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1 1
1
2
3
19
22
1.1 31
1.2 38
1.3.1 43
1.3.2 (1) 58
1.3.2 (2) 74
1.3.3 78
1.4 82
1.5 87
1.6 92
2 94
94
95
96
98
102
2.1 108
2.2 124
1
( 28 )
1.
2.
3.
4.
2
1.
2. r x
y
r
r =2x
1yRRy,x
Dr = 2xx
Rr = 0yy
Rr = 0xx
x y
x y
“ ”
x, y z
y
3.
y y x x y 2
x
y x
Y
4.
2
3
1. 4 2
4 2
4 2 2
2 4
“ ” “ ” “ ”
“ ” “ ”
“ ” “ ”
2. x y x > y, y = x + 1, y = x2
(x, y) (x, y) x > y x, y
10 y = x + 1
x, y
1.
r1 = {(– 4, 2), (–2, –1), (0, –1), (6, 3)}
r2 = {(3, 5), (5, 3), (–1, 0), (3, 0)}
r1 r2
– 4
– 2
0
6
2
–1
3
r1
3
5
–1
5
3
0
r2
3
4
r1
r2
r1
r2
r1
r2
2.
r = {(x, y) y2 = x + 1}
r (
r1 (3, –2) (3, 2) r1
(3, 2) (3, –2) r
r
3.
Y Y
x
y Y
4
5
r
y x
x y y = 2x
1
x = 2 y ( 0) x x 2 y
2
{x x 2} x R
y = 2x
1
y 0 ( )
0
{y y 0}
x y y
x
y =2x
1
x – 2 =y
1
x = 2y
1
5
6
y 0 x
{y y 0}
y x (x, y)
(1) r = {(x, y) R R y = x} (x R, y R)
(2) r = {(x, y) R R y = –3} (x R, y = –3)
(3) r = {(x, y) R R y = 5x – 2} (x R, y = R)
(4) r = {(x, y) R R y = 7 – 2x} (x R, y = R)
(5) r = {(x, y) R R y = x2} (x R, y 0)
(6) r = {(x, y) R R y = (x – 3)2} (x R, y 0)
(7) r = {(x, y) R R y = x2 – 8} (x R, y –8)
(8) r = {(x, y) R R y = x } (x R, y 0)
(9) r = {(x, y) R R y = x + 5} (x R, y 5)
(10) r = {(x, y) R R y = x – 7} (x R, y –7)
(11) r = {(x, y) R R y = x
1} (x 0, y 0)
(12) r = {(x, y) R R y = 5x
1} (x 5, y 0)
(13) r = {(x, y) R R y = 5x } (x –5, y 0)
(14) r = {(x, y) R R y = x7 } (x 7, y 0)
(15) r = {(x, y) R R y = 2x } (x R, y 0)
(16) r = {(x, y) R R y = 5x 2 } (x R, y 5 )
(17) r = {(x, y) R R y = 2x7 } ( 7 < x < 7 , 0 < y < 7 )
(18) r = {(x, y) R R y = 2x5 } (x –2, y 5)
6
7
y = x x
1) x
2) x
3) x
x
{x x 0}
y x (x, y) r
r = {(x, y) y = x2, –2 x 2}
r {y 0 y 4}
4 ( .4 – .6)
1. (linear function)
1) X Y
(1) (x, y) 1
x y (x, y) = (1, 1) 1
(x, y) 1 x > 0 y > 0
7
8
(2) 2, 3 4
(x, y)
1
2
3
4
x > 0, y > 0
x < 0, y > 0
x < 0, y < 0
x > 0, y < 0
(3) X Y
X y (x, 0)
Y x (0, y)
2) y = ax + c
(1) y = ax + c y = x y = ax
a 0 a 1 y = ax
y = x
Y
X (1, 1)
Y
X
y = ax, a > 0
0
Y
X
y = ax, a < 0
0
2
2–2–2
8
9
(2) y = x y = x + c
1
y = x y = x + c c > 0
y = x
y = x + 1
y = x + 2
y = x + 3
2
y = x y = x + c c < 0
y = x
y = x – 1
y = x – 2
y = x – 3
Y X
y = x + c c 0 y = x
3) y = ax + c a, c 0
y = ax + c y = x
y = 3x + 1 y
(1) y = x
0
Y
X
y = x
1
2
3
Y
X0
–3
–2
–1
Y
X0
9
10
(2) y = 3x
(3) y = 3x + 1
y = 3x + 1 y = 3x X (3
1, 0) Y
(0, 1)
4) y = ax + c
Y X y = 3x + 1
y = 3x + 1 y = 3x, y = 3x + 2, y = 3x – 1
