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8/7/2019 TRI CT10 TrigonometricGraphs
1/21
Trigonometry Rapid Learning Series - 10
Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com 1
Rapid Learning CenterChemistry :: Biology :: Physics :: Math
Ra id Learnin Center Presents
Teach Yourself
Trigonometry Visually in 24 Hours
1/42 http://www.RapidLearningCenter.com
Functions
Trigonometry Rapid Learning Series
Rapid Learning Center
www.RapidLearningCenter.com/
Rapid Learning Inc. All Rights Reserved
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.
8/7/2019 TRI CT10 TrigonometricGraphs
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Trigonometry Rapid Learning Series - 10
Rapid Learning Inc. All rights reserved. :: http://www.RapidLearningCenter.com 2
Learning Objectives
By completing this tutorial, you will learn concepts of:
trigonometric functions
The general form of the
sine and cosine curves
Amplitudes and periods of
trigonometric functions
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Transformations
Previous Content
New Content
Trigonometric Graphs
involveinvolve
Concept Map
Asymptotes
Amplitude
Vertical Shift
Horizontal Shift
Transformations
of the include
Trigonometric Functions
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Sine
Cosine
Tangent Cotangent
Secant
Cosecant
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Trigonometry Rapid Learning Series - 10
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Cosine
Graphs
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Key points can be used to plot the
graph of a function.
Definition: Key Points
,
choose key points that correspond
to the:
maximum values of a function
minimum values of a function
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x-intercepts of a function (anx-intercept is a point at which the
value of a function is equal to
zero)
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Trigonometry Rapid Learning Series - 10
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Use five key points to plot the graph ofy= sin x.
Graph: y= sin x
Point x y Type
A 0 0 Intercept
B /2 1 Maximum
C 0 InterceptA
B
C E
y= sin x
x
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- n mum
E 2 0 Intercept
D
Use five key points to plot the graph ofy= cos x.
y= cos x
Graph: y = cos x
A
B
C
D
E
x
1
o n x y ype
A 0 1 Maximum
B /2 0 Intercept
C -1 Minimum
D 3/2 0 Intercept
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E 2
1 Maximum
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Trigonometry Rapid Learning Series - 10
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Amplitude One-half of the vertical distance betweenthe maxima and minima of a sine or cosine graph.
Definition: Amplitude
=
x 2|a|
=
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x 2|a|
The general forms of the sine and cosine functions are:
y= d+ asin(bx c) y= d+ acos(bx c)
Identifying the Amplitude
a, b, c, and dare constants (numbers whose values do
not change).
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The absolute value of a is the amplitude of the
function.
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Trigonometry Rapid Learning Series - 10
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Find the equation for
the sine graph shown. x 2a
Example: Amplitude
Solution:
Vertical distance between the max and min:
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3 (-3) = 3 + 3 = 6
Amplitude: |a| = 6/2 = 3
Equation: y= 3 sin x
Fundamental Period The smallest interval over which
a periodic function repeats itself.
Identifying the Period
y= d+ asin(bx c) y= d+ acos(bx c)
The constant b is directly related to the period of
sine and cosine.
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period = 2b
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Trigonometry Rapid Learning Series - 10
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Find the equation for the
cosine graph shown.
y
period
Example: Period
Solution:
period = 4 b= 2/4
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period = 2/b
4 = 2/b
b=
Equation: y= cos(x/2)
Transformations
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Transformations - Outline
Vertical translations
Transformations on the trigonometric functions include:
Horizontal translations
Changes in amplitude
and period
Phase shifts
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Vertical Translation An upward or downward shift in
the graph of a function.
Vertical Translation
Function Upward Shift Downward Shift
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y= d+ asin(bx c) y= d+ acos(bx c)
Identifying Vertical Translations
The constant d determines the magnitude and
direction of the vertical shift of a sine or cosine graph.
yy
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d dxx
Plot the graph of y= 1 + sin x.
Example: Vertical Translation
We begin by plotting y= sin x.
Since d= 1, shift y= sin x upward by 1 unit.
y
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y= x
x1
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Horizontal Translation A shift to the left or right of the
graph of a function.
Horizontal Translation
Function Shift to the Left Shift to the Right
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y= d+ asin(bx c) y= d+ acos(bx c)
Identifying Horizontal Translations
The ratio c/bis called the phase shift. The ratio c/b
determines the magnitude and direction of the
horizontal translation.
yy
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xx
c/b
c/b
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Plot the graph of y= cos(x+ /5).
Solution:
Example: Horizontal Translation
y= cos(x+ /5)
y= cos(x (-/5))
b= 1 , c= -/5
= -
x
/5
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-
Since c/bis negative, shift y= cos xto the left by/5 units.
Plot the graph of y= 1 + 2cos(2x+ /3).
Example: Multiple Transformations
Solution:
Amplitude: |a| = 2
Period:
2/b= 2/2 =
Horizontal Shift:
y
x
/6
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c/b= (-/3)/2 = -/6
Vertical Shift: d= 1
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Draw the graph of y= 3e-0.1xsin x.
