WISKUNDE
GRAAD 10
NOG OEFENINGE
  
Grafieke van hiperbole : antwoorde.
  
MATHEMATICS
GRADE 10
MORE EXERCISES
  
Graphs of hyperbolas : answers.
  
   Die vertikale asimptoot : x = 0
  
   The vertical asymptote : x = 0
  
   Die horisontale asimptoot : q = 0 en dus is y = 0;
  
   The horizontal asymptote : q = 0 and therefore y =0.
  
   a = 4 en dus positief sodat die kromme in
  
   kwadrante I en III lê.
  
   a = 4 and thus positive so that the curve is in
  
   quadrants I and III
  
   x = 0 en y = 0 sodat daar geen afsnitte op
  
   die asse is nie.
  
   x = 0 and y = 0 so that there are no intercepts
  
   on the axes.
  
            
  
   Die vertikale asimptoot : x = 0
  
   The vertical asymptote : x = 0
  
   Die horisontale asimptoot : q = 0 en dus is y = 0;
  
   The horizontal asymptote : q = 0 and therefore y =0.
  
   a = −6 en dus negatief sodat die kromme in
  
   kwadrante II en IV lê.
  
   a = −6 and thus negative so that the curve is in
  
   quadrants II and IV
  
   x = 0 en y = 0 sodat daar geen afsnitte op
  
   die asse is nie.
  
   x = 0 and y = 0 so that there are no intercepts
  
   on the axes.
  
            
  
   Die vertikale asimptoot : x = 0
  
   The vertical asymptote : x = 0
  
   Die horisontale asimptoot : q = 1 en dus is y = 1;
  
   The horizontal asymptote : q = 1 and therefore y = 1.
  
   a = 2 en dus positief sodat die kromme in
  
   kwadrante I en III lê.
  
   a = 2 and thus positive so that the curve is in
  
   quadrants I and III
  
  
                                                  2
   X-afsnit / X-intercept  :  0 = — + 1
                                                  x
   
                                X x  :  0 = 2 + x
  
                                        −2 = x
   
                        X-afsnit / X-intercept is A (−2 ; 0)
   
   
            
  
  
  
  
  
   Die vertikale asimptoot : x = 0
  
   The vertical asymptote : x = 0
  
   Die horisontale asimptoot : q = −3 en dus is y = −3
  
   The horizontal asymptote : q = −3 and therefore y = −3
  
   a = 6 en dus positief sodat die kromme in
  
   kwadrante I en III lê.
  
   a = 6 and thus positive so that the curve is in
  
   quadrants I and III
  
  
                                                   6
   X-afsnit / X-intercept  :   0 = — − 3
                                                   x
   
                                X x  :  0 = 6 − 3x
  
                                        3x = 6
   
                                          x = 2
   
                        X-afsnit / X-intercept is A (2 ; 0)
   
   
            
  
  
  
  
  
  
  
   Die vertikale asimptoot : x = 0
  
   The vertical asymptote : x = 0
  
   Die horisontale asimptoot : q = 5 en dus is y = 5
  
   The horizontal asymptote : q = 5 and therefore y = 5
  
   a = −8 en dus positief sodat die kromme in
  
   kwadrante II en IV lê.
  
   a = −8 and thus positive so that the curve is in
  
   quadrants II and IV
  
  
                                                         8
   X-afsnit / X-intercept  :   0 = 5 − —
                                                         x
   
                                X x  :  0 = 5x − 8
  
                                        5x = 8
   
                                          x = 8/5   = 1,6
   
                        X-afsnit / X-intercept is A (1,6 ; 0)
   
  
            
  
  
  
  
  
  
  
   Die vertikale asimptoot : x = 0
  
   The vertical asymptote : x = 0
  
   Die horisontale asimptoot : q = 4 en dus is y = 4
  
   The horizontal asymptote : q = 4 and therefore y = 4
  
   a = −2 en dus positief sodat die kromme in
  
   kwadrante II en IV lê.
  
   a = −2 and thus positive so that the curve is in
  
   quadrants II and IV
  
  
                                                         2
   X-afsnit / X-intercept  :   0 = 4 − —
                                                         x
   
                                X x  :  0 = 4x − 2
  
                                           2 = 4x
   
                                           x = 2/4   = 0,5
   
                        X-afsnit / X-intercept is A (0,5 ; 0)
   
