WISKUNDE
GRAAD 10
NOG OEFENINGE
Lineêre funksies en hulle grafieke : antwoorde.
MATHEMATICS
GRADE 10
MORE EXERCISES
Linear functions and their graphs : answers.
f : y = mx + c
By / At B(0;4) : 4 = m(0) + c
4 = c
y = mx + 4
By / At A(−2;0) : 0 = m(−2) + 4
2m = 4
m = 2
y = 2x + 4
g : y = px + q
By / At C(0;−4) : −4 = p(0) + q
−4 = q
y = px − 4
By / At D(2;0) : 0 = p(2) − 4
4 = 2p
p = 2
∴ y = 2x − 4
f : y = mx + c
By Y-afsnit / At Y-intercept : −2 = m(0) + c
c = −2
y = mx − 2
By X-afsnit / At X-intercept : 0 = m(−7) − 2
7m = −2
m = −2/7
∴ y = −2x/7 − 2
g : y = px + q
By Y-afsnit / At Y-intercept : 6 = p(0) + q
q = 6
y = px + 6
By X-afsnit / At X-intercept : 0 = p(5) + 6
−5p = 6
p = −1,2
∴ y = −1,2x + 6
f : y = mx + c
By / At A(0;1) : 1 = m(0) + c
c = 1
y = mx + 1
By / At B(2;5) : 5 = m(2) + 1
4 = 2m
m = 2
∴ y = 2x + 1
g : y = px + q
By C (0;8) : 8 = p(0) + q
q = 8
y = px + 8
By D (3;−4) : −4 = p(3) + 8
−12 = 3p
p = −4
∴ y = −4x + 8
f : y = mx + c
By / At B(0;6) : 6 = m(0) + c
c = 6
y = mx + 6
By / At A(−2;0) : 0 = m(−2) + 6
2m = 6
m = 3
∴ y = 3x + 1
g : y = px + q
By B (0;6) : 6 = p(0) + q
q = 6
y = px + 6
By Q (1;1) : 1 = p(1) + 6
−5 = p
∴ y = −5x + 6
3.4.1 g : y = −5x + 6
By / At C(v;0) : 0 = −5(v) + 6
5v = 6
v = 6/5 = 1,2
C is die punt / the point (1,2 ; 0)
3.4.2 g : y = −5x + 6
D en P het dieselfde x-koördinaat, −1 . . . DP || Y-as
D and P have the same x-coordinate, −1 . . .
DP || Y-axis
By / At D (−1;d) : d = −5(−1) + 6
d = 11
D is die punt / the point (−1 ; 11)
3.4.3 lengte van DP is die verskil in y-koördinate
length of DP is the difference in y-coordinates.
lengte / DP = yD − yP
= 11 − 3
= 8 eenhede / units
f : y = mx + c
By / At P(−2;3) : 3 = m(−2) + c
−2m + c = 3 . . . (1)
By / At Q(4;9) : 9 = m(4) + c
4m + c = 9 . . . (2)
Los op vir m en c / Solve for m and c
(2) − (1) : 6m = 6
m = 1
In / Into (2) : 4(1) + c = 9
c = 5
∴ y = x + 5
Y-afsnit / Y-intercept is 5
∴ B(0 ; 5)
X-afsnit / X-intercept ;
: 0 = x + 5
;
x = −5
∴ A(−5 ; 0)
g : y = px + q
By / At R(−1;9) : 9 = p(−1) + q
−p + q = 9 . . . (1)
By / At S(3;−3) : −3 = p(3) + q
3p + q = −3 . . . (2)
(2) − (1) : 4p = −12
p = −3
In / Into (1) : −(−3) + q = 9
q = 6
∴ y = −3x + 6
Y-afsnit / Y-intercept is 6
∴ C(0 ; 6)
X-afsnit / X-intercept ;
: 0 = −3x + 6
;
x = 2
∴ D(2 ; 0)
3.5.2 AD = xD − xA
= 2 − (−5)
= 7
CB = yC − yB
= 6 − 5
= 1