Solution:y
Example: Exponential Decay
y= asin bx
a= 3e-0.1x b= 1
Amplitude:
|a| = 3e-0.1x
plot 3e-0.1x and -3e-0.1x
x
3e-0.1x
-3e-0.1x
3e-0.1x
-3e-0.1x
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Period:
2/b = 2/1 = 2
*Remove the exponential curve.
Cotangent
Graphs
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Asymptote Any line that a function approachesclosely without ever intersecting.
Vertical asymptote A vertical line that a function
Definition: Asymptote
approaches closely without ever intersecting.
As a function approaches a vertical asymptote it will
either become:
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or
increasingly
positive
increasingly
negative
The function y= tan x has a vertical asymptote at each
odd integer multiple of/2 (, -3/2, -/2, /2, 3/2, ).
Asymptotes of Tangent and Cotangent
The function y= cot x has a vertical asymptote at each
multiple of (, -3, -2, -, , 2, 3,).
y y
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x
Asymptotes of Tangent
x
Asymptotes of Cotangent
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To graph y= tan x:
1. Draw a pair of consecutive vertical
How-to: Graph y= tan x
asymptotes.
2. Tabulate and plot 7 key points.
3 key points will be located
between the left asymptote and the
midpoint
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1 key point will be the midpoint
between the asymptotes
3 key points will be located
between the midpoint and the right
asymptote
x y Type
Use key points to plot the graph ofy= tan x.
Graph: y= tan x
- .
B -1.5707 -10381 normal
C -1.57 -1256 normal
D -1.5 -14.10 normal
E 0 0 intercept
H
GF
y H
GF
y
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F 1.5 14.10 normal
G 1.57 1256 normal
H 1.5707 10381 normal
I /2 Undef. asymptote
E
DC
B
x
-/2 /2
E
DC
B
x
-/2 /2
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Consider the function y= a tan(bx).
The period of this function will be /b.
Graph ofy= a tan(bx)
,
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
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Draw the graph of y= 2 tan(x/2).
Solution:
Example: Tangent Transformation
a= 2 b=
Period:
/b = /(1/2)
= 2
Consecutive asymptotes:
y
x
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x= - and
Behavior:
a> 0 increasing interval (-, )
-
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Trigonometry Rapid Learning Series - 10
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To graph y= cot x:
1. Draw a pair of consecutive vertical
How-to: Graph y= cot x
asymptotes.
2. Tabulate and plot 7 key points.
3 key points will be located
between the left asymptote and
the midpoint
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1 key point will be the midpoint
between the asymptotes
3 key points will be located
between the midpoint and the
right asymptote
x y Type
Use key points to plot the graph ofy= cot x.
Graph: y= cot x
A 0 Undef. asymptote
B 0.001 999 normal
C 0.01 99.9 normal
D 0.1 9.97 normal
E /2 0 intercept F
EDC
B
y
xF
EDC
B
y
x
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F 3.1 -24.0 normal
G 3.14 -627 normal
H 3.141 -1687 normal
I Undef. asymptote
H
G
0
H
G
0
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Consider the function y= a cot(bx).
The period will be /b.
Graph ofy= acot(bx)
,
will be decreasing.
If the constant a is negative, the graph of a period
will be increasing.
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Draw the graph of y= -3 cot(x/6).
Solution:y
Example: Cotangent Transformation
a= -3 b= 1/6
Period:
/b= /(1/6)
= 6
Consecutive as m totes:
x3
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x= 0 and 6
Behavior:
a< 0 increasing interval (0, 6)
0 6
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Cosecant
Graphs
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Use five key points to plot the graph of y= csc x.
Use the fact that csc xis the reciprocal of sin x.
Graph: y= csc x
x sin x y= csc x
0 0 Undefined/
Asymptote
/2 1 1
Minimum
0 Undefined/
y
x
min
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3/2 -1 -1
Maximum
2 0 Undefined/
asymptote
max
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Trigonometry Rapid Learning Series - 10
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Draw the graph of y= csc(2x /3).
Solution: y
Example: Cosecant Transformation
y= csc(bx c)
b= 2 c= /3
Period:
2/b= 2/2 =
Horizontal shift:
x
/6
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c/b= (/3)/2 = /6
Vertical asymptotes:
x = 2/3, 7/6, 5/3, 13/6
Use five key points to plot the graph of y= sec x.
Use the fact that sec xis the reciprocal of cos x.
Graph: y= sec x
x cos x y= sec x
0 1 1
Minimum
/2 0 Undefined/
Asymptote
-1 -1
y
x
minmin
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Maximum
3/2 0 Undefined/
Asymptote
2 1 1
minimum
max
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Trigonometry Rapid Learning Series - 10
CongratulationsYou have successfully completed
the core tutorial
Graphs of Trigonometric
Functions
Rapid Learning Center
Rapid Learning CenterChemistry :: Biology :: Physics :: Math
a s ex
Step 1: Concepts Core Tutorial (Just Completed)
Step 2: Practice Interactive Problem DrillStep 3: Recap Super Review Cheat Sheet
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