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