Graphs of the form y = a sin x o
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Transcript of Graphs of the form y = a sin x o
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Graphs of the form y = a sin xo
Trigonometry Graphs Trigonometry Graphs w
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Nat 5
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
![Page 4: Graphs of the form y = a sin x o](https://reader035.fdocuments.net/reader035/viewer/2022062314/56814178550346895dad6548/html5/thumbnails/4.jpg)
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Nat 5
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
![Page 5: Graphs of the form y = a sin x o](https://reader035.fdocuments.net/reader035/viewer/2022062314/56814178550346895dad6548/html5/thumbnails/5.jpg)
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Nat 5
Tangent Graphs Tangent Graphs
Key Features
Key Features
(Period is 180o)
Zeros (Roots) at 0 and 180o
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Work through N5 TJ
Ex 16.1 , 16.2 and 16.3
(Page 157)
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Trig GraphsTrig Graphs
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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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Nat 5
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
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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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Nat 5
Sine Graph Sine Graph
2
4
6
-6
-4
-2
090o 180o 270o 360o
y = 5sinxo
y = 4sinxo
y = sinxo
y = -6sinxo
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Nat 5
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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Nat 5
CosineCosine
1
2
3
-3
-2
-1
090o 180o 270o 360o
y = cosxo
y = 2cosxo
y = 3cosxo
y = 0.5cosxo
y = -cosxo
What effect does the
number at the front have on the graphs ?
Demo
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Nat 5
Cosine Graph Cosine Graph
2
4
6
-6
-4
-2
090o 180o 270o 360o
y = cosxo
y = 4cosxo
y = 6cosxo
y = cosxo
y = -cosxo
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Now try N5 TJ
Ex 16.4
(Page 161)
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Trig GraphsTrig Graphs
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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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Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To investigate graphs of the form
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
AmplitudeAmplitudePeriodPeriodRoots.Roots.
![Page 19: Graphs of the form y = a sin x o](https://reader035.fdocuments.net/reader035/viewer/2022062314/56814178550346895dad6548/html5/thumbnails/19.jpg)
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Nat 5
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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Nat 5
Sine Graph Sine Graph
1
2
3
-3
-2
-1
090o 180o 270o 360o
y = sinxo
y = sin2xo
y = sin4xo
y = sin0.5xo
What effect does the
number in front of x have on the graphs ?
Demo
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Nat 5
Trigonometry Graphs Trigonometry Graphs
y = a sin (bx)
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.
How many times it repeats
itself in 360o
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Nat 5
CosineCosine
1
2
3
-3
-2
-1
090o 180o 270o 360o
y = cosxo
y = cos2xo
y = cos3xo
What effect does the
number at the front have on the graphs ?
![Page 23: Graphs of the form y = a sin x o](https://reader035.fdocuments.net/reader035/viewer/2022062314/56814178550346895dad6548/html5/thumbnails/23.jpg)
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Nat 5
Trigonometry Graphs Trigonometry Graphs
y = a cos (bx)
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.
How many times it repeats
itself in 360o
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Nat 5
Trigonometry Graphs Trigonometry Graphs
y = a tan (bx)
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.
How many times it repeats
itself in 180o
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y = 0.5sin2xo
y = 2sin4xo
y = -3sin0.5xo
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Nat 5
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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Nat 5
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
![Page 27: Graphs of the form y = a sin x o](https://reader035.fdocuments.net/reader035/viewer/2022062314/56814178550346895dad6548/html5/thumbnails/27.jpg)
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Now try N5 TJ
Ex 16.5
(Page 163)
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Nat 5
Trig GraphsTrig Graphs
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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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Learning IntentionLearning Intention Success CriteriaSuccess Criteria
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
AmplitudeAmplitudePeriodPeriodRoots.Roots.
![Page 30: Graphs of the form y = a sin x o](https://reader035.fdocuments.net/reader035/viewer/2022062314/56814178550346895dad6548/html5/thumbnails/30.jpg)
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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
![Page 31: Graphs of the form y = a sin x o](https://reader035.fdocuments.net/reader035/viewer/2022062314/56814178550346895dad6548/html5/thumbnails/31.jpg)
DEMO
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Trig GraphsTrig Graphs
1
2
3
-3
-2
-1
090o 180o 270o 360o
Combinationsy = cos2xo + 1
y = -2cos2xo - 1Higher
Write down the equations for the graphs shown?
![Page 32: Graphs of the form y = a sin x o](https://reader035.fdocuments.net/reader035/viewer/2022062314/56814178550346895dad6548/html5/thumbnails/32.jpg)
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Now try N5 TJ
Ex 16.6
(Page 165)
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Nat 5
Trig GraphsTrig Graphs
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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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Nat 5
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Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To investigate graphs of the form
y = asin(xo + b)
y = acos(xo + b)
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Nat 5
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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Nat 5
Phase AnglePhase Angle
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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Nat 5
Phase AnglePhase Angle
1
-1
090o 180o 270o 360o
y = sin(x - 45)o
45o
To the right “-”45o
By how much do we have to move the
standard sine curve so it fits on the other sine
curve?
Demo
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Nat 5
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
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70o
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Nat 5
Phase Angle Phase Angle
1
-1
090o 180o 270o 360o
y = cos(x - 70)o
160o
To the right “-”
By how much do we have to move the
standard cosine curve so it fits on the other
cosine curve?
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56o
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Nat 5
Phase Angle Phase Angle
1
-1
090o 180o 270o 360o
y = cos(x + 56)o
34o
To the left “+”
By how much do we have to move the
standard cosine curve so it fits on the other
cosine curve?
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Nat 5
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 > 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.
![Page 41: Graphs of the form y = a sin x o](https://reader035.fdocuments.net/reader035/viewer/2022062314/56814178550346895dad6548/html5/thumbnails/41.jpg)
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Now try N5 TJ
Ex 16.7
(Page 168)
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Nat 5
Phase AnglePhase Angle
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