0
Y
X
y = 3xy = 3x + 1
1
–1
y = x
0
Y
X
y = 3x
10
11
2. (Quadratic function)
y = x2
y = x2
1. y = ax2
1) a > 1 y = 2x2, y = 3x
2, y = 5x
2
y = ax2 a > 1
y = x2
2) 0 < a < 1 y = 2x2
1, y = 2x
3
1, y = 2x
5
1
y = ax2 0 < a < 1
y = x2
Y
X0
Y
X0
Y
X0
11
12
3) y = ax2, a < 0 y = –x
2 y = ax
2
a > 0 a < 0 2
2. y = x2 + c, c > 0
y = x2 + c, c > 0 y = x
2
y = x2
y = x2 + 1
y = x2 + 2
y = x2 + 3
3. y = x2 – c, c > 0
y = x2 – c, c > 0 y = x
2
y = x2
y = x2 – 1
y = x2 – 2
y = x2 – 3
4. y = (x + c)2 , c > 0
y = (x + c)2 , c > 0 y = x
2
y = x2
y = (x + 1)2
y = (x + 2)2
y = (x + 3)2
Y
X0
–3 –2 –1
Y
X0
3
21
X
Y
0
–3–2
–1
Y
X0
12
13
5. y = (x – c)2 , c > 0
y = (x – c)2 , c > 0 y = x
2
y = x2
y = (x – 1)2
y = (x – 2)2
y = (x – 3)2
y = ax2, y = (x – a)
2 y = x
2 + c
y = a(x – h)2 + k
y = 2(x – 1)2 + 2 y = x
2
1) y = x2
2) y = (x – 1)2 y = x
2
1 2 3
Y
X0
Y
X1
y = (x – 1)2
Y
X
y = x2
0
0
13
14
3) y = 2(x – 1)2 y = (x – 1)
2
4) y = 2(x – 1)2 + 1 y = 2(x – 1)
2
y = a(x – h)2 + k
1) a > 0, h > 0, k > 0 y = 3)1x(2
1 2
a > 0, h > 0, k < 0 y = 3(x – 3)2 – 3
a > 0, h < 0, k > 0 y = 2(x + 1)2 + 1
a > 0, h < 0, k < 0 y = (x + 2)2 – 2
2) a < 0, h > 0, k > 0 y = – 3)1x(2
1 2
a < 0, h > 0, k < 0 y = –3(x – 3)2 – 3
a < 0, h < 0, k > 0 y = –2(x + 1)2 + 1
a < 0, h < 0, k < 0 y = –(x + 2)2 – 2
Y
X
Y
X1
1
y = 2(x – 1)2
y = 2(x – 1)2 + 1
10
0
14
15
ax2 + bx + c = 0 a 0
a
ax2 + bx + c = 0 a(x + h)
2 + k = 0 x =
a2
b
y
1. y = ax2
ax2 = 0
a > 0 a < 0
y = 0 x = 0
ax2 = 0 x = 0
2. y = a(x – c)2 y = a(x + c)
2 c > 0
a(x – c)2 = 0 a(x + c)
2 = 0
y = a(x – c)2, c > 0 y = a(x + c)
2, c > 0
c1 c2 c3
Y
X0 –c3 –c2 –c1
Y
X0
Y
X0
Y
X0
a < 0
Y
X0
15
16
a(x – c)2 = 0 x = c
a(x + c)2 = 0 x = –c
3. ax2 + bx + c = 0, a 0
x = a2
ac4bb 2
y = ax2 + bx + c a(x – h)
2 + k
(h, k) a
a > 0 (–h, –k) (0, –k) (h, –k)
(–h, k) (0, k) (h, k) a < 0
a
X
(–h, –k) (h, –k)
Y
a > 0
0
(h, k)
Y
X
(–h, k)
(h, –k)(–h, –k)
a > 0 a > 0
a < 0 a < 0
0
Y
X
(–h, k) (h, k)
a < 0
0
16
17
x + a < 0 x + a > 0
1 x + 3 8
1 x + 3 8
x – 5 0
y = x – 5 y
y x – 5 0 x 5
2 y1 = x + 3
y2 = 8
y1 y2
y1 = y2 y1 y2 x = 5
y1 x + 3 y2 8 x 5
Y
X
–550
y = x – 5
y2 = 8
Y
X3
5–3
y1 = x + 3
0
17
18
2 x2 – 2x – 3 < 0
1) x2
x2
x2 – 2x – 3
x2-2x-3 (x-c)
2 X
2) X
X
x2 – 2x – 3 = 0
(x – 3)(x + 1) = 0 x = –1, 3
1) X (–1, 0) (3, 0)
x2 – 2x – 3 < 0 –1 < x < 3
x, y (x, y)
–1
Y
X30
18
19
1.
1) {(0, 1), (1, –2), (2, 0), (3, 2)}
2) {(0, –1), (1, –2), (1, 1), (2, 2), (3, 0)}
2.