3.6.1 f : y = mx + c
By / At P(−1;−11) : −11 = m(−1) + c
−m + c = −11 . . . (1)
By / At Q(2;−2) : −2 = m(2) + c
2m + c = −2 . . . (2)
(2) − (1) : 3m = 9
m = 3
In / Into (1) : −(3) + c = −11
c = −8
∴ y = 3x − 8
Y-afsnit / Y-intercept is −8
∴ A(0 ; −8)
X-afsnit / X-intercept ;
: 0 = 3x − 8
;
x = 8/3
∴ B(8/3 ; 0)
g : y = px + q
By / At R(−3;5) : 5 = p(−3) + q
−3p + q = 5 . . . (1)
By / At S(2;−5) : −5 = p(2) + q
2p + q = −5 . . . (2)
(2) − (1) : 5p = −10
p = −2
In / Into (1) : −3(−2) + q = 5
q = −1
∴ y = −2x − 1
Y-afsnit / Y-intercept is −1
∴ D(0 ; &minus1)
X-afsnit / X-intercept ;
: 0 = −2x − 1
;
x = −1/2
∴ C(−1/2 ; 0)
3.6.2 lengte / QS = yQ − yS
=(−2) − (−5)
= 3
3.6.3 lengte / length TV = yT − yV
8 = (1) − yV
yV = 1 − 8
= −7
xV = xT = 3
V is die punt / the point (3 ; −7)
4.1 y = mx + c
By / At A(0;3)
: 3 = m(0) + c
c = 3
y = mx + 3
By / At B(2;5)
: 5 = m(2) + 3
2 = 2m
m = 1
∴ y = x + 3
Vr / Qu 4
4.2 y = mx + c
By / At C(0;−5)
: −5 = m(0) + c
c = −5
y = mx − 5
By / At D(−3;−11)
: −11 = m(−3) − 5
3m = 6
m = 2
∴ y = 2x − 5
Vr / Qu 4
4.3 y = mx + c
By / At M(−2;24)
: 24 = m(−2) + c
−2m + c = 24
. . . (1)
By / At N(3;−6)
: −6 = m(3) + c
3m + c = −6
. . . (2)
(2) − (1)
: 5m = −30
m = −6
In / Into (1)
:
−2(−6) + c = 24
c = 12
∴ y = −6x + 12
Vr / Qu 4
4.4 y = mx + c
By / At P(−6;2)
: 2 = m(−6) + c
−6m + c = 2
. . . (1)
By / At Q(3;−7)
: −7 = m(3) + c
3m + c = −7
. . . (2)
(2) − (1)
:
9m = −9
m = −1
In / Into (2)
:
3(−1) + c = −7
c = −4
∴ y = −x − 4
Vr / Qu 4
4.5
y = mx + c
4.6
y = mx + c
8
8
2
9
9
2
At M(— ; 0) :
0 = m(—) + c
At P(— ; —) :
—— = m(—) + c
5
5
5
2
2
5
verwyder breuke × met KGV
verwyder breuke × met KGV
remove fractions × LCM
remove fractions × LCM
× 5 :
0 = 8m + 5c . . . (1)
× 10 :
45 = 4m + 10c . . . (1)
8
1
1
8
At N(−1 ; 13) :
13 = −m + c . . . (2)
At Q(−—;− —) :
− — = m(− —) + c
5
2
2
5
(2) × 5 :
65 = − 5m + 5c . . . (3)
× 10 :
− 5 = − 16m + 10c . . . (2)
(1) − (3) :
−65 = 13m
(1) − (2) :
50 = 20m
5
m = − 5
m = ——
2
5
In / Into (2) :
13 = −(−5) + c
In / Into (1) :
45 = 4(——) + 10c
2
13 + 5 = c
(2 × 45) − (4 × 5) = (2 × 10)c
c = 18
20c = 70
7
c = ——
2
5
7
y = − 5x + 18
y = — x + ——
2
2
2y = 5x + 7
Antwoord / Answer 5.1
f(x) : y = mx + c
f(x) : y = mx + c
At P(−5 ; −19) :
At R(−5 ; 53) :
−19 = m(−5) + c
53 = m(−5) + c
−5m + c = −19 . . . (1)
−5m + c = 53 . . . (1)
At Q(8 ; 20) :