  
            
  
  
  
  
  
  
  
   Vertikale asimptoot / Vertical asymptote  : x = 0
  
   Horisontale asimptoot / Horizontal asymptote  : y = 0
  
                                                     a
   Vergelyking / Equation  :    y = —
                                                     x
  
                                                      a
                 By / At  A(2;4)  :   4 = —
                                                      2
  
                                   X 2 :   8 = a
  
                                                       8
   Vergelyking / Equation  :    y = —
                                                       x
  
   Horisontale asimptoot / Horizontal asymptote  : x = 0
  
   Vertikale asimptoot / Vertical asymptote  : y = 0
  
                                                     a
   Vergelyking / Equation  :  y = —
                                                     x
  
                                                         a
               By / At  B(−3;4)  :    4 = ——
                                                       −3
  
                                 X −3 :  −12 = a
  
                                                        −12
   Vergelyking / Equation  :    y = ———
                                                           x
  
   Vertikale asimptoot / Vertical asymptote  : x = 0
  
   Horisontale asimptoot / Horizontal asymptote  : y = 1
  
   Horisontale asimptoot / Horizontal asymptote   q = 1
  
                                                       a
   Vergelyking / Equation  :    y = — + 1
                                                       x
  
                                                      a
                 By / At  P(1;3)  :   3 = — + 1
                                                      1
  
                                   X 1 :   3 = a + 1
  
                                              2 = a
  
                                                       2
   Vergelyking / Equation  :    y = — + 1
                                                       x
  
   Vertikale asimptoot / Vertical asymptote  : x = 0
  
   Horisontale asimptoot / Horizontal asymptote  : y = −3
  
   Horisontale asimptoot / Horizontal asymptote   q = −3
  
                                                     a
   Vergelyking / Equation  :  y = — − 3
                                                     x
  
                                                         a
                 By / At  P(1;4)  :    4 = —— − 3
                                                         1
  
                                    X 1 :    4 = a − 3
  
                                                7 = a
  
                                                       7
   Vergelyking / Equation  :    y = — − 3
                                                       x
  
Antwoord / Answer 3.  

  3.1  Vertikale asimptoot / Vertical asymptote  :   x = 0
  
         Horisontale asimptoot / Horizontal asymptote      :   y = 1
                                                                                           q = 1
  
  
                                                           a
  3.2  Vergelyking / Equation  :  y = — + 1
                                                           x
  
                                                               a
                       By / At  P(3;2)  :    2 = —— + 1
                                                               3
  
                                          X 3 :    6 = a + 3
  
                                                      3 = a
  
                                                              3
          Vergelyking / Equation  :    y = — + 1
                                                              x
  
  
                                                           3
  3.3  Vergelyking / Equation  :  y = — + 1
                                                           x
  
                                                              3
         X-afsnit / X-intercept      :   0 = —— + 1
                                                              x
  
                                           X x :   0 = 3 + x
  
                                                    −3 = x
  
         X-afsnit / X-intercept  A is (−3 ; 0)
  
  
  
  
                                                           3
  3.4  Vergelyking / Equation  :  y = — + 1
                                                           x
  
                                                               3
                       By / At  B(b;4)  :    4 = —— + 1
                                                               b
  
                                          X b :  4b = 3 + b
  
                                                    3b = 3
  
                                                      b = 1
  
  
  
                                                           3
  3.5  Vergelyking / Equation  :  y = — + 1
                                                           x
  
                                                                 3
                       By / At  C(−1;c)  :    c = —— + 1
                                                               −1
  
                                                        c = −3 + 1
  
                                                        c = −2
  
  
  
  
  
  3.6  Definisieversameling  /  Domain   { x |  x ∈ ℜ }
  
          Waardeversameling  /  Range   { y |  y ≠ 1;  y ∈ ℜ }
  
Vraag / Question 3.  

  
Antwoord / Answer 4.  

