WISKUNDE
GRAAD 12
NOG OEFENINGE
  
Funksies en grafieke van die derde graad : antwoorde.
  
MATHEMATICS
GRADE 12
MORE EXERCISES
  
Functions and graphs of the third degree : answers.
  
1.1  Y-afsnit / Y-intercept is (0 ; −12)
1.3  x3 - 3x2 - 13x + 15 = (x + 1)(x2 + 4x  − 12)
                                       = (x + 1)(x + 6)(x − 2)
  X-afsnitte / X-intercepts : (−6 ; 0), (−1 ; 0)
                                    en / and (2 ; 0)
1.4  By draaipunte is f'(x) = 0 / At the turning points
        f'(x) = 0.
        f'(x) = 3x2 + 10x - 8
                 3x2 + 10x - 8 = 0
                 3x2 + 10x - 8 = 0
          x = 3,31 OF / OR x = − 1,31 . . . formule / formula
        f(3,31) = (3,31)3 - 3(3,31)2 - 13(3,31) + 15
                     = −24,63
        f(−1,31) =(-1,31)3 - 3(-1,31)2 - 13(-1,31) + 15
                      = 24,63
      Draaipunte / Turning points (3,31 ; −24,63 ) en / and
       (-1,31 ; 24,63)
1.2  Gebruik die faktorstelling. / Use the factor theorem.
       f(−1) = (−1)3 + 5(−1)2 - 8(−1) − 12
              = −1 + 5 + 8 − 12 = 0
        (x + 1) is 'n faktor / a factor.
  
1.5  
  
1.6.1  −6 < x < −1 en / and  x > 2
1.6.2  x < −4  en / and  x > 0,67
  
  
  
  
2.1  Y-afsnit / Y-intercept is (0 ; 4)
        f(−1) =  (−1)3 − 3x(−1)2 + 4
                 = −1 − 3 + 4 = 0
        x + 1 is 'n faktor. / a factor.
        x3 − 3x2 + 4 = (x + 1)(x2 − 4x  + 4)
                                     = −(x + 1)(x − 2)(x − 2)
  X-afsnitte / X-intercepts : (−1 ; 0), (2 ; 0)
                                    en / and (2 ; 0)
   
   
   
   
   
   
2.2  By draaipunte is f'(x) = 0 / At the turning points
        f'(x) = 0.
        f'(x) = 3x2 − 6x
                 3x2 − 6x = 0
                  3x(x −2) = 0
          x = 0 OF / OR x = 2
        f(0) = (0)3 − 3x(0)2 + 4
                  = 4
        f(2) = (2)3 − 3x(0)2 + 4
               = 0
      Draaipunte / Turning points (0 ; 4 ) en / and
       (2 ; 0)
   
   
2.3  
  
2.4.1  x > −1  en / and  x ≠ 2
2.4.2  0 < x < 2
  
  
  
  
3.1  Y-afsnit / Y-intercept is (0 ; 3)
        f(−1) =  −(−1)3 + (−1)2 + 5x(−1) + 3
                 = 1 + 1 − 5 + 3 = 0
        x + 1 is 'n faktor. / a factor.
        −x3 + x2 + 5x + 3 = −(x + 1)(x2 − 2x  − 3)
                                     = −(x + 1)(x + 1)(x − 3)
  X-afsnitte / X-intercepts : (−1 ; 0), (−1 ; 0)
                                    en / and (3 ; 0)
   
   
   
   
   
   
3.2  By draaipunte is f'(x) = 0 / At the turning points
        f'(x) = 0.
        f'(x) = −3x2 + 2x + 5
                 −3x2 + 2x + 5 = 0
                  3x2 − 2x − 5 = 0
                 (3x - 5)(x + 1) = 0
          x = 5/3 OF / OR x = −1
        f(5/3) = −(5/3)3 + (5/3)2 + 5x(5/3) + 3
                  = 9,48
        f(−1) = −(−1)3 + (−1)2 + 5x(−1) + 3
               = 0
      Draaipunte / Turning points (−1 ; 0 ) en / and
       (5/3 ; 9,48)
   
