1.
The sketch below is of the graph of y = x2
y
O
x
On the axes provided, sketch the following graphs.
The graph of y = x2 is shown dotted on each set of axes to act as a guide.
(a)
y = x2 + 2
y
O
x
(1)
(b)
y = (x – 2)2
y
O
x
(1)
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(c)
1
y  x2
2
y
O
x
(1)
(Total 3 marks)
2.
The diagrams, which are not drawn to scale, show the graph of y = x2 and four
other graphs A, B, C and D.
A, B, C and D represent four different transformations of y = x2.
Find the equation of each of the graphs A, B, C and D.
y
Graph A
y = x2
O
Answer
3
x
Graph A is y = .........................
y
Graph B
y = x2
–3
Answer
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O
x
Graph B is y = .........................
2
y
y = x2
O
x
Graph C
Answer
Graph C is y = .........................
y
3
y = x2
O
x
Graph D
Answer
Graph D is y = .........................
(Total 4 marks)
3.
The graph y = x2 is transformed as shown.
y
y
Not drawn
accurately
y = x2
O
x
(–3,0) O
x
Write down the equation of the transformed graph.
Answer y = ....................................................
(1)
(Total 1 mark)
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4.
The diagram shows the graph of y = x2 for 2  x  2.
y
5
–5
5 x
0
–5
Each of the graphs below is a transformation of this graph.
Write down the equation of each graph.
y
5
(a)
–5
y
5
(b)
–5
5 x
0
–5
0
5 x
–5
y
5
–5
0
5 x
(c)
–5
Answer (a) y = ....................................................
(1)
Answer (b) y = ....................................................
(1)
Answer (c) y = ....................................................
(1)
(Total 3 marks)
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4
5.
This is the graph of y = sin x.
y
1
0
0°
90°
180°
270°
360° x
–1
On the grid below, sketch the graph of y = sin (x - 90)
y
1
0
0°
90°
180°
270°
360° x
–1
(Total 2 marks)
6.
This is the graph of y = cos x for 0°  x  360°
y
1
90
O
180
270
x
360
–1
(a)
On the axes below draw the graph of y = cos(x – 90) for 0°  x  360°
y
1
O
–1
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90
180
270
360
x
(2)
5
(b)
Write down a possible equation of the following graph.
y
1
90
O
180
270
x
360
–1
Answer ……………………………………………
(1)
(Total 3 marks)
7.
This is the graph of y = cos x for 0°  x  360°
y
2
1
0
90
180
270
360 x
–1
–2
Write the equation of each of the transformed graphs.
In each case the graph of y = cos x is shown dotted to help you.
(a)
y
2
1
0
90
180
270
360 x
–1
–2
Equation y = ..............................................................
(1)
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(b)
y
2
1
0
90
180
270
360 x
–1
–2
Equation y = ..............................................................
(1)
(c)
y
2
1
0
90
180
270
360 x
–1
–2
Equation y = ..............................................................
(1)
(d)
y
2
1
0
90
180
270
360 x
–1
–2
Equation y = ..............................................................
(1)
(Total 4 marks)
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8.
The diagram shows the graph of y = sin x° for 0 < x < 360
y
3
2
1
0
90
180
270
360 x
270
360 x
–1
–2
–3
On the axes below sketch the following graphs.
(a)
y = 2 sin x° for 0 < x < 360
y
3
2
1
0
90
180
–1
–2
–3
(1)
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(b)
y = sin 2x° for 0 < x < 360
y
3
2
1
0
90
180
270
360 x
–1
–2
–3
(1)
(c)
y = 2 + sin x° for 0 < x < 360
y
3
2
1
0
90
180
270
360 x
–1
–2
–3
(1)
(Total 3 marks)
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9.
The graph of y = sin x for 0°  x  360° is shown on the grid below.
The point P(90, 1) lies on the curve.
y
2
1
O
P
90
180
270
360
x
360
x
–1
–2
On both of the grids that follow, sketch the graph of the transformed function.
In both cases write down the coordinates of the transformed point P.
(a)
y = sin (x – 45)
y
2
1
O
90
180
270
–1
–2
P (......................., ......................)
(2)
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10
(b)
y = 2sinx
y
2
1
O
90
180
270
360
x
–1
–2
P (......................., ......................)
(2)
(Total 4 marks)
10.
This is the graph of y = sin x for 0°  x  360°
y
3
2
1
0
x
90
180
270
360
–1
–2
–3
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11
Draw the graphs indicated for 0°  x  360°
In each case the graph of y = sinx is shown to help you.
(a)
y = 2 sinx
y
3
2
1
0
x
90
180
270
360
–1
–2
–3
(1)
(b)
y = – sinx
y
3
2
1
0
x
90
180
270
360
–1
–2
–3
(1)
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(c)
y = sin 2x
y
3
2
1
0
x
90
180
270
360
–1
–2
–3
(1)
(Total 3 marks)
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13
1.
(a)
Parallel curve translated up y axis
‘2’ need not be marked, needs to look
symmetrical
B1
(b)
Parallel curve translated in positive direction along x axis
Must 'sit on' x axis and look
symmetrical
B1
(c)
Curve through (0,0) nearer to x axis than original
Must look symmetrical
B1
[3]
2.
Graph A is y = (x – 3)2
Graph B is y (x + 3)2
Graph C is y = –x2
Graph D is y = 3 – x2
B1
B1
B1
B1
[4]
3.
(a)
(x + 3)2
B1
[1]
4.
(a)
y = (x + 3)2
B1
(b)
y = x2 – 2
B1
(c)
y = 0.5x2
B1
[3]
5.
Curve through (0,– 1) (90,0)
(180,1) (270,0) (360,– 1)
B1 if any 180° span correct
B2
[2]
6.
(a)
attempt at translation of graph
 
B1
9 0
0
accurate ie, through correct points
B1
Must be 0° to 360°
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(b)
(y =) – sin × or – cos (x – 90)
or cos (x + 90)
B1
[3]
7.
(a)
y = cosx + 1
B1
y = 1 + cosx
(b)
y = 2cosx
B1
(c)
y = cos2x
B1
(d)
y = cos(90 – x), y = cos (x +270)
y = cos (x – 90) or y = sin x
B1
[4]
8.
(a)
Correct sketch
B1
(b)
Correct sketch
B1
(c)
Correct sketch
B1
[3]
9.
(a)
(b)
Attempt at translation of 45° to the right
M1
P = (135,1)
A1
Attempt at sine curve of twice the amplitude of the original
M1
P = (90, 2)
A1
[4]
10.
(a)
curve through (0,0) (90,2) (180,0) (270,-2) (360,0)
B1
(b)
curve through (0,0) (90,–1) (180,0) (270,1) (360,0)
B1
(c)
curve through (0,0) (45,1) (90,0)
(135,–1) (180,0) (225,1) (270,0) (315,–1) (360,0)
B1
[3]
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