Transcript Volume of Prisms

Trigonometry Graphs
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Nat 5
Creation of BASIC Trig Graphs
Graphs of the form y = a sin xo
Graphs of the form y = a sin bxo
Graphs of the form y = a sin bxo + c
Phase angle y = a sin(x + b)
Exam Type Questions
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Nat 5
Trig Graphs
Creation of a sine graph
Sine Graph
Creation of a cosine graph
Cosine Graph
Creation of a tan graph
Tan Graph
Let’s investigate
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Graphs
Key Features
Sine Graph
Max value occurs at x = 90
Zeros (Root) at 0, 180o and 360o
Nat 5
o
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Mini value occurs at x = 270o
Key Features
(Period is every 360o)
Maximum value of 1 - AMPLITUDE
Minimum value of -1
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Cosine Graphs
Zeros (Roots) at 90 and 270
Key Features
o
Max value occurs at x = 0o and 360o
Nat 5
Minimum value occurs at x = 180o
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o
Key Features
(Period is 360o)
Maximum value of 1 - AMPLITUDE
Minimum value of -1
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Key Features
Tangent Graphs
Zeros (Roots) at 0 and 180o
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Nat 5
Key Features
(Period is 180o)
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Trig Graphs
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Nat 5
Work through N5 TJ
Ex 16.1 , 16.2 and 16.3
(Page 157)
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Starter
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Nat 5
1.
Factorise the following.
2x2 + 7x + 6
2.
A TV is reduced by 20% to £200.
What was the original price.
Q3. Solve (2x -1)(x -1) = 0
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Sine & Cosine Graph
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Nat 5
Learning Intention
Success Criteria
1. To investigate graphs of
the form
1. Identify the key points for
various trig graphs including
Amplitude
Period
Roots.
y = a sin xo
y = a cos xo
created by Mr. Lafferty
Key Features
Sine Graph
Max value at x = 90
Zeros at 0, 180o and 360o
Nat 5
o
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Minimum value at x = 270o
Key Features
(repeats itself every 360o)
Maximum value of 1
Minimum value of -1
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What effect
does the number
at the front
have on the
Nat 5 graphs ?
Sine Graph
y = 2sinxo
y = 3sinxo
3
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y = sinxo
y = 0.5sinxo
y = -sinxo
2
1
0
90o
180o
270o
360o
-1
-2
-3
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Demo
Sine Graph
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Nat 5
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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Sine Graph
y = 4sinxo
y = sinxo
Nat 5
y = -6sinxo
6
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y = 5sinxo
4
2
0
90o
180o
-2
-4
-6
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270o
360o
Cosine Graphs
Zeros at 90 and 270
Key Features
o
o
Max value at x = 0o and 360o
Nat 5
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Minimum value at x = 180o
Key Features
(repeats itself every 360o)
Maximum value of 1
Minimum value of -1
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What effect
does the number
at the front
have on the
Nat 5 graphs ?
Cosine
y = 2cosx
y = cosxo
y = 3cosxo
3
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o
y = 0.5cosxo
y = -cosxo
2
1
0
90o
180o
270o
360o
-1
-2
-3
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Demo
Cosine Graph
y = 4cosx
y = cosxo
o
Nat 5
y = 6cosxo
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6
y = cosxo
y = -cosxo
4
2
0
90o
180o
-2
-4
-6
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270o
360o
Trig Graphs
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Nat 5
Now try N5 TJ
Ex 16.4
(Page 161)
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Starter
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Nat 5
1.
Calculate y (y + y )
2.
Factorise 4ab + 36x
3.
4 w

w 2w
4
-2
2
6
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5
Trig Graphs
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Nat 5
Learning Intention
Success Criteria
1. To investigate graphs of
the form
1. Identify the key points for
various trig graphs including
Amplitude
Period
Roots.
y = a sin bxo
y = a cos bxo
created by Mr. Lafferty
Period of a Function
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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
Let’s investigate the function
y = sin bx
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What effect
does the number
in front of x
have on the
Nat 5 graphs ?
Sine Graph
y = sin2xo
y = sin4xo
3
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y = sinxo
y = sin0.5xo
2
1
0
90o
180o
270o
360o
-1
-2
-3
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Demo
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Nat 5
Trigonometry Graphs
y = a sin (bx)
How many times
it repeats
itself in 360o
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.
created by Mr. Lafferty
What effect
does the number
at the front
have on the
Nat 5 graphs ?