   
2.3  
  
3.4.1  x > 3
3.4.3  −1 < x < 5/3
  
3.4.2  −1 < x < 5/3
  
  
4.1  Y-afsnit / Y-intercept is (0 ; −9)
        f(1) =  −(1)3 + (1)2 + 9x(1) − 9
               = −1 + 1 + 9 − 0 = 0
        x − 1 is 'n faktor. / a factor.
        −x3 + x2 + 9x − 9 = −(x − 1)(x2 − 9)
                                     = −(x − 1)(x + 3)(x − 3)
  X-afsnitte / X-intercepts : (−3 ; 0), (1 ; 0)
                                    en / and (3 ; 0)
   
   
   
   
   
4.2  By draaipunte is f'(x) = 0 / At the turning points
        f'(x) = 0.
        f'(x) = −3x2 + 2x + 9
                 −3x2 + 2x + 9 = 0
                  x = −1,43  OF / OR  x = 2,10     formule
        f(−1,43) = −(−1,43)3 + (−1,43)2 + 9x(−1,43) − 9
                       = −16,90
        f(2,10) = −(2,10)3 + (2,10)2 + 9x(2,10) − 9
                    = 5,05
      Draaipunte / Turning points (−1,43 ; −16,90 )
      en / and (2,10 ; 5,05)
   
4.3  
  
4.4  By buigpunt / At point of inflection f"(x) = 0)
       f"(x) = −6x + 2
        −6x + 2 = 0
                   x = 1/3
            f(1/3) = −(1/3)3 + (1/3)2 + 9x(1/3) − 9
                      = −5,93
  Buigpunt / Point of inflection is (1/3 ; −5,93)
    
4.5.1  x ≤ −3   OF / OR  1 ≤ x ≤ 3
4.5.2  x < −1,43  OF / OR  x > 2,10
4.5.3  −1,43 < x < 2,10
4.5.4  x < 1/3
  
  
  
  
5.1  Y-afsnit / Y-intercept is (0 ; −15)
        C is die punt (0 ; −15).
        f(−1) =  (−1)3 + 3x(−1)2 − 13x(−1) − 15
                 = −1 + 3 + 13 − 15 = 0
        x + 1 is 'n faktor. / a factor.
        x3 + 3x2 − 13x − 15 = (x + 1)(x2 + 2x − 15)
                                          = (x + 1)(x + 5)(x − 3)
        A(−5 ; 0), B(−1 ; 0) en / and D(3 ; 0)
        
   
   
   
   
5.2  By draaipunte P en Q is f'(x) = 0 / At the turning points
        P and Q f'(x) = 0.
        f'(x) = 3x2 + 6x − 13
                 −3x2 + 6x − 13 = 0
                  x = −3,31  OF / OR  x = 1,31     formule
        f(−3,31) = (−3,31)3 + 3x(−3,31)2 − 13x(−3,31) − 15
                       = 24,63
        f(1,31) = (1,31)3 + 3x(1,31)2 − 13x(1,31) − 15
                    = −24,63
      Draaipunte / Turning points P(−3,31 ; 24,63)
      en / and Q(1,31 ; −24,63)
   
5.3.1  −5 < x < −1  en / and  x > 3
5.3.2  −3.31 < x < 1,31
5.3.3  −3.31 < x < 1,31
   
5.4  f(x) = 0 het 2 ongelyke negatiewe wortels en
       een positiewe wortel.
       f(x) = 0 has 2 unequal negative roots and
       one positive root.
  
5.5  Alle punte skuif 25 eenhede opwaarts sodat
        draaipunt Q nou die punt (1,31 ; −24,63 + 25),
        d.i. (1,31 ; 0,37) is. g(x) sny die X-as in een punt
        en het dus net een reële wortel.
5.5  All points are shifted 25 units upwards so that
        turning point Q now is the point
        (1,31 ; −24,63 + 25), i.e (1,31 ; 0,37).
        g(x) intersects the X-axis in only one place
        and therefore has only one real root.
  
