graphs of trig functions. y = sin x max sin x = 1 when x = 90° min sin x = -1 when x = 270° graph...

Graphs of Trig FunctionsGraphs of Trig Functions

90 180 270 360

-1

-0.5

0.5

1

x

y

90 180 270 360

-1

-0.5

0.5

1

x

y

90 180 270 360

-1

-0.5

0.5

1

x

y

90 180 270 360

-1

-0.5

0.5

1

x

y

90 180 270 360

-1

-0.5

0.5

1

x

y

yy = sin = sin xx

90 180 270 360

-1.5

-1

-0.5

0.5

1

1.5

x

y

MaxMax sin sin xx = 1 when = 1 when xx = 90 = 90°°

MinMin sin sin xx = -1 when = -1 when xx = 270 = 270°°

Graph of Graph of yy = sin = sin xx

90 180 270 360

-2

-1.5

-1

-0.5

0.5

1

1.5

2

x

y

90 180 270 360

-2

-1.5

-1

-0.5

0.5

1

1.5

2

x

y

yy = sin = sin xx

MaxMax 2sin 2sin xx = 2 when = 2 when xx = 90 = 90°°

MinMin 2sin 2sin xx = -2 when = -2 when xx = 270 = 270°°

Graph of Graph of yy = 2sin = 2sin xx

yy = 2sin = 2sin xx

Am

pli

tud

e =

2A

mp

litu

de

= 2

MaxMax 5sin 5sin xx = 5 when = 5 when xx = 90 = 90°°

MinMin 5sin 5sin xx = -5 when = -5 when xx = 270 = 270°°

Graph of Graph of yy = 5sin = 5sin xx

90 180 270 360

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

90 180 270 360

-5

-4

-3

-2

-1

1

2

3

4

5

x

yyy = 5sin = 5sin xx

yy = sin = sin xx

Am

pli

tud

e =

5A

mp

litu

de

= 5

90 180 270 360

-1.5

-1

-0.5

0.5

1

1.5

x

y

90 180 270 360

-1.5

-1

-0.5

0.5

1

1.5

x

y

yy = sin = sin xx

The The negativenegative multiplier reflects multiplier reflects the initial graph in the the initial graph in the xx-axis-axis

yy = -sin = -sin xx

Graph of Graph of yy = -sin = -sin xx

Am

pli

tud

e =

1A

mp

litu

de

= 1

90 180 270 360

-1.5

-1

-0.5

0.5

1

1.5

x

y

90 180 270 360

-1.5

-1

-0.5

0.5

1

1.5

x

y

yy = sin = sin xx

The negative multiplier reflects The negative multiplier reflects the initial graph in the the initial graph in the xx-axis-axis

Graph of Graph of yy = -1 = -1·5·5sin sin xx

yy = -1 = -1·5·5sin sin xx

Am

pli

tud

e =

A

mp

litu

de

=

11 ··55

90 180 270 360

-1.5

-1

-0.5

0.5

1

1.5

x

y

yy = cos = cos xx

MaxMax cos cos xx = 1 when = 1 when xx = 0 = 0° or 360° or 360°°

MinMin cos cos xx = -1 when = -1 when xx = 180 = 180°°

Graph of Graph of yy = cos = cos xx

90 180 270 360

-3

-2

-1

1

2

3

x

y

yy = cos = cos xx

90 180 270 360

-3

-2

-1

1

2

3

x

y

yy = 3cos = 3cos xx

Graph of Graph of yy = 3cos = 3cos xx andand y = -y = -3cos3cos x x

90 180 270 360

-3

-2

-1

1

2

3

x

y

yy = -3cos = -3cos xx

Write down the equations of the following graphs Write down the equations of the following graphs

90 180 270 360

-2

-1

1

2

x

y

90 180 270 360

-4

-3

-2

-1

1

2

3

4

x

y

90 180 270 360

-2

-1

1

2

x

y

90 180 270 360

-3

-2

-1

1

2

3

x

y

1.

3.

2.

4.

90 180 270 360

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

90 180 270 360

-4

-3

-2

-1

1

2

3

4

x

y

90 180 270 360

-3

-2

-1

1

2

3

x

y

90 180 270 360

-10

-8

-6

-4

-2

2

4

6

8

x

y

6.5.

8.7.

