Graphs of the form y = a sin x o

created by Mr. Laffertycreated by Mr. Lafferty

Graphs of the form y = a sin xo

Trigonometry Graphs Trigonometry Graphs w

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Graphs of the form y = a sin bxo

Phase angle y = a sin(x + b)

Graphs of the form y = a sin bxo + c

Exam Type Questions

Creation of BASIC Trig Graphs

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Nat 5Trig Graphs Trig Graphs

Let’s investigate Graphs

Creation of a sine graph Sine Graph

Creation of a cosine graph Cosine Graph

Creation of a tan graph Tan Graph


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Nat 5Sine Graph Sine Graph

Key Features

(Period is every 360o)

Maximum value of 1 - AMPLITUDE

Minimum value of -1

Key Features

Zeros (Root) at 0, 180o and 360o

Max value occurs at x = 90o

Mini value occurs at x = 270o


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Cosine Graphs Cosine Graphs

Key Features

Key Features

(Period is 360o)

Maximum value of 1 - AMPLITUDE

Minimum value of -1

Zeros (Roots) at 90o and 270o

Max value occurs at x = 0o and 360o

Minimum value occurs at x = 180o


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Tangent Graphs Tangent Graphs

Key Features

Key Features

(Period is 180o)

Zeros (Roots) at 0 and 180o


Work through N5 TJ

Ex 16.1 , 16.2 and 16.3

(Page 157)

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Trig GraphsTrig Graphs


StarterStarter

2 2x +7x +6

Q3. Solve (2x -1)(x -1) = 0

1. Factorise the following.

2. A TV is reduced by 20% to £200.

What was the original price.

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Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1.1. Identify the key points for Identify the key points for various trig graphs various trig graphs includingincluding

AmplitudeAmplitudePeriodPeriodRoots.Roots.

1. To investigate graphs of the form

y = a sin xo

y = a cos xo

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Sine & Cosine Graph Sine & Cosine Graph


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Nat 5Sine Graph Sine Graph

Key Features

(repeats itself every 360o)

Maximum value of 1

Minimum value of -1

Key Features

Zeros at 0, 180o and 360o

Max value at x = 90o

Minimum value at x = 270o


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Sine Graph Sine Graph

1

2

3

-3

-2

-1

090o 180o 270o 360o

y = sinxo

y = 2sinxo

y = 3sinxo

y = 0.5sinxo

y = -sinxo

What effect does the

number at the front have on the graphs ?

Demo


Sine Graph Sine Graph w

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y = a sin (x)

For a > 1 stretches graph in the y-axis direction

For 0 < a < 1 compresses graph in the y - axis direction

For a negative flips graph in the x – axis.


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2

4

6

-6

-4

-2

090o 180o 270o 360o

y = 5sinxo

y = 4sinxo

y = sinxo

y = -6sinxo


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Cosine Graphs Cosine Graphs

Key Features

Key Features

(repeats itself every 360o)Maximum value of 1

Minimum value of -1

Zeros at 90o and 270o

Max value at x = 0o and 360o

Minimum value at x = 180o


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CosineCosine

1

2

3

-3

-2

-1

090o 180o 270o 360o

y = cosxo

y = 2cosxo

y = 3cosxo

y = 0.5cosxo

y = -cosxo



Demo


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Cosine Graph Cosine Graph

2

4

6

-6

-4

-2

090o 180o 270o 360o

y = cosxo

y = 4cosxo

y = 6cosxo

y = cosxo

y = -cosxo


Now try N5 TJ

Ex 16.4

(Page 161)

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StarterStarter

6

+ 36x

4 w3.

w 2w

4 - 2 5

2

1. Calculate y (y +y )

2. Factorise 4ab

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y = a sin bxo

y = a cos bxo

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Trig Graphs Trig Graphs

1.1. Identify the key points for Identify the key points for various trig graphs various trig graphs includingincluding



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When a pattern repeats itself over and over, it is said to be periodic.

Sine function has a period of 360o

Period of a FunctionPeriod of a Function

Let’s investigate the function

y = sin bx


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1

2

3

-3

-2

-1

090o 180o 270o 360o

y = sinxo

y = sin2xo

y = sin4xo

y = sin0.5xo


number in front of x have on the graphs ?

Demo


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Trigonometry Graphs Trigonometry Graphs

y = a sin (bx)




How many times it repeats

itself in 360o


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CosineCosine

1

2

3

-3

-2

-1

090o 180o 270o 360o

y = cosxo

y = cos2xo

y = cos3xo




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y = a cos (bx)





itself in 360o


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y = a tan (bx)





itself in 180o

y = 0.5sin2xo

y = 2sin4xo

y = -3sin0.5xo


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Trig Graph Trig Graph

1

2

3

-3

-2

-1

090o 180o 270o 360o

Write down equations for

graphs shown ?

Combinations

Demo


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CosineCosine

1

2

3

-3

-2

-1

090o 180o 270o 360o

Write down equations for the graphs shown?

Combinations

y = 1.5cos2xo

y = -2cos2xo

y = 0.5cos4xo


Now try N5 TJ

Ex 16.5

(Page 163)

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StarterStarter

2

f

2. Sketch the f unction y = (x +5) + 1

1. Make f the subject of the formula

4 w = +1

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1. We are learning how to sketch graphs of the type

y = asinxo + b

y = acosxo + b

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y = asinxy = asinxoo + b + b

1.1. Identify and sketch the key Identify and sketch the key points for various trig points for various trig graphs includinggraphs including



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Trig Graph Trig Graph

1

2

3

-3

-2

-1

090o 180o 270o 360o

Write down equations for

graphs shown ?

CombinationsHigher

y = 0.5sin2xo + 0.5

y = 2sin4xo- 1

Demo

DEMO


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1

2

3

-3

-2

-1

090o 180o 270o 360o

Combinationsy = cos2xo + 1

y = -2cos2xo - 1Higher

Write down the equations for the graphs shown?


Now try N5 TJ

Ex 16.6

(Page 165)

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StarterStarter

2

x + 6x + 2

3. Sketch the f unction y = 2sin4x

1. Make b the subject of the formula

b c =

a

2. Use the quadratic formula to solve

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y = asin(xo + b)

y = acos(xo + b)

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Phase AnglePhase Angle

1.1. Identify and sketch the key Identify and sketch the key points for trig graphs of points for trig graphs of the form the form

y = asin(xo + b)

y = acos(xo + b)


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1

-1

0 90o 180o 270o 360o-60o

y = sin(x + 60)o

To the left “+”60o

By how much do we have to move the

standard sine curve so it fits on the other sine

curve?


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1

-1

090o 180o 270o 360o

y = sin(x - 45)o

45o

To the right “-”45o


standard sine curve so it fits on the other sine

curve?

Demo


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Phase Angle Phase Angle

y = sin (x + b)

For c > 0 moves graph to the left along x – axis

For c < 0 moves graph to the right along x – axis

Moves graph along x - axis

70o


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1

-1

090o 180o 270o 360o

y = cos(x - 70)o

160o

To the right “-”


standard cosine curve so it fits on the other

cosine curve?

56o


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1

-1

090o 180o 270o 360o

y = cos(x + 56)o

34o

To the left “+”


standard cosine curve so it fits on the other

cosine curve?


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Summary of work So farSummary of work So far

y = a sin (x + b)

For b > 0 moves graph to the left along x – axis

For b < 0 moves graph to the right along x – axis



For a - negative flips graph in the x – axis.


Now try N5 TJ

Ex 16.7

(Page 168)

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Nat 5


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Graphs of the form y = a sin x o

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