t3_graphs of trigonometric functions-part 1

8/9/2019 T3_graphs of Trigonometric Functions-Part 1

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Graphs of Trigonometric

functions


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Introduction

To graphy = a sinx andy = a cosx, we can find

y-coordinates on the graphs by multiplying

y-coordinates on the graphs ofy = sinx andy = cosxby

a.

The next two slides show the special casesy = 2 sinx

andy = sinx:


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Exercise: Draw the graph of 2cos(x)Exercise: Draw the graph of 2cos(x)


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Example

Sketch the graph of the equation

y = 2 sinx.

Solution The graph ofy = 2 sinx sketched on

the next slide can be obtained by

first sketching the graph ofy = sinx and then

multiplyingy-coordinates by 2.

An alternative method is to reflect the graph ofy = 2 sinx through thex-axis:


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Solution


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Amplitude

The largesty-coordinate is the amplitude of the

graph or, equivalently, the amplitude of the

function fgiven byf(x) = a sinx.

Similar remarks and techniques apply

ify = a cosx.

For example, we find the amplitude and sketch

the graph ofy = 3 cosx:


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Step 1. Find the amplitude | A |.

Step 2. Solve Bx+ C= 0 and Bx+ C= 2T :

Bx+ C = 0 and Bx+ C = 2T

x = CB x

= CB

+2TB

Phase shift Period

Phase shift = CB

Period =2 TB

The graph completes one full cycle asBx+ Cvaries from 0 to2T that is, asxvaries over the interval

-

CB

, CB

+2TB

Step 3. Graph one cycle over the interval-

CB

, CB

+2TB

.

Step 4. Extend the graph in step 3 to the left or right as desired.

raphing y sin( ) and y os( X )


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Period of sine graph

B changes the length of one cycle

`B ` > 1, graph is compressed

`B ` < 1, graph is stretched

Period = 2T

`B`

Y= A sin(Bx+C)


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Using B to graph period

Make sure B is positive before you begin graphing

y = sin (-Bx) is equivalent to

y = -sin (Bx)

y = cos (-Bx) is equivalent to

y = cos (Bx)


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Graph of y = A sin (Bx)

Compare the graph of y = sin (2x), y = sin x and

y = 2sin(2x)


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Phase Shift of graph

C shifts graph horizontally

C > 0 moves graph to left

C < 0 moves graph to right Phase shift = -C

B

y = A sin(Bx+C)


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Example:

Find the amplitude, the period and the phase shift and sketch

the graph of

)2

2sin(3T

! xy

Solution:

The equation is of the form y = A sin(Bx+C) with A = 3, B = 2

and C= T/2

Thus the amplitude is |a|=3 and the period is TTT 2/2||/2 b

Phase shift: -C/B= -T/4


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Interval containing one cycle:

Interval containing one sine wave can be found by solving the

Following inequality:

]4/3,4/[intervalon theoccurs3amplitudeofwavesineoneThus

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TT

TT

TT

ee

ee

x

x


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Vertical movement of the graph

D shifts the graph vertically

Compare the graph of

y = sin (x) withy = sin (x) 2

y = sin (x) + 2


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Analyzing the process of breathing

The rhythmic process of breathing consists of alternating periods of

inhaling and exhaling. One complete cycle normally takes every 5 seconds.

IfF(t) denotes the air flow rate at time t(in liters per second) and if the

maximum flow rate is 0.6 liter per second, find a formula of the form F(t) =

Asin(Bt) that fits this information.

Solution:

)5

2sin(6.0)(

ormulatheusgivesThis

.6.0lete,oamplitudethetoscorrespondratelomaximumtheince

5

2or5

2

henceandseconds,5isperiodn theapplicatioIn this

/2isoperiodthen the0someor)sin()(i

ttF

A

B

BFBBtAtF

T

TT

T

!

!

!!

"!

Exercise: Sketch the graph ofF(t)


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Exercise:

Sketch the graph of

2)32cos(2 !

T

xy


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2)3

2cos(2 !T

xy

t3_graphs of trigonometric functions-part 1

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