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Trigonometry Rapid Learning Series - 10

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Graphs of Trigonometric Functions

Trigonometry Rapid Learning Series

Wayne Huang, Ph.D. Mark Cowan, Ph.D. Diop El Moctar, Ph.D. Poornima Gowda, Ph.D. Daniel Deaconu, Ph.D. Fabio Mainardi, Ph.D. Theresa Johnson, M.Ed. Jessica Davis, M.S. Wendy Perry, M.A. Cesar Anchiraico, M.S.

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Trigonometry Rapid Learning Series - 10

Learning Objectives

By completing this tutorial, you will learn concepts of:

The basic graphs of the trigonometric functions The general form of the sine and cosine curves Amplitudes and periods of trigonometric functions Transformations

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Concept Map

Previous Content New Content

Trigonometric Graphs

involve involve

Transformations

of the include

Asymptotes

Trigonometric Functions Vertical Shift Sine Si Cosine Tangent

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Cosecant Secant Cotangent

Amplitude p

Horizontal Shift

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Trigonometry Rapid Learning Series - 10

Sine and Cosine Graphs

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Definition: Key Points

Key points can be used to plot the graph of a function. In drawing the graph of a function, function choose key points that correspond to the: maximum values of a function minimum values of a function x-intercepts of a function (an x-intercept is a point at which the value of a function is equal to zero)

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Trigonometry Rapid Learning Series - 10

Graph: y = sin x

Use five key points to plot the graph of y = sin x.

y = sin x B C A D E x

Point A B C D E

x 0 /2 3/2 3 /2 2

y 0 1 0 -1 1 0

Type Intercept Maximum Intercept Minimum Mi i Intercept

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Graph: y = cos x

Use five key points to plot the graph of y = cos x.

Point A B C D E Type Maximum Intercept Minimum Intercept Maximum C

1

x 0 /2 3/2 2

y 1 0 -1 0 1

y = cos x A B D x E

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Trigonometry Rapid Learning Series - 10

Definition: Amplitude

Amplitude ­ One-half of the vertical distance between the maxima and minima of a sine or cosine graph.

y = a sin x

x 2|a|

y = a cos x

x 2|a|

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Identifying the Amplitude

The general forms of the sine and cosine functions are: y = d + a sin(bx ­ c) y = d + a cos(bx ­ c)

a, b, c, and d are constants (numbers whose values do not change).

The absolute value of &quot;a&quot; is the amplitude of the function.

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Trigonometry Rapid Learning Series - 10

Example: Amplitude

Find the equation for the sine graph shown.

x

2a

Solution: Vertical distance between the max and min: 3 ­ (-3) = 3 + 3 = 6 Amplitude: |a| = 6/2 = 3 Equation: y = 3 sin x

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Identifying the Period

Fundamental Period ­ The smallest interval over which a periodic function repeats itself. y = d + a sin(bx ­ c) y = d + a cos(bx ­ c)

The constant &quot;b&quot; is directly related to the period of sine and cosine. period = 2 b

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Trigonometry Rapid Learning Series - 10

Example: Period

y

Find the equation for the cosine graph shown.

period

x

Solution: period = 4 period = 2/b

4 = 2/b

b = 2/4 b=½ Equation: y = cos(x/2)

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Transformations

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Trigonometry Rapid Learning Series - 10

Transformations - Outline

Transformations on the trigonometric functions include: Vertical t a s at o s e t ca translations Horizontal translations Changes in amplitude and period Phase shifts Exponential damping

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Vertical Translation

Vertical Translation ­ An upward or downward shift in the graph of a function. Function Upward Shift Downward Shift

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Trigonometry Rapid Learning Series - 10

Identifying Vertical Translations

y = d + a sin(bx ­ c) y = d + a cos(bx ­ c)

The constant &quot;d&quot; determines the magnitude and direction of the vertical shift of a sine or cosine graph.

y y

d

x

d

x

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Example: Vertical Translation

Plot the graph of y = 1 + sin x.

Solution: We begin by plotting y = sin x. Since d = 1, shift y = sin x upward by 1 unit.

y

y = 1 + sin x

1

x

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Trigonometry Rapid Learning Series - 10

Horizontal Translation

Horizontal Translation ­ A shift to the left or right of the graph of a function. Function Shift to the Left Shift to the Right

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Identifying Horizontal Translations

y = d + a sin(bx ­ c) y = d + a cos(bx ­ c)

The ratio c/b is called the phase shift. The ratio c/b determines the magnitude and direction of the horizontal translation.

y y

c/b x c/b

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x

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Trigonometry Rapid Learning Series - 10

Example: Horizontal Translation

Plot the graph of y = cos(x + /5). Solution: y = cos(x + /5) y = cos(x ­ (-/5)) b = 1 , c = -/5 c/b = -/5

/5

y

x

Since c/b is negative, shift y = cos x to the left by /5 units.

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Example: Multiple Transformations

Plot the graph of y = 1 + 2cos(2x + /3).

Solution: Amplitude: |a| = 2 Period: 2/b = 2/2 = Horizontal Shift: c/b = (-/3)/2 = -/6 Vertical Shift: d = 1

y

/6

x

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Trigonometry Rapid Learning Series - 10

Example: Exponential Decay

Draw the graph of y = 3e-0.1x sin x. Solution: y = a sin bx a = 3e-0.1x b=1

y

3e-0.1x x -3e-0.1x

Amplitude: |a| = 3e-0.1x plot 3e-0.1x and -3e-0.1x Period: 2/b = 2/1 = 2 *Remove the exponential curve.