Cosine
y = cosx
y = cos2xo
y = cos3xo
3
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o
2
1
0
90o
180o
-1
-2
-3
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270o
360o
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Nat 5
Trigonometry Graphs
y = a cos (bx)
How many times
it repeats
itself in 360o
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.
created by Mr. Lafferty
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Nat 5
Trigonometry Graphs
y = a tan (bx)
How many times
it repeats
itself in 180o
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.
created by Mr. Lafferty
Write down
equations for
graphs shown ?
Trig Graph
Combinations
Nat 5
y = 0.5sin2xo
y = 2sin4xo
y = 3sin0.5xo
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3
2
1
0
90o
180o
270o
360o
-1
-2
-3
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Demo
Write down
equations for the
graphs shown?
Cosine
Combinations
Nat 5
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3
y = 1.5cos2xo
y = -2cos2xo
y = 0.5cos4xo
2
1
0
90o
180o
-1
-2
-3
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270o
360o
Trig Graphs
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Nat 5
Now try N5 TJ
Ex 16.5
(Page 163)
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Starter
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Nat 5
1.
Make f the subject of the formula
4
w = +1
f
2.
Sketch the function y = (x + 5) + 1
2
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y = asinxo + b
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Nat 5
Learning Intention
Success Criteria
1. We are learning how to
sketch graphs of the type
y = asinxo + b
1. Identify and sketch the key
points for various trig graphs
including
Amplitude
Period
Roots.
y = acosxo + b
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Write down
equations for
graphs shown ?
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Higher
Demo
y = 0.5sin2x
Trig Graph
o
+ 0.5
o- 1
y
=
2sin4x
Combinations
3
2
1
0
90o
180o
-1
-2
-3
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270o
360o
Write down the
equations for the
graphs shown?
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Higher
DEMO
Trig Graphsy = cos2x + 1
o
o
Combinationsy = -2cos2x - 1
3
2
1
0
90o
180o
-1
-2
-3
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270o
360o
Trig Graphs
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Nat 5
Now try N5 TJ
Ex 16.6
(Page 165)
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Starter
Nat 5
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1.
Make b the subject of the formula
c=
2.
b
a
Use the quadratic formula to solve
x2 + 6x + 2
3.
Sketch the function y = 2sin4x
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Phase Angle
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Nat 5
Learning Intention
Success Criteria
1. To investigate graphs of
the form
y = asin(xo + b)
1. Identify and sketch the key
points for trig graphs of the
form
y = asin(xo + b)
y = acos(xo + b)
y = acos(xo + b)
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Phase Angle
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By how much do we have
to move the standard sine
curve so it fits on the
Nat 5
other sine curve?
y = sin(x + 60)o
1
To the left “+”
60o
-60o
0
90o
180o
-1
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270o
360o
Phase Angle
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By how much do we have
to move the standard sine
curve so it fits on the
Nat 5
other sine curve?
y = sin(x - 45)o
1
0
To the right “-”
45o
45o
90o
180o
270o
360o
-1
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Demo
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Nat 5
Phase Angle
y = sin (x + b)
Moves graph
along x - axis
For c > 0 moves graph to the left along x – axis
For c < 0 moves graph to the right along x – axis
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Phase Angle
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By how much do we have
to move the standard
cosine curve so it fits on
Nat 5 the other cosine curve?
y = cos(x - 70)o
1
0
To the right “-”
70o
o
90o 160 180o
-1
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270o
360o
Phase Angle
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By how much do we have
to move the standard
cosine curve so it fits on
Nat 5 the other cosine curve?
1
0
y = cos(x + 56)o
To the left “+”
56o
34o
90o
180o
-1
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270o
360o
Summary of work So far
Nat 5
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y = a sin (x + b)
For a > 1 stretches graph
in the y-axis direction
For b > 0 moves graph to
the left along x – axis
For 0 < a < 1 compresses graph For b < 0 moves graph to
in the y - axis direction
the right along x – axis
For a - negative flips graph
in the x – axis.
created by Mr. Lafferty
Phase Angle
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Nat 5
Now try N5 TJ
Ex 16.7
(Page 168)
created by Mr. Lafferty