  
6.1  Y-afsnit / Y-intercept is (0 ; −12)
        D is die punt the point (0 ; −12).
        f(−1) =  (−1)3 + 2x(−1)2 − 11x(−1) − 12
                 = −1 + 2 + 11 − 12 = 0
        x + 1 is 'n faktor. / a factor.
        x3 + 3x2 − 13x − 15 = (x + 1)(x2 + x − 12)
                                          = (x + 1)(x + 4)(x − 3)
        A(−4 ; 0), B(−1 ; 0) en / and C(3 ; 0)
        
   
   
   
6.2  By draaipunte P en Q is f'(x) = 0 / At the turning
        points P and Q f'(x) = 0.
        f'(x) = 3x2 + 4x − 11
                  3x2 + 4x − 11 = 0
                  x = −2,69  OF / OR  x = 1,36     formule
        f(−2,69) = (−2,69)3 + 2x(−2,69)2 − 11x(−2,69) − 12
                       = 12,6
        f(1,36) = (1,36)3 + 2x(1,36)2 − 11x(1,36) −12
                    = −20,75
      Draaipunte / Turning points P(−2,69 ; 12,6)
      en / and Q(1,36 ; −20,75)
  
6.3.1    −1 < x < 3  en / and  x < −4
6.3.2    x < −2,69  en / and  x > 1,36
6.3.3    x < −4  of / or  x > 1,36
6.3.4  Buigpunt as / point of inflection if f"(x) = 0
                     f"(x) = 6x + 4
                        6x + 4 = 0
                        x = −2/3
          f(x) is konkaaf na onder as / concave down
         if x < −2/3
   
6.4  f(x) = 0 het 3 ongelyke reële wortels, 2 negatief
       en een positief.
       f(x) = 0 has 3 real unequal roots, 2 negative and
       one positive.
  
6.5  P word nou die punt (−2,69 ; 0).
        P now becomes the point (−2,69 ; 0).
        g(x) = 0 het dus 2 gelyke negatiewe wortels en
        een positiewe wortel.
        g(x) = 0 thus has 2 equal negative roots and
        one positive root.
  
  
7.1  Die X-afsnitte is / The X-intercepts are (−2 ; 0),
        (3 ; 0) en/and (6 ; 0)
        Dus/Thus f(x) = p(x + 2)(x − 3)(x − 6)
                               = p(x + 2)(x2 − 9x + 18)
                               = px3 − 7px2 + 36
       By Y-afsnit / At Y-intercept 
             p(0 + 2)(0 - 3)(0 - 6) = 36
                                        36p = 36
                                            p = 1
        a = 1, b = −7p = −7, c = 0 en d = 36
        f(x) = x3 − 7x2 + 36
   
7.2  By draaipunte is f'(x) = 0 / At the turning
        points f'(x) = 0.
        f'(x) = 3x2 − 14x
                  3x2 − 14x = 0
                  x(3x − 14) = 0
                  x = 0  of/or  x =14/3
        f(14/3) = (14/3)3 − 7x(14/3)2 + 36
                    = −14,81
      B is die punt / is the point (4,67 ; −14,81)
  
  
  
8.1  Die X-afsnitte is / The X-intercepts are (−5 ; 0),
        (1 ; 0) en/and (4 ; 0)
        Dus/Thus f(x) = p(x + 5)(x − 1)(x − 4)
                               = p(x − 1)(x2 + x − 20)
                               = px3 − 21px + 20p
       By / At D    :   p(−4)3 − 21p(−4) + 20p = 40
                                                                40p = 40
                                                                     p = 1
        a = p = 1, b = 0, c = −21 en/and d = 26
        f(x) = x3 − 21x + 40
   
8.2  By draaipunte is f'(x) = 0 / At the turning
        points f'(x) = 0.
        f'(x) = 3x2 − 21
                  3x2 − 21 = 0
                             x2 = 7
                               x = ±√7
        f'(x) > 0 as/if  x < −√7 of/or  x > +√7
  
  
  
  
  
8.3  f(x) het 'n lokale maksimum by x = −√7 / f(x) has a local maximum at x = −√7
        f(x) = (−√7)3 − 21(−√7) + 40
              = 57,04
        Die lokale maksimum is die punt (−√7 ; 57,04) / The local maximum is the point (−√7 ; 57,04)
  
