Block 3

Trigonometric Graphs

What is to be learned?

• A reminder of how to draw and identify trig graphs.

• Take it a bit further.

90 180 270 360

1

0

-1

Y = sinx

Maximum Value = 1

Minimum Value = -1

90 180 270 360

1

0

-1

Y = cosx

Maximum Value = 1

Minimum Value = -1

90 180 270 360

7

0

-7

Y = 7sinx

Maximum Value = 7

Minimum Value = -7

Range = Max - Min

Range = 7 – (-7)

= 14

→range = 14

Range

90 180 270 360

4

0

-4

Y = 4cosx

Maximum Value = 4

Minimum Value = -4

→range = 8

90 180 270 360

8

0

-8

Y = - 8sinx

Maximum Value = 8

Minimum Value = -8

“Opposite” to Sin x

90 180 270 360

6

0

-6

Y = - 6cosx

Maximum Value = 6

Minimum Value = -6

“Opposite” to Cos x

900 1800 2700 3600 900 1800 2700 3600

3

-3

6

-6

Write the Equations

1. 2.

y = -3sinx y = -6cosx

y = 9sinx y = cosx

3. 4.

9

-9

1

-1

900 1800 2700 3600 900 1800 2700 3600

90 180 270 360

1

0

-1

Y = sin x

540450

Period of graph is 3600

Cycle starts again

Also applies to Y = cos x

Between 00 and 3600 there is 1 cycle

Taking it Further

90 180 270 360

1

0

-1

Y = sin 2x

Period of graph is 1800

There are 2 cycles between 00 and 3600

Combining these rulesDraw y = 6sin2x

Max 6

Min -6

2 cycles

Period = 360 ÷ 2 = 1800

90 180 270 360

6

0

-6

Y = 6sin 2x

Recognising Graph

Max 8

Min -8

4 cycles

90 180 270 360

8

0

-8

Y = 8cos4x

Cosine

900 1800 2700 3600 900 1800 2700 3600

900 1800 2700 3600 900 1800 2700 3600

7

-7

5

-5

3

- 3

2

-2

Write the Equations

1. 2.

3. 4.

y = 7sin2x y = 5cos2x

y = 3cos4x y = 2sin3x

Changing the Scale

Nice for Drawing Graphs y = 4 Sin 6xCycles?Period

6360 ÷ 6 = 600

15 30 45 60

4

0

-4

300 600 900 1200

7

0

-7

Not so nice for recognising graphs

Period = 1200

No of Cycles in 360? 360 ÷ 120 = 3y = 7 cos 3x

2400 3600

Find equation of graph below.CyclesMax 7Negative sin

360 ÷ 60 = 6

15 30 45 60

7

0

-7

y = -7sin6x

Remember rules for y = (x – 3 )2 + 5

Same rules for trig graphs!

3 units to right Up 5

Extra Trig Graph Rules

90 180 270 360

4

0

-4

Y = 4cos (x – 450)

450

Y = 4cosx 450 to right

Sketch Normal Graph

Move each point right/left

y =4cos(x – 450)

90 180 270 360

11

0

-11

Recognising Sin Graph300 to right

y = 11 sin(x – 300)

300

90 180 270 360

13

0

-13

Recognising Cos Graph200 to left

y = 13 cos(x + 200)

-200

90 180 270 360

11

0

-11

A Bit of ConfusionSin Graph300 to left

y = 11 sin(x + 300)

-300 600

Cos Graph600 to right

y = 11 cos(x – 600)

Both correct

6

-6

y = 6cos(x + 300)

-300

Identify this graph

900 1800 2700 3600

90 180 270 360

1

0

-1

Y = sinx + 2

Y = sinx

2

3

90 180 270 360

4

0

-4

Y = 4cosx + 6

8

12

range = 8

Graph Type

y = 4cosx

2

6

10

-2

Equation?

90 180 270 3600

No Maximum (or minimum)

What about y = Tanx ???Goes to infinity

Cycle completePeriod is 1800

90 180 270 3600

Changing the period

Cycle completeNormal Period is 1800

2 cycles

y = tan2x

90 180 270 3600

y = -Tanx

AlsoCan now use radians!

90 180 270 360

1

0

-1

Y = sinx

π/2 π 3π/2 2π

Trigonometric GraphsFollow all the same rules as other function graphs.

Range is handy for identifying (max – min)

e.g. for y = 7sinx →range = 14

π/2π 2π

2

0

-2

y = 2cos(x – π/4)

4

6

y = 2cosx

Sketch y = 2cos(x – π/4) + 1

y = 2cos(x – π/4) + 1

3π/2

0

-2

-4

Sketch y = 3sin(x + π/4) – 1

Y = 3sinx

2

4

Y = 3sin(x + π/4)

Y = 3sin(x + π/4) – 1

Key Question

2π3π/2ππ/2