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Tangent and Cotangent Graphs

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Trigonometry Rapid Learning Series - 10

Definition: Asymptote

Asymptote ­ Any line that a function approaches closely without ever intersecting. Vertical asymptote ­ A vertical line that a function approaches closely without ever intersecting. As a function approaches a vertical asymptote it will either become:

or

increasingly positive

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increasingly

negative

Asymptotes of Tangent and Cotangent

The function y = tan x has a vertical asymptote at each odd integer multiple of /2 (..., -3/2, -/2, /2, 3/2, ...). The function y = cot x has a vertical asymptote at each multiple of (..., -3, -2, -, , 2, 3,...).

y y

x

x

Asymptotes of Tangent

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Asymptotes of Cotangent

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Trigonometry Rapid Learning Series - 10

How-to: Graph y = tan x

To graph y = tan x: 1. Draw a pair of consecutive vertical asymptotes. t t 2. Tabulate and plot 7 key points. 3 key points will be located between the left asymptote and the midpoint 1 key point will be the midpoint between the asymptotes 3 key points will be located between the midpoint and the right asymptote

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Graph: y = tan x

Use key points to plot the graph of y = tan x.

x A B C D E F G H I

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y Undef. Undef -10381 -1256 -14.10 0 14.10 1256 10381 Undef.

Type asymptote normal normal normal intercept normal normal normal asymptote

-/2 /2 C B D E x y F H G

-/2 /2 -1.5707 -1.57 -1.5 0 1.5 1.57 1.5707 /2

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Trigonometry Rapid Learning Series - 10

Graph of y = a · tan(bx)

Consider the function y = a · tan(bx). The period of this function will be /b. If the constant &quot;a&quot; is positive, the graph of a period positive will be increasing. (Figure 1) If the constant &quot;a&quot; is negative, the graph of a period will be decreasing. (Figure 2)

Figure 1 Figure 2 g

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Example: Tangent Transformation

Draw the graph of y = 2 tan(x/2). Solution: a=2 Period: /b = /(1/2) = 2

x

b=½

y

Consecutive asymptotes: y p x = - and Behavior: a &gt; 0 increasing interval (-, )

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-

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Trigonometry Rapid Learning Series - 10

How-to: Graph y = cot x

To graph y = cot x: 1. Draw a pair of consecutive vertical asymptotes. t t 2. Tabulate and plot 7 key points. 3 key points will be located between the left asymptote and the midpoint 1 key point will be the midpoint between the asymptotes 3 key points will be located between the midpoint and the right asymptote

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Graph: y = cot x

Use key points to plot the graph of y = cot x.

x A B C D E F G H I

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y Undef. 999 99.9 9.97 0 -24.0 -627 -1687 Undef.

Type asymptote normal normal normal intercept normal normal normal asymptote

0 C B D E F x G H y

0 0.001 0.01 0.1 /2 3.1 3.14 3.141

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Trigonometry Rapid Learning Series - 10

Graph of y = a cot(bx)

Consider the function y = a · cot(bx). The period will be /b. If the constant &quot;a&quot; is positive, the graph of a period positive will be decreasing. If the constant &quot;a&quot; is negative, the graph of a period will be increasing.

Figure 1

Figure 2

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Example: Cotangent Transformation

Draw the graph of y = -3 cot(x/6). Solution: a = -3 b = 1/6 Period: /b = /(1/6) = 6 Consecutive asymptotes: x = 0 and 6

0 Behavior: a &lt; 0 increasing interval (0, 6) 6 y

3

x

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Trigonometry Rapid Learning Series - 10

Secant and Cosecant Graphs

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Graph: y = csc x

Use five key points to plot the graph of y = csc x. Use the fact that csc x is the reciprocal of sin x.

x 0 /2 3/2 2 sin x 0 1 0 -1 0 y = csc x Undefined/ Asymptote 1 Minimum Undefined/ Asymptote -1 Maximum Undefined/ asymptote

y min x max

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Trigonometry Rapid Learning Series - 10

Example: Cosecant Transformation

Draw the graph of y = csc(2x ­ /3). Solution: y = csc(bx ­ c) b=2 c = /3 Period: 2/b= 2/2 = Horizontal shift: c/b = (/3)/2 = /6 Vertical asymptotes: x = 2/3, 7/6, 5/3, 13/6

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y /6

x

Graph: y = sec x

Use five key points to plot the graph of y = sec x. Use the fact that sec x is the reciprocal of cos x.

x 0 /2 3/2 2 cos x 1 0 -1 0 1 y = sec x 1 Minimum Undefined/ Asymptote -1 Maximum Undefined/ Asymptote 1 minimum

y min x

min

max

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Trigonometry Rapid Learning Series - 10

Example: Secant Transformation

Draw the graph of y = -1 + 2 sec x. Solution: y = d + a · sec x a=2 d = -1

x 1 y

Amplitude: |a| = |2| = 2 Vertical shift: d = -1 down 1 unit Vertical asymptotes: x = /2, 3/2, 5/2, 7/2

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Learning Summary

The vertical shift of a trigonometric function is determined by the value of d.

The period of a sine or cosine function is 2/b.

The period of a tangent or cotangent function is /b.

The amplitude is determined by |a| in the general form equations of sine and cosine.

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The horizontal shift of a trigonometric function is determined by the ratio c/b.

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Trigonometry Rapid Learning Series - 10

Congratulations

You have successfully completed the core tutorial

Graphs of Trigonometric Functions

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