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

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									                  GCSE MATHEMATICS Higher Tier, topic sheet.
                      TRANSFORMATION OF GRAPHS
                          WRITE YOUR ANSWERS ON THESE SHEETS


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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Answers.

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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