1) 3y = 2x + 4
2) y = 4 – x2
3.
1) f(1) f(0) f(x) = –2x – 7
2) f(0) f(4) f(x) = 3 – x
3) f(–2) f(0) f(x) = x + 4
4) f(–1) f(2) f(x) =
4.
1) 2)
3) 4)
2x , x 0
2x + 1 , x < 0
Y
X
y3
0
Y
X0
y4
y1
Y
X1
0–1
Y
X0
y2
–2–1
19
20
5.
1) f = {(–3, 0), (–1, 4), (0, 2), (2, 2), (4, –1)}
2) y = 5x
1
6. A = {–1, 0, 1}
y
1) y = x
2) y = 1x
7.
1) y = –x2 – 2 2) y = x
2 + 2x + 3 3) y = 2 x
8.
1) y = (x – 4)2 – 3 2) y = –(x – 4)
2 + 3
3) y = (x + 4)2 – 3 4) y = –(x + 4)
2 + 3
( ) ( )
( ) ( )
Y
X
Y
X
Y
X
Y
X
–3
3
–3
3
– 4
4
– 44
0 0
0 0
20
21
9. 120
10. 2 –3
11. 500
125
12.
( )
10 1 – 2
20 3 – 4
30 5 – 7
13.
1) y = –x2 – 2 2) y = x
2 – 4x
14.
1) y = x + 1 2) y = –x2 + 4
15.
1) 2x + 1 < 3 2) x2 + 4x – 5 < 0 3) x – 2 > 0
21
22
1 2
1. 1)
{(0, 1), (1, –2), (2, 0), (3, 2)}
2)
1 1
{(0, –1), (1, –2), (1, 1), (2, 2), (3, 0)}
2. 1) 3y = 2x + 4 y = 3
4x
3
2
y = 3
4x
3
2
Y y =3
4x
3
2 1
3y = 2x + 4
0
1
2
3
–2
0
1
2
0
1
2
3
–2
–1
0
1
2
Y
X
y = 3
4x
3
2
0
22
23
2) y = 4 – x2 y = 4 – x
2
Y y = 4 – x2
1
y = 4 – x2
3. 1) f(x) = –2x – 7
f(1) = –2(1) – 7 = –9
f(0) = –2(0) – 7 = –7
2) f(x) = x3
f(0) = 3 – 0 = 3
f(4) = 3 – 4 = 1
3) f(x) = 4x
f(–2) = –2 + 4 = 6
f(0) = 0 + 4 = 4
4)
f(x) =
f(–1) = 2(–1) + 1 = –1
f(2) = 2(2) = 4
2x, x 0
2x + 1, x < 0
Y
X0
y = 4 – x2
4
2–2
23
24
4. 1)
Y y1 1
y1
y1
y1 {y y –1}
2) Y
y2 1
x y
y2
y2
y2
3)
Y y3 1
y3
y3 {x x R, x –1}
y3 {y y = 2}
4)
Y y4 1 (0, 0)
(0, –2)
y4 x
0 y
y4 {x x 0}
y4 {y y 0 y = –2}
Y
X0
y4
Y
X0
y2
Y
X
y3
0
y1
Y
X10–1
2
–1
–2
24
25
5. 1) f = {(–3, 0), (–1, 4), (0, 2), (2, 2), (4, –1)}
{–3, –1, 0, 2, 4}
{–1, 0, 2, 4}
2) f(x) = 5x
1
{x x R x –5}
{y y R y 0}
6. 1) y = x
r = {(x, y) y = x x A}
r = {(–1, 1), (0, 0), (1, 1)}
2) y = 1x
r = {(x, y) y = 1x x A}
r = {(–1, 0), (0, 1), (1, 2 )}
7. 1) y = –x2 – 2
{y y R y –2}
2) y = x2 + 2x + 3
= (x2 + 2x + 1) + 2
= (x + 1)2 + 2
{y y R y 2}
Y
X0
–2
Y
X0
2
–1
25
26
3) y = 2 x
{y y R y 0}
8. 1) ( ) 2) ( ) 3) ( ) (4) ( )
9. ABCD 120
f(x) ABCD
f(x) = x(60 – x)
f(x) = 60x – x2
y = 60x – x2 x =
a2
b
x = 2
60 = 30 y = 30(60-30) = 900
x = 30 y = 900
120 900
A B
D C
x x
60 – x
60 – x
Y
X0
Y
X0
900
30 60
(30, 900)
26
27
10. x y
x y 2
x – y = 2 y = x – 2
1 xy = –3
x(x – 2) = –3
x2 – 2x + 3 = 0
(x2 – 2x + 1) + 2 = 0
(x – 1)2 + 2 = 0
y = (x – 1)2 + 2
y X
2
–3
2 x y –3