  4.1  Vertikale asimptoot / Vertical asymptote  :   x = 0
  
         Horisontale asimptoot / Horizontal asymptote      :   y = −2
                                                                                           q = −2
  
  
                                                           a
  4.2  Vergelyking / Equation  :  y = — − 2
                                                           x
  
                                                               a
                     By / At  R(2;−4)  :   −4 = —— − 2
                                                               2
  
                                        X 2 :    −8 = a − 4
  
                                                    −4 = a
  
                                                              −4
          Vergelyking / Equation  :    y = —— − 2
                                                                x
  
  
                                                           −4
  4.3  Vergelyking / Equation  :  y = —— − 2
                                                             x
  
                                                              −4
         X-afsnit / X-intercept      :   0 = —— − 2
                                                               x
  
                                           X x :   0 = −4 − 2x
  
                                                    −2 = x
  
         X-afsnit / X-intercept  A is (−2 ; 0)
  
  
  
  
                                                            −4
  4.4  Vergelyking / Equation  :  y = —— − 2
                                                             x
  
                                                            −4
                     By / At  Q(−3;t)  :    t = —— − 2
   
                                      X −3 :  −3t = −4 + 6
  
                                                    3t = −2
  
                                                            −2
                                                      t = ——
                                                              3
  
  
                                                            −4
  4.5  Vergelyking / Equation  :  y = —— − 2
                                                             x
  
                                                                −4
                       By / At  R(r;−4)  :    −4 = —— − 2
                                                                   r
  
                                            X r :   −4r = −4 − 2r
  
                                                      −2r = −4
  
                                                        r = 2
  
  
  
  4.6  Definisieversameling  /  Domain   { x |  x ∈ ℜ }
  
          Waardeversameling  /  Range   { y |  y ≠ −2;  y ∈ ℜ }
  
Vraag / Question 4.  

  
Antwoord / Answer 5.  

  5.1  Vertikale asimptoot / Vertical asymptote  :   x = 0
  
         Horisontale asimptoot / Horizontal asymptote      :   y = 3
                                                                                           q = 3
  
  
                                                           a
  5.2  Vergelyking / Equation  :  y = — + 3
                                                           x
  
                                                              −5
         X-afsnit / X-intercept      :   0 = —— + 3
                                                               x
  
                                           X x :   0 = −5 + 3x
  
                                                    5 = 3x
  
                                                   5
                                                  — = x
                                                   3
  
                                                       5
         X-afsnit / X-intercept  A is (— ; 0)
                                                       3
  
  
                                                            −5
  5.3  Vergelyking / Equation  :  y = —— + 3
                                                             x
  
  
                                                           −5
                  By / At  B(b;−2)  :   −2 = —— + 3
                                                              b
  
  
                               X b :  −2b = −5 + 3b
  
                                         −5b = −5
  
                                              b = 1
  
  
  
  
  
  
                                                            −5
  5.4  Vergelyking / Equation  :  y = —— + 3
                                                             x
  
                                                            −5
                     By / At  C(2;c)  :    c = —— + 3
                                                            2
   
                                       X 2 :  2c = −5 + 6
  
                                                  2c = 1
  
                                                            1
                                                    c = ——
                                                            2
  
  
  
  5.5 .  Definisieversameling  /  Domain  
  
                                           { x |  x ≠ 0;  x ∈ ℜ }
  
             Waardeversameling  /  Range  
  
                                           { y |  y ≠ 3;  y ∈ ℜ }
  
  
  
  
  
  
  
  
  
  
Antwoord / Answer 6.  

  6.1  Vertikale asimptoot / Vertical asymptote  :   x = 0
  
         Horisontale asimptoot / Horizontal asymptote      :   y = −5
                                                                                           q = −5
  
  
                                                            −6
  6.2  Vergelyking / Equation  :  y = —— − 5
                                                             x
  
                                                              −6
         X-afsnit / X-intercept      :   0 = —— − 5
                                                               x
  
                                          X x :   0 = −6 − 5x
  
                                                    5x = −6
  
                                                             −6
                                                      x = ——
                                                               5
  
                                                        −6
         X-afsnit / X-intercept  A is (—— ; 0)
                                                          5
  
  
                                                            −6
  6.3  Vergelyking / Equation  :  y = —— − 5
                                                             x
  
  
                                                            −6
                       By / At  B(b;3)  :   3 = —— − 5
                                                              b
  
  
                                          X b :  3b = −6 − 5b
  
                                                    8b = −6
  
                                                              −3
                                                      b = ——
                                                               4
  
  
  
  
                                                            −6
  6.4  Vergelyking / Equation  :  y = —— − 5
                                                             x
  
                                                            −6
                     By / At  C(2;c)  :    c = —— − 5
                                                            2
   
                                       X 2 :  2c = −6 − 10
  
                                                  2c = −16
  
                                                    c = −8
  
  
  
  6.5 .  Definisieversameling  /  Domain  
  
                                           { x |  x ≠ 0;  x ∈ ℜ }
  
  
             Waardeversameling  /  Range  
  
                                           { y |  y ≠ −5;  y ∈ ℜ }
  
  
  
  
  
  
Antwoord / Answer 7.  