9.1  Die X-afsnitte is / The X-intercepts are (−4 ; 0),
        (−4 ; 0) en/and (3 ; 0)
        Dus/Thus f(x) = p(x + 4)(x + 2)(x − 3)
                               = p(x − 1)(x2 + x − 20)
                               = px3 + 3px2 − 10px − 24p
       By Y-afsnit / At Y-intercept    :  
                                     p(0 + 4)(0 + 2)(0 − 3) = 24
                                                                −24p = 24
                                                                       p = −1
        a = p = −1, b = −3, c = 10 en/and d = 24
        f(x) = −x3 − 3x2 + 10x + 24
   
9.2.1   f(x) < 0 as/if −4 < x < −2 en/and x > 3
9.2.2  By draaipunte is f'(x) = 0 /
          At the turning points f'(x) = 0.
           f'(x) = −3x2 − 6x + 10
                  −3x2 − 6x + 10 = 0
                             x = −3,08 of/or x = 1,08   . . . formule
              f'(x) < 0 as / if  x < −3,08 of / or   x > 1,08
  
  
  
  
  
  
10.1  P is 'n punt op f(x) en dus is f(−2) = 14 en
          f'(−2) = 0
          f(−2) = (−2)3 + b(−2)2 c(−2) + d = 14
                   = −8 + 4b − 2c + d
                   4b − 2c + d = 22
        Maar / But A(0 ; 6) is die Y-afsnit / the Y-intercept
        en dus / and therefore d = 6.
        Dus / Therefore  4b − 2c = 16   . . . (1)
        f'(x) = 3x2 + 2bx + c
        f'(−2) = 3(−2)2 + 2b(−2) + c
                        12 − 4b + c = 0
                               −4b + c = −12. . . (2)
      (1) + (2) :                  −c = 4
                                          c = −4
      In / Into (1)  :    4b − 2(−4) = 16
                                             4b = 8
                                               b = 2
       b = 2,  c = −4  en / and  d = 6
       f(x) = x3 + 2x2 − 4x + 6
   
  
  
  
  
  
10.2.1   f(x) < 0 as/if x < −3,58
10.2.2  By draaipunte is f'(x) = 0 /
             At the turning points f'(x) = 0.
             f'(x) = 3x2 + 4x − 4
                        3x2 + 4x − 4 = 0
                     (3x − 2)(x + 2) = 0
                             x = 2/3   of/or   x = −2
              f'(x) < 0 as / if   −2 < x < 0,67
  
10.3  Bepaal die koördinate van die tweede
          draaipunt Q. / Determine the coordinates of
          the second turning point Q.
          f(2/3) = (2/3)3 + 2(2/3)2 − 4(2/3) + 6
                    = 4,52
          As draaipunt Q 'n punt op die X-as is, sal f(x)
          positiewe wortels besit. Dus as h ≤ 4,52,
          sal g(x) = 0 positiewe wortels besit.
          As h < −6, sal g(x) = 0 net een
          positiewe wortel besit.
  
          If turning point Q is a point on the X-axis, f(x)
          will have positive roots. Thus if h ≤ 4,52,
          g(x) = 0 will have positive roots. If h < −6,
          g(x) = 0 will have only one positive root.
  
11.1  NB 3/11 stel die breuk 3 gedeel deur 11 voor.
          By draaipunte P en Q is f'(x) = 0.
          f'(x) = 3ax2 + 2bx + c
          By P is / At P f(−1) = 24 en/and f'(−1) = 0
          f(−1) = a(−1)3 + b(−1)2 + c(−1) + d
                               −a + b − c + d = 24  . . . (1)
          f'(−1) = 3a(−1)2 + 2b(−1) + c
                                     3a − 2b + c = 0  . . . (2)
  