At S(4 ; −28) :
20 = m(8) + c
−28 = m(4) + c
8m + c = 20 . . . (2)
4m + c = −28 . . . (2)
(2) − (1) :
13m = 39
(2) − (1) :
9m = −81
m = 3
m = −9
In / Into (2) :
8(3) + c = 20
In / Into (1) :
−5(−9) + c = 53
c = −4
c = 8
∴ y = 3x − 4
∴ y = −9x + 8
Antwoord / Answer 5.2
Antwoord / Answer 5.3
A is die Y-afsnit van f(x) /
C is die Y-afsnit van g(x) /
A is the Y-intercept of f(x)
C is the Y-intercept of g(x)
Y-afsnit / intercept = −4
Y-afsnit / intercept = 8
A is die punt / the point (0 ; −4)
C is die punt / the point (0 ; 8)
B is die X-afsnit van f(x) /
D is die X-afsnit van g(x) /
B is the X-intercept of f(x)
D is the X-intercept of g(x)
X-afsnit / intercept 0 = 3x −4
X-afsnit / intercept 0 = −9x + 8
4
8
x = ——
x = ——
3
9
4
8
B is die punt / the point (— ; 0)
D is die punt / the point (— ; 0)
3
9
Antwoord / Answer 5.4
Antwoord / Answer 5.5
By / At E is f(x) = g(x)
lengte CA d(CA) = verskil in y-koördinate /
3x − 4 = −9x + 8
length of CA d(CA) = difference in y-coordinates
3x + 9x = 8 + 4
12x = 12
d(CA) = yC − yA
x = 1
= 8 − (−4)
In / Into (1) : y = 3(1) − 4
= 12
= − 1
E is die punt / point (1 ; −1)
Antwoord / Answer 5.6
Antwoord / Answer 5.7
5.6.1
d(TV) = yT − yV
5.7.1
f(x) = y = 0 by die X-afsnit / at the X-intercept.
= −9x + 8 − (3x − 4)
4
∴ x = —
= −12x + 12
3
5.6.2
d(TV) = 60 : −12x + 12 = 60
5.7.2
g(x) > 0 as die y-waardes groter is as nul, positief is / if the y-values are greater than 0, i.e. positive.
−12x = 60 − 12
−12x = 48
∴ vir alle punte bokant die X-as. / ∴ for all points above the x-axis.
x = −4
By / At T :
y = −9(−4) + 8
8
∴ x < —
= 44
9
T is die punt / the point (−4 ; 44)
5.7.3
f(x) = g(x) by hulle snypunt, E / at their point of
intersection, E.
By / At V :
y = 3(−4) − 4
= −16
∴ f(x) = g(x) as / if x = 1
V is die punt / the point (−4 ; −16)
5.7.4
f(x) ≥ g(x) as f(x) se y-koördinate groter is as die van g(x) /
if the y-coordinates of f(x) is greater than the y-coordinates of g(x)
∴ f(x) ≥ g(x) as / if x ≥ 1
5.7.5
f(x) × g(x) = 0
as f(x) = 0 of as g(x) = 0 of
as f(x) = g(x) = 0 /
if f(x) = 0 or if g(x) = 0 or
if f(x) = g(x) = 0
8
4
∴ f(x) × g(x) = 0 as / if
x = — of / or x = —
9
3
5.7.6
f(x) × g(x) > 0
as f(x) en g(x) albei positief of negatief is /
if f(x) and g(x) are both positive or negative
∴ f(x) × g(x) > 0 as / if
xD < x < xB
8
4
∴ f(x) × g(x) > 0 as / if
— < x < —
9
3