  7.1  Vertikale asimptoot / Vertical asymptote  :   x = 0
  
         Horisontale asimptoot / Horizontal asymptote      :   y = −5
                                                                                           q = −5
  
  
                                                           a
  7.2  Vergelyking / Equation  :  y = — − 5
                                                           x
  
                                                                a
                     By / At  P(0,5;3)  :   3 = —— − 5
                                                              0,5
  
                                       X 0,5 :  1,5 = a − 2,5
  
                                                       4 = a
  
                                                                4
          Vergelyking / Equation  :    y = —— − 5
                                                                x
  
  
  
  
                                                             4
  7.3  Vergelyking / Equation  :  y = —— − 5
                                                             x
  
                                                               4
         X-afsnit / X-intercept      :   0 = —— − 5
                                                               x
  
                                           X x :   0 = 4 − 5x
  
                                                    5x = 4
  
                                                           4
                                                    x = —
                                                           5
  
         X-afsnit / X-intercept  A is (0,8 ; 0)
  
  
                                                             4
  7.4  Vergelyking / Equation  :  y = —— − 5
                                                             x
  
                                                                4
                     By / At  B(b;−3)  :   −3 = —— − 5
                                                                b
   
                                         X b :  −3b = 4 − 5b
  
                                                     2b = 4
  
                                                       b = 2
  
  
                                                            4
  7.5  Vergelyking / Equation  :  y = —— − 5
                                                             x
  
                                                              4
                  By / At  C(−1,6;c)  :   c = —— − 5
                                                           −1,6
  
                                     X −1,6 :  −1,6c = 4 + 8
  
                                                     −1,6c = 12
  
                                                             c = −7,5
  
  
  7.6  Definisieversameling  /  Domain   { x |  x ≠ 0;  x ∈ ℜ }
  
          Waardeversameling  /  Range      { y |  y ≠ −5;  y ∈ ℜ }
  
Vraag / Question 7.  

  
Antwoord / Answer 8.  

  8.1  Vertikale asimptoot / Vertical asymptote  :   x = 0
  
         Horisontale asimptoot / Horizontal asymptote      :   y = −3
                                                                                           q = −3
  
  
                                                           a
  8.2  Vergelyking / Equation  :  y = — − 3
                                                           x
  
                                                                    a
                  By / At  P(4;−3,5)  :   −3,5 = —— − 3
                                                                    4
  
                                          X 4 :    −14 = a − 12
  
                                                        −2 = a
  
                                                                    2
          Vergelyking / Equation  :    y = −  —— − 3
                                                                    x
  
  
  
  
  
  
                                                                2
  8.3  Vergelyking / Equation  :  y = − —— − 3
                                                                x
  
                                                                  2
         X-afsnit / X-intercept      :   0 = − —— − 3
                                                                  x
  
                                           X x :   0 = −2 − 3x
  
                                                    3x = −2
  
                                                            −2
                                                      x = —
                                                             3
  
                                               −2         X-afsnit / X-intercept  A is (—— ; 0)
                                                3
  
  
                                                             2
  8.4  Vergelyking / Equation  :  y = —— − 3
                                                             x
  
                                                            2
                     By / At  B(2;b)  :   b = —— − 3
                                                            2
   
                                                   b = 1 − 3
  
                                                     b = −2
  
                                                         
  
  
                                                            2
  8.5  Vergelyking / Equation  :  y = —— − 3
                                                             x
  
                                                                 2
                     By / At  C(c;−1)  :   −1 = —— − 3
                                                                 c
  
                                           X c :  −c = 2 − 3c
  
                                                     2c = 2
  
                                                             c = 1
  
  
  8.6  Definisieversameling  /  Domain   { x |  x ≠ 0;  x ∈ ℜ }
  
          Waardeversameling  /  Range      { y |  y ≠ −3;  y ∈ ℜ }
  
Vraag / Question 8.  

  
  
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