What is the difference?

http://faculty.valenciacc.edu/ashaw/

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Rev.S08

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.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

0

3π/4

3π/2

9π/4

3π

2x/3

0

π/2

π

3π/2

2π

cos 2x/3

1

0

−1

0

1

How to Graphy = cos (2x/3) over one period?- The period is 3π.
- Divide the interval into four equal parts.
- Obtain key points for one period.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

How to Graphy = cos(2x/3) over one period? (Cont.)

- The amplitude is 1.
- Join the points and connect with a smooth curve.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

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 1Find 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 2. The points will be maximum points, minimum points, and x-intercepts.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

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.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

0

π/8

π/4

3π/8

π/2

4x

0

π/2

π

3π/2

2π

sin 4x

0

1

0

−1

0

−2 sin 4x

0

−2

0

2

0

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

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

How to Graph y = −2 sin(4x)?(Cont.)

- Plot the points and join them with a sinusoidal curve with amplitude 2.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

What is a Phase Shift?

- In trigonometric functions, a horizontal translation is called a phase shift.
- In the equation
- the graph is shiftedπ/2 units to the right.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

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.

http://faculty.valenciacc.edu/ashaw/

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Rev.S08

π/3

5π/6

4π/3

11π/6

7π/3

x−π/3

0

π/2

π

3π/2

2π

sin (x−π/3)

0

1

0

−1

0

How to Graph y = sin (x−π/3) by Using Horizontal Translation or Phase Shift?(Cont.)http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

How to Graph y = 3 cos(x+π/4) by Using Horizontal Translation or Phase Shift?

- Find the interval.
- Divide into four equal parts.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

−π/4

π/4

3π/4

5π/4

7π/4

x + π/4

0

π/2

π

3π/2

2π

cos(x + π/4)

1

0

−1

0

1

3 cos (x + π/4)

3

0

−3

0

3

How to Graph y = 3 cos(x+π/4) by Using Horizontal Translation or Phase Shift?http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

0

π/6

π/3

π/2

2π/3

3x

0

π/2

π

3π/2

2π

−2 sin 3x

0

−2

0

2

0

2 − 2 sin 3x

2

0

2

4

2

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.

http://faculty.valenciacc.edu/ashaw/

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Rev.S08

How to Graph y = 2 − 2 sin 3x by Using Vertical Translation or Vertical Shift?(Cont.)

- Plot the points and connect.

http://faculty.valenciacc.edu/ashaw/

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Rev.S08

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.)

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

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 < 0. The horizontal translation (phase shift) is d units to the right if d > 0 and |d| units to the left if d < 0.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

How to Graph y = −1 + 2 sin (4x + π)?

- Step 2:Divide the interval.
- Step 3 Table

- Write the expression in the form c + a sin b(x−d) by rewriting 4x + πas
- Step 1

http://faculty.valenciacc.edu/ashaw/

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Rev.S08

−π/4

−π/8

0

π/8

π/4

x + π/4

0

π/8

π/4

3π/8

π/2

4(x + π/4)

0

π/2

π

3π/2

2π

sin 4(x + π/4)

0

1

0

−1

0

2 sin 4(x + π/4)

0

2

0

−2

2

−1 + 2sin(4x + π)

−1

1

−1

−3

−1

How to Graph y = −1 + 2 sin (4x + π)?(Cont.)http://faculty.valenciacc.edu/ashaw/

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Rev.S08

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.

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Rev.S08

Let’s Take a Look at Other Circular Functions.

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Rev.S08

Use as a Guide

y = a csc bx

y = a sin bx

y = a sec bx

y = cos bx

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 1Graph the corresponding reciprocal function as a guide, using a dashed curve.

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Rev.S08

Guidelines for Sketching Graphs of Cosecant and Secant Functions Continued

- Step 2Sketch 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.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

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.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

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.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

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:

http://faculty.valenciacc.edu/ashaw/

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Rev.S08

Guidelines for Sketching Graphs of Tangent and Cotangent Functions continued

- Step 2Sketch the two vertical asymptotes found in Step 1.
- Step 3Divide the interval formed by the vertical asymptotes into four equal parts.
- Step 4Evaluate the function for the first-quarter point, midpoint, and third-quarter point, using the x-values found in Step 3.
- Step 5Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

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: 0Third quarter: π/8

http://faculty.valenciacc.edu/ashaw/

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Rev.S08

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

0

π/8

2x

−π/4

0

π/4

tan 2x

−1

0

1

http://faculty.valenciacc.edu/ashaw/

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Rev.S08

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.

http://faculty.valenciacc.edu/ashaw/

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Rev.S08

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.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

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

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.S08

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