90 180 270 360

-4

-3

-2

-1

1

2

3

4

5

6

x

y

Graph of Graph of yy = sin = sin x x ±± constantconstant

90 180 270 360

-4

-3

-2

-1

1

2

3

4

5

6

x

y

y y = sin = sin xx + 2 + 2

90 180 270 360

-4

-3

-2

-1

1

2

3

4

5

6

x

y

yy = sin = sin xx - 1 - 190 180 270 360

-4

-3

-2

-1

1

2

3

4

5

6

x

yyy = sin = sin xx + 5 + 5

90 180 270 360

-4

-3

-2

-1

1

2

3

4

5

6

x

y

yy = sin = sin xx - 3 - 3

Adding or subtracting a constant to or from Adding or subtracting a constant to or from sin sin xx moves moves the graph vertically by the amount of the constantthe graph vertically by the amount of the constant

90 180 270 360

-4

-3

-2

-1

1

2

3

4

5

6

x

y

90 180 270 360

-4

-3

-2

-1

1

2

3

4

5

6

x

y

90 180 270 360

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

90 180 270 360

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

90 180 270 360

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

90 180 270 360

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

90 180 270 360

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

Graph of Graph of yy = cos = cos x x ±± constantconstant

yy = cos = cos xx + 6 + 6

yy = cos = cos xx + 4 + 4

yy = cos = cos xx - 1 - 1

yy = cos = cos xx - 4 - 4

90 180 270 360

-1

-0.5

0.5

1

x

y

Graphs of Multiple AnglesGraphs of Multiple Angles

yy = sin = sin xx

90 180 270 360

-1

-0.5

0.5

1

x

y

yy = sin 2 = sin 2xx

Max and Min values remain unchanged at +1 and -1Max and Min values remain unchanged at +1 and -1

Two complete cycles between 0Two complete cycles between 0°° and 360 and 360°°

Period = 180Period = 180°°

90 180 270 360

-1

-0.5

0.5

1

x

y


yy = sin = sin xx


Three complete cycles between 0Three complete cycles between 0°° and 360 and 360°°


90 180 270 360

-1

-0.5

0.5

1

x

y

yy = sin 3 = sin 3xx

90 180 270 360

-1

-0.5

0.5

1

x

y


yy = sin = sin xx


Half a complete cycle between 0Half a complete cycle between 0°° and 360 and 360°°


90 180 270 360

-1

-0.5

0.5

1

x

y

yy = sin 0 = sin 0·5·5xx



Four complete cycles between 0Four complete cycles between 0°° and 360 and 360°°


90 180 270 360

-1

-0.5

0.5

1

x

y

yy = cos = cos xx

90 180 270 360

-1

-0.5

0.5

1

x

y

yy = cos 4 = cos 4xx

Write down the Write down the equationsequations and and periodperiod of these graphs of these graphs

90 180 270 360

-1

1

x

y

90 180 270 360

-1

1

x

y

90 180 270 360

-1

1

x

y

90 180 270 360

-1

1

x

y

Summary of Graphs so farSummary of Graphs so far

yy = = aa sin sin bbxx ±± cc or or yy = = aa cos cos bbxx ±± cc

Where Where aa, , bb and and cc are constants are constants

aa is the is the amplitudeamplitude, height above centre line of graph,, height above centre line of graph, if if aa is is negativenegative the graph is reflected in the the graph is reflected in the xx-axis-axis

bb is the number of complete cycles between 0is the number of complete cycles between 0° and 360°° and 360°

cc moves the graph vertically up or down from moves the graph vertically up or down from xx-axis-axis

Mixed examples – combined effectsMixed examples – combined effects

90 180 270 360

-3

-2

-1

1

2

3

x

y

Curve rises from origin so is a Curve rises from origin so is a sinesine graph graphAmplitude is 3 soAmplitude is 3 so y = 3 sin ?y = 3 sin ?xxThere are 2 complete cycles from 0There are 2 complete cycles from 0° to 360° so ° to 360° so

y = 3sin 2xy = 3sin 2x

90 180 270 360

-2

2

4

6

8

10

x

y

Curve falls from Curve falls from yy-axis so is a -axis so is a cosinecosine graph graph

Centre line half way between -2 and 10 = Centre line half way between -2 and 10 = 44

SoSo yy = ? cos ? = ? cos ?xx + 4 + 4

Amplitude is 6, soAmplitude is 6, so yy = 6 cos ? = 6 cos ?xx + 4 + 4

There is only one complete cycle from 0There is only one complete cycle from 0° to 360° so° to 360° so

yy = 6cos = 6cos xx + 4 + 4

90 180 270 360

-5

-4

-3

-2

-1

1

x

y

Curve falls from Curve falls from yy-axis so is a -axis so is a cosinecosine graph graph

Centre line half way between -5 and 1 = Centre line half way between -5 and 1 = -2-2

Amplitude is 3, soAmplitude is 3, so yy = 3 cos ? = 3 cos ?xx - 2 - 2

SoSo yy = ? cos ? = ? cos ?xx - 2 - 2

There are three complete cycles from 0There are three complete cycles from 0° to 360° so° to 360° so

yy = 3cos 3 = 3cos 3xx - 2 - 2

90 180 270 360

-1

1

2

3

x

y

Find the equations of the following graphsFind the equations of the following graphs

90 180 270 360

-2

-1

1

2

x

y 90 180 270 360

-9

-8

-7

-6

-5

-4

-3

-2

-1

xy

90 180 270 360

-2

-1

1

2

3

4

5

6

7

8

x

y

graphs of trig functions. y = sin x max sin x = 1 when x = 90° min sin x = -1 when x = 270° graph...

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