# GCSE MATHEMATICS Higher Tier, topic sheet. TRANSFORMATION OF GRAPHS by acm31250

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```									                  GCSE MATHEMATICS Higher Tier, topic sheet.
TRANSFORMATION OF GRAPHS

1.   Diagram 1 shows a sketch of y = x².
y

O               x
Diagram 1

Diagrams 2, 3 and 4 show sketches of the graph of y = x² after it has been moved parallel to
the x-axis, or parallel to the y-axis, or parallel to both.

y                             y                                      y

3                                                                  3

O          x                  O 2            x                    O 2               x
Diagram 2                       Diagram 3                           Diagram 4

Write down the equations of the graphs shown in

(a)     Diagram 2,

(b)     Diagram 3,

(c)     Diagram 4.

2.   The diagram shows a sketch of the function y = f (x).                y
The curve cuts the x-axis at (0, 0) and (3, 0).                       6
A (1, 2) is a point on the curve.
4

A (1, 2)
2

0                                    x
–1                1        2       3   4
–2
The function is transformed.
–4
On each grid below, sketch the transformed function.
State the co-ordinates of A on each new curve.                       –6

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y
(a)   y = f (x) – 2
6
A is at (……….. , …………)
4

2

0                   x
–1        1   2   3   4
–2

–4

–6

(b)   y = 2f (x)                    y
6
A is at (……….. , …………)
4

2

0                   x
–1        1   2   3   4
–2

–4

–6

(c)   y = f (2x)                    y
6
A is at (……….. , …………)
4

2

0                   x
–1        1   2   3   4
–2

–4

–6

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3.   The sketch below is of y = x2.
y

x
O

On the axes provided, sketch the following graphs.
y = x2 is shown dotted on each set of axes.

In each case p is a positive integer greater than 1.

(a)   Sketch y = px2
y

x
O

(b)   (i)    Sketch y = x2 + p
y

x
O
(ii)   Write down the coordinates of the point where the graph crosses the y-axis.

(c)   (i)    Sketch y = (x + p)2
y

x
O

(ii)   Write down the coordinates of the point where the graph touches the x-axis.

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4.    The diagram shows the graph of y = cos x for 0°         x     360°
y
3

2

1

0
90º     180º           270º           360º     x

–1

–2

–3

On the axes below sketch the graphs of

(a)   y = 2 cos x for 0°     x       360°                            (b)         y = cos2x for 0°     x    360°
y                                                        y
3                                                        3

2                                                        2

1                                                        1

0                                                        0
90º        180º           270º    360º   x                      90º          180º        270º         360º   x

–1                                                       –1

–2                                                       –2

–3                                                       –3

(c)   y = 2 + cos x for 0°       x    360°      y
3

2

1

0
90º          180º          270º        360º   x

–1

–2

–3
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5.   The diagram shows a sketch of the graph of y = f(x) for – 3 x                     3
y
5
4
3
2
1

–5 –4 –3 –2 –1         1       2    3    4     5x
–1
–2
–3
–4
–5

(a)   Draw each of the graphs with these equations.

(i)    y = f(x) + 2                                           (ii)        y = f(x – 2)
y                                                                       y
5                                                                      5
4                                                                      4
3                                                                      3
2                                                                      2
1                                                                      1

–5 –4 –3 –2 –1       1   2   3   4    5x                        –5 –4 –3 –2 –1              1   2   3   4    5x
–1                                                              –1
–2                                                                  –2
–3                                                                  –3
–4                                                                  –4
–5                                                                  –5

(b)   The graph below is a transformation of the graph of y = f(x).
Write down, in terms of f(x), the equation of the graph.
y
5
4
3
2
1

–5 –4 –3 –2 –1       1   2   3    4   5x
–1
–2
–3
–4
y
–5
5

6.   The diagram shows the graph of y = x2 for 2            x       2.

–5                   0                       5 x

–5
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Each of the graphs below is a transformation of this graph.

Write down the equation of each graph.
y                                                    y
(a)                5                                 (b)                5

–5            0                  5 x                  –5            0             5 x

–5                                                    –5

y
c)                 5

–5           0                  5 x

–5

7.   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

P
1

O                90              180                 270          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.

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(a)   y = sin (x – 45)

y

2

1

O                  90            180           270      360    x

–1

–2

(b)   y = 2sinx

y

2

1

O               90            180           270      360       x

–1

–2

8.   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°)

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y
1

0
0°    90°     180°    270°     360° x
–1

9.     The graph y = x2 is transformed as shown.

y                                         y

y = x2                                                                 Not drawn
accurately

O                x                    (–3,0) O     x

Write down the equation of the transformed graph.

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1.    (a)     y = x2 + 3.
(b)     y = (x 2)2.
(c)     y = (x 2)2 + 3.

2.    (a)     A = (1, 0).
(b)     A = (1, 4).
(c)     A = (0.5, 2).

3.    (b)     (ii)     (0, p).
(c)     (ii)     ( p, 0).

5.    (b)     y=3      f(x).

6.    (a)     y = (x + 3)2.
(b)     y = x2 2.
y = ( 1 x) .
2
(c)           2

7.    (a)     P = (135, 1).
(b)     P = (90, 2).

9.    y = (x + 3)2.

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