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Regions In The Plane
To draw a region, first you have to sketch the curve.
Regions In The Plane
To draw a region, first you have to sketch the curve.
1) If < or > use a dotted line
Regions In The Plane
To draw a region, first you have to sketch the curve.
1) If < or > use a dotted line
2) If  or  use a solid line
Regions In The Plane
To draw a region, first you have to sketch the curve.
1) If < or > use a dotted line
2) If  or  use a solid line
3) Circle the point of intersection if it is NOT included (i.e. one is < or >)
Regions In The Plane
To draw a region, first you have to sketch the curve.
1) If < or > use a dotted line
2) If  or  use a solid line
3) Circle the point of intersection if it is NOT included (i.e. one is < or >)
To calculate the region, test a point NOT on the curve.
Regions In The Plane
To draw a region, first you have to sketch the curve.
1) If < or > use a dotted line
2) If  or  use a solid line
3) Circle the point of intersection if it is NOT included (i.e. one is < or >)
To calculate the region, test a point NOT on the curve.
e.g. (i ) y  x  3
Regions In The Plane
To draw a region, first you have to sketch the curve.
1) If < or > use a dotted line
2) If  or  use a solid line
3) Circle the point of intersection if it is NOT included (i.e. one is < or >)
To calculate the region, test a point NOT on the curve.
e.g. (i ) y  x  3
                                            y        y  x3
                                              3

                                             –3              x
Regions In The Plane
To draw a region, first you have to sketch the curve.
1) If < or > use a dotted line
2) If  or  use a solid line
3) Circle the point of intersection if it is NOT included (i.e. one is < or >)
To calculate the region, test a point NOT on the curve.
e.g. (i ) y  x  3
                                            y        y  x3
          test (0,0)
             03                              3

                                             –3              x
Regions In The Plane
To draw a region, first you have to sketch the curve.
1) If < or > use a dotted line
2) If  or  use a solid line
3) Circle the point of intersection if it is NOT included (i.e. one is < or >)
To calculate the region, test a point NOT on the curve.
e.g. (i ) y  x  3
                                            y        y  x3
          test (0,0)
             03                              3

                                             –3              x
(ii ) x 2  y 2  9
(ii ) x 2  y 2  9        y

                            3
                                x2  y 2  9




                      –3              3        x




                           –3
(ii ) x 2  y 2  9        y
    test (0,0)
                            3
    02  02  9                 x2  y 2  9
          09


                      –3              3        x




                           –3
(ii ) x 2  y 2  9        y
    test (0,0)
                            3
    02  02  9                 x2  y 2  9
          09


                      –3              3        x




                           –3
(ii ) y  4  x 2
(ii ) y  4  x 2        y

                         2   y  4  x2




                    –2           2        x
(ii ) y  4  x 2        y
      test (0,0)
    0  40    2         2   y  4  x2
    02


                    –2           2        x
(ii ) y  4  x 2        y
      test (0,0)
    0  40    2         2   y  4  x2
    02


                    –2           2        x
(ii ) y  4  x 2        y
      test (0,0)
    0  40    2         2   y  4  x2
    02


                    –2           2        x
(iv) y  x 2 and y  3 x  4
(iv) y  x 2 and y  3 x  4   y
                                   y  x2




                                            x
(iv) y  x 2 and y  3 x  4   y
                                   y  x2
 y  x2
test (0,1)
  0  12
  0 1
                                            x
(iv) y  x 2 and y  3 x  4   y
                                   y  x2
 y  x2
test (0,1)
  0  12
  0 1
                                            x
(iv) y  x 2 and y  3 x  4   y
                                   y  x2
 y  x2
test (0,1)                            y  3x  4
  0  12
  0 1
                                            x
(iv) y  x 2 and y  3 x  4   y
                                   y  x2
 y  x2         y  3x  4
test (0,1)      test (0,0)            y  3x  4
  0  12        0 04
  0 1          04

