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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 xintercepts of a function (an xintercept 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 Onehalf of the vertical distance between the maxima and minima of a sine or cosine graph.
y = a sin x
x 2a
y = a cos x
x 2a
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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 "a" 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 "b" 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 "d" 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 = 3e0.1x sin x. Solution: y = a sin bx a = 3e0.1x b=1
y
3e0.1x x 3e0.1x
Amplitude: a = 3e0.1x plot 3e0.1x and 3e0.1x 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
Howto: 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 "a" is positive, the graph of a period positive will be increasing. (Figure 1) If the constant "a" 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 > 0 increasing interval (, )
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Trigonometry Rapid Learning Series  10
Howto: 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 "a" is positive, the graph of a period positive will be decreasing. If the constant "a" 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 < 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
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Graphs of Trigonometric Functions
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