  
11.1  NB. 3/11 represents the fraction 3 divided by 11.
          At turning points P and Q f'(x) = 0.
          f'(x) = 3ax2 + 2bx + c
          By Q is / At Q f(11/3) = 2020/27 and f'(11/3) = 0
          f(11/3) = a(11/3)3 + b(11/3)2 + c(11/3) + d
          (1331/27)a + (121/9)b + (11/3)c + d = (2020/27)
            1331a + 363b + 99c + 27d = 2020  . . . (3)
          f'(11/3) = 3a(11/3)2 + 2b(11/3) + c
                               121a − 22b + 3c = 0  . . . (4)
  
  (1) X 27 :      −27a  + 27b − 27c + 27d = 648  . . . (5)
  (3) − (5) :        1358a  + 336b + 126c = 1372  . . . (6)
  (2) X 126 :              378a  − 252b + 126c = 0  . . . (7)
  (6) − (7) :                       980a  + 558b = 1372  . . . (8)
  (4) X 42 :              5082a  + 924b + 126c = 0  . . . (9)
  (6) − (7) :                  3724a  + 558b = −1372  . . . (10)
  (6) − (7) :                  2744a  = −2744
                                             a = −1
  In / Into (8) :        980(−1) + 588b = 1372
                                             b = 2352/588 = 4
  In / Into (2) :        (−1) − 2(4) + c = 0
                                                    c = 11
  In / Into (1) :        −(−1) + 4 − 11 + d = 24
                                                    d = 30
      a = −1,   b = 4,   c = 11,   d = 30   :  f(x) = −x3 + 4x2 + 11x + 30
  
11.2  Y-afsnit / Y-intercept is (0 ; 30).
11.3.1   f'(x) > 0 as / if x < −1  of/or  x > 11/3
  
  
11.3.2   f(x) is stygend as / rising if −1 < x < 11/3
  
12.1  NB 7/3 stel die breuk 7 gedeel deur 3 voor.
          By draaipunte A en B is f'(x) = 0.
          f'(x) = 3ax2 + 2bx + c
          By A is / At A f(1) =1 en/and f'(1) = 0
          f(1) = a(1)3 + b(1)2 + c(1) + d
                               a + b + c + d = 1  . . . (1)
          f'(1) = 3a(1)2 + 2b(1) + c
                                     3a + 2b + c = 0  . . . (2)
  
  
12.1  NB. 7/3 represents the fraction 7 divided by 3.
          At turning points A and B f'(x) = 0.
          f'(x) = 3ax2 + 2bx + c
          By B is / At B f(7/3) = 59/27 and f'(7/3) = 0
          f(7/3) = a(7/3)3 + b(7/3)2 + c(7/3) + d
          (343/27)a + (49/9)b + (7/3)c + d = (59/27)
            343a + 147b + 63c + 27d = 59  . . . (3)
          f'(7/3) = 3a(7/3)2 + 2b(7/3) + c
                               49a + 14b + 3c = 0  . . . (4)
  
  (1) X 27 :      27a  + 27b + 27c + 27d = 27  . . . (5)
  (3) − (5) :             316a  + 120b + 36c = 32  . . . (6)
  (2) X 36 :                 108a  + 72b + 36c = 0  . . . (7)
  (6) − (7) :                          208a  + 48b = 32  . . . (8)
  (4) X 12 :               588a  + 168b + 36c = 0  . . . (9)
  (9) − (6) :                     272a  + 48b = −32  . . . (10)
  (10) − (8) :                     64a  = −64
                                               a = −1
  In / Into (8) :        208(−1) + 48b = 32
                                                    b = 240/78 = 5
  In / Into (2) :        3(−1) + 2(5) + c = 0
                                                       c = −7
  In / Into (1) :        (−1) + 5 + (−7) + d = 1
                                                           d = 4
      a = −1,   b = 5,   c = −7,   d = 4   :  f(x) = −x3 + 5x2 − 7x + 4
  
12.2  Y-afsnit / Y-intercept is (0 ; 4).
12.3.1   f'(x) < 0 as / if x < 1  of/or  x > 7/3
  
  
11.3.2   f(x) is stygend as / rising if 1 < x < 7/3
  
  
Na bo Oefeninge - Graad 12 Oefeninge - Graad 10 Oefeninge - Graad 11 Tuisblad
  
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