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

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Mac 1114 l.jpg
MAC 1114

  • Module 4

  • Graphs of the Circular Functions

Rev.S08


Learning objectives l.jpg
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 l.jpg
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 l.jpg
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/

Click link to download other modules.

Rev.S08


Example of a periodic function l.jpg
Example of a Periodic Function

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

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Rev.S08


Example of another periodic function l.jpg
Example of Another Periodic Function

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Rev.S08


What is the amplitude of a periodic function l.jpg
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


How to graph y 3 sin x l.jpg

x

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 2 x l.jpg
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 2 x cont l.jpg
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/

Click link to download other modules.

Rev.S08


Period of a periodic function l.jpg
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


How to graph y cos 2 x 3 over one period l.jpg

x

0

3π/4

3π/2

9π/4

2x/3

0

π/2

π

3π/2

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 graph y cos 2 x 3 over one period cont l.jpg
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 l.jpg
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 l.jpg
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


How to graph y 2 sin 4 x l.jpg

x Continued

0

π/8

π/4

3π/8

π/2

4x

0

π/2

π

3π/2

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 4 x cont l.jpg
How to Graph Continuedy = −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 l.jpg
What is a Phase Shift? Continued

  • 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 3 by using horizontal translation or phase shift l.jpg
How to Graph Continuedy = 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/

Click link to download other modules.

Rev.S08


How to graph y sin x 3 by using horizontal translation or phase shift cont l.jpg

x Continued

π/3

5π/6

4π/3

11π/6

7π/3

x−π/3

0

π/2

π

3π/2

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 4 by using horizontal translation or phase shift l.jpg
How to Graph Continuedy = 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


How to graph y 3 cos x 4 by using horizontal translation or phase shift22 l.jpg

x Continued

−π/4

π/4

3π/4

5π/4

7π/4

x + π/4

0

π/2

π

3π/2

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


How to graph y 2 2 sin 3 x by using vertical translation or vertical shift l.jpg

x Continued

0

π/6

π/3

π/2

2π/3

3x

0

π/2

π

3π/2

−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/

Click link to download other modules.

Rev.S08


How to graph y 2 2 sin 3 x by using vertical translation or vertical shift cont l.jpg
How to Graph Continued y = 2 − 2 sin 3x by Using Vertical Translation or Vertical Shift?(Cont.)

  • Plot the points and connect.

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

Click link to download other modules.

Rev.S08


Further guidelines for sketching graphs of sine and cosine functions l.jpg
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 l.jpg
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 4 x l.jpg
How to Graph Functions (Cont.)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

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Rev.S08


How to graph y 1 2 sin 4 x cont l.jpg

x Functions (Cont.)

−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

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 4 x cont29 l.jpg
How to Graph Functions (Cont.)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.

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

Click link to download other modules.

Rev.S08


Let s take a look at other circular functions l.jpg
Let’s Take a Look at Other Circular Functions. Functions (Cont.)

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

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Rev.S08


Cosecant function l.jpg
Cosecant Function Functions (Cont.)

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

Click link to download other modules.

Rev.S08


Secant function l.jpg
Secant Function Functions (Cont.)

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

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Rev.S08


Guidelines for sketching graphs of cosecant and secant functions l.jpg

To Graph Functions (Cont.)

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.

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

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Rev.S08


Guidelines for sketching graphs of cosecant and secant functions continued l.jpg
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 l.jpg
How to Graph Functions Continuedy = 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 l.jpg
    How to Graph Functions Continuedy = 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


    Tangent function l.jpg
    Tangent Function Functions Continued

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

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    Rev.S08


    Cotangent function l.jpg
    Cotangent Function Functions Continued

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

    Click link to download other modules.

    Rev.S08


    Guidelines for sketching graphs of tangent and cotangent functions l.jpg
    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/

    Click link to download other modules.

    Rev.S08


    Guidelines for sketching graphs of tangent and cotangent functions continued l.jpg
    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 2 x l.jpg
    How to Graph Functions continuedy = 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/

    Click link to download other modules.

    Rev.S08


    How to graph y tan 2 x cont l.jpg
    How to Graph Functions continuedy = 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/

    Click link to download other modules.

    Rev.S08


    How to graph y tan 2 x cont43 l.jpg
    How to Graph Functions continuedy = 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/

    Click link to download other modules.

    Rev.S08


    What have we learned l.jpg
    What have we learned? Functions continued

    • 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 l.jpg
    Credit Functions continued

    • 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/

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    Rev.S08


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