                                            x
(iv) y  x 2 and y  3 x  4   y
                                   y  x2
 y  x2         y  3x  4
test (0,1)      test (0,0)            y  3x  4
  0  12        0 04
  0 1          04

                                            x
(iv) y  x 2 and y  3 x  4   y
                                   y  x2
 y  x2          y  3x  4
test (0,1)       test (0,0)           y  3x  4
  0  12         0 04
  0 1           04
   point of intersection                    x
      x 2  3x  4
(iv) y  x 2 and y  3 x  4    y
                                    y  x2
 y  x2            y  3x  4
test (0,1)         test (0,0)          y  3x  4
  0  12           0 04
  0 1             04
   point of intersection                     x
      x 2  3x  4
      x 2  3x  4  0
    x  1 x  4   0
     x  1 or x  4
(iv) y  x 2 and y  3 x  4             y
                                                y  x2
 y  x2            y  3x  4
test (0,1)         test (0,0)                         y  3x  4
  0  12           0 04                    (4,16)
  0 1             04          (1,1)
   point of intersection                                   x
      x 2  3x  4
      x 2  3x  4  0
    x  1 x  4   0
     x  1 or x  4
(iv) y  x 2 and y  3 x  4                 y
                                                           y  x2
 y  x2            y  3x  4
test (0,1)         test (0,0)                                    y  3x  4
  0  12           0 04                               (4,16)
  0 1             04              (1,1)
   point of intersection                                              x
      x 2  3x  4
      x 2  3x  4  0
    x  1 x  4   0
     x  1 or x  4



   Exercise 3F; 3ac, 4d, 5cg, 6cd, 7, 10, 12a, 15a, 19, 20, 21a, 23*
(iv) y  x 2 and y  3 x  4             y
                                                y  x2
 y  x2            y  3x  4
test (0,1)         test (0,0)                         y  3x  4
  0  12           0 04                    (4,16)
  0 1             04          (1,1)
   point of intersection                                   x
      x 2  3x  4
      x 2  3x  4  0
    x  1 x  4   0
     x  1 or x  4

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11X1 T03 05 regions in the plane (2011)

  • 1. Regions In The Plane To draw a region, first you have to sketch the curve.
  • 2. Regions In The Plane To draw a region, first you have to sketch the curve. 1) If < or > use a dotted line
  • 3. Regions In The Plane To draw a region, first you have to sketch the curve. 1) If < or > use a dotted line 2) If  or  use a solid line
  • 4. Regions In The Plane To draw a region, first you have to sketch the curve. 1) If < or > use a dotted line 2) If  or  use a solid line 3) Circle the point of intersection if it is NOT included (i.e. one is < or >)
  • 5. Regions In The Plane To draw a region, first you have to sketch the curve. 1) If < or > use a dotted line 2) If  or  use a solid line 3) Circle the point of intersection if it is NOT included (i.e. one is < or >) To calculate the region, test a point NOT on the curve.
  • 6. Regions In The Plane To draw a region, first you have to sketch the curve. 1) If < or > use a dotted line 2) If  or  use a solid line 3) Circle the point of intersection if it is NOT included (i.e. one is < or >) To calculate the region, test a point NOT on the curve. e.g. (i ) y  x  3
  • 7. Regions In The Plane To draw a region, first you have to sketch the curve. 1) If < or > use a dotted line 2) If  or  use a solid line 3) Circle the point of intersection if it is NOT included (i.e. one is < or >) To calculate the region, test a point NOT on the curve. e.g. (i ) y  x  3 y y  x3 3 –3 x
  • 8. Regions In The Plane To draw a region, first you have to sketch the curve. 1) If < or > use a dotted line 2) If  or  use a solid line 3) Circle the point of intersection if it is NOT included (i.e. one is < or >) To calculate the region, test a point NOT on the curve. e.g. (i ) y  x  3 y y  x3 test (0,0) 03 3 –3 x
  • 9. Regions In The Plane To draw a region, first you have to sketch the curve. 1) If < or > use a dotted line 2) If  or  use a solid line 3) Circle the point of intersection if it is NOT included (i.e. one is < or >) To calculate the region, test a point NOT on the curve. e.g. (i ) y  x  3 y y  x3 test (0,0) 03 3 –3 x
  • 10. (ii ) x 2  y 2  9
  • 11. (ii ) x 2  y 2  9 y 3 x2  y 2  9 –3 3 x –3
  • 12. (ii ) x 2  y 2  9 y test (0,0) 3 02  02  9 x2  y 2  9 09 –3 3 x –3
  • 13. (ii ) x 2  y 2  9 y test (0,0) 3 02  02  9 x2  y 2  9 09 –3 3 x –3
  • 14. (ii ) y  4  x 2
  • 15. (ii ) y  4  x 2 y 2 y  4  x2 –2 2 x
  • 16. (ii ) y  4  x 2 y test (0,0) 0  40 2 2 y  4  x2 02 –2 2 x
  • 17. (ii ) y  4  x 2 y test (0,0) 0  40 2 2 y  4  x2 02 –2 2 x
  • 18. (ii ) y  4  x 2 y test (0,0) 0  40 2 2 y  4  x2 02 –2 2 x
  • 19. (iv) y  x 2 and y  3 x  4
  • 20. (iv) y  x 2 and y  3 x  4 y y  x2 x
  • 21. (iv) y  x 2 and y  3 x  4 y y  x2 y  x2 test (0,1) 0  12 0 1 x
  • 22. (iv) y  x 2 and y  3 x  4 y y  x2 y  x2 test (0,1) 0  12 0 1 x
  • 23. (iv) y  x 2 and y  3 x  4 y y  x2 y  x2 test (0,1) y  3x  4 0  12 0 1 x
  • 24. (iv) y  x 2 and y  3 x  4 y y  x2 y  x2 y  3x  4 test (0,1) test (0,0) y  3x  4 0  12 0 04 0 1 04 x
  • 25. (iv) y  x 2 and y  3 x  4 y y  x2 y  x2 y  3x  4 test (0,1) test (0,0) y  3x  4 0  12 0 04 0 1 04 x
  • 26. (iv) y  x 2 and y  3 x  4 y y  x2 y  x2 y  3x  4 test (0,1) test (0,0) y  3x  4 0  12 0 04 0 1 04 point of intersection x x 2  3x  4
  • 27. (iv) y  x 2 and y  3 x  4 y y  x2 y  x2 y  3x  4 test (0,1) test (0,0) y  3x  4 0  12 0 04 0 1 04 point of intersection x x 2  3x  4 x 2  3x  4  0  x  1 x  4   0 x  1 or x  4
  • 28. (iv) y  x 2 and y  3 x  4 y y  x2 y  x2 y  3x  4 test (0,1) test (0,0) y  3x  4 0  12 0 04 (4,16) 0 1 04 (1,1) point of intersection x x 2  3x  4 x 2  3x  4  0  x  1 x  4   0 x  1 or x  4
  • 29. (iv) y  x 2 and y  3 x  4 y y  x2 y  x2 y  3x  4 test (0,1) test (0,0) y  3x  4 0  12 0 04 (4,16) 0 1 04 (1,1) point of intersection x x 2  3x  4 x 2  3x  4  0  x  1 x  4   0 x  1 or x  4 Exercise 3F; 3ac, 4d, 5cg, 6cd, 7, 10, 12a, 15a, 19, 20, 21a, 23*
  • 30. (iv) y  x 2 and y  3 x  4 y y  x2 y  x2 y  3x  4 test (0,1) test (0,0) y  3x  4 0  12 0 04 (4,16) 0 1 04 (1,1) point of intersection x x 2  3x  4 x 2  3x  4  0  x  1 x  4   0 x  1 or x  4