xy = –3 y = x
3
y1 = x – 2 y2 = x
3
Y
X0
2
1
x y1 = x – 2 y2 = x
3
–3 –5 1
–2 – 42
3
–1 –3 3
0 –2
1 –1 –3
2 02
3
3 1 –1
Y
X0
3
2
–2
–3
y1 y2
2 –3
27
28
11. y
x
y = 500 + 125x
12. x
10 , 1 x 2
f(x) = 20 , 3 x 4
30 , 5 x 7
13. 1) y = –x2 – 2 y
(0, –2)
2) y = x2 – 4x y = (x
2 – 4x + 4) – 4 = (x – 2)
2 – 4
y
(2, – 4)
Y
X0
(0, –2)
Y
X0
– 4
2
(2, – 4)
28
29
14. 1) x + 1 = 0
y = x + 1
y
x y = 0
x + 1 = 0
2) –x2 + 4 = 0
y = –x2 + 4 y
y x
x
–x2 + 4 = 0
x2
= 4
x = –2, 2
–x2 + 4 = 0 x = –2, 2
15. 1) 2x + 1 < 3
2x – 2 < 0 2(x – 1) < 0
y1 = 2(x – 1)
y1
Y
X01
y = x + 1
Y
X0
4
y1
2–2y2
29
30
y1 < 0 x < 1
2x + 1 < 3 x < 1
2) x2 + 4x – 5 < 0
y = x2 + 4x – 5
= (x2 + 4x + 4) – 5 – 4
= (x + 2)2 – 9
y = (x + 2)2 – 9
X
x2 + 4x – 5 = 0
(x + 5) (x – 1) = 0
x = –5, 1
x2 + 4x – 5 < 0 –5 < x < 1
3) x – 2 > 0
y = x – 2
y
y > 0 x 2
1
Y
X0
Y
X0–5 1
Y
X
0
y1
2
y1 = 2x – 2
–2
–9
30
31
1.1
1. 1) x y
x y
1
2
3
4
5
6
7
6.00 .
6.03 .
6.01 .
6.05 .
6.06 .
6.02 .
0.01 .
2) 155
. . . . 155 .
155 . ( .)
( .)
3) 2
1
A h
A = h102
1 A = 5h
.
.
.
.
40
42
38
31
32
2. 1) 35
1.5
x ( )
y ( )
y = 35 + 1.5x
2) a
ABCD a
ABCD x
x2 = a
2 + a
2
x2
= 2a2
x = a2
x = a2
3) 200
3
x ( )
y ( )
y = 200 + 3x
A Ba
a
CD
x
32
33
3. A B
1)
2)
3)
b
1
4)
4. 1)
2) a b 1
3)
4) a 1
abc
123
A B
abc
123
A B
abcd
A B1234
abcd
A B1234
33
34
5. 1) {(2, 10), (3, 15), (4, 20)}
2) {(–7, 3), (–2, 1), (–2, 4), (0, 7)}
1
3) {(–2, 1), (0, 1), (2, 1), (4, 1), (–3, 1)}
4) {(5, 0), (3, –1), (0, 0), (5, –1), (3, –2)}
1
234
101520
A B
–7–2
0
1347
A B
–3–2
0
2
4
1
A
B
0
3
5
–2
–1
0
A B
34
35
6. A = {a, b, c} B = {1, 2}
A B
(1) (2)
(3) (4)
7. A B 3 A
B
A = {a, b, c} B = {1, 2, 3}
(1) (2)
(3) (4)
(5) (6)
abc
1
2
A B
abc
123
A B
abc
123
A B
abc
123
A B
abc
123
A B
abc
123
A B
abc
123
A B
abc
1
2
A B
abc
1
2
A B
abc
1
2
A B
35
36
8. Df = {–2, –1, 0, 1, 2} f
1) f(x) = x2
f = {(–2, 4), (–1, 1), (0, 0), (1, 1), (2, 4)}
2) f(x) = 1x
x22
f = {(–2, 5
4), (–1, –1), (0, 0), (1, 1) (2,
5
4)}
3) f(x) = 2x
f = {(–2, 0), (–1, 1), (0, 2 ), (1, 3 ), (2, 2)}
4) f(x) = x + 1
f = {(–2, 1), (–1, 0), (0, 1), (1, 2), (2, 3)}
9. 1)
f(a) = 2
f(b) = 4
f(c) = 3
f(d) = 1
2)
f(a) = 1
f(b) = 4
f(c) = 1
f(d) = 3
10. 1) Y 1
2) Y 1
3) Y 1
4) Y 1
a
b
c
d
1
2
3
4
a
b
c
d
1
2
3
4
36
37
11. 1)
{y y –2}
2)
{y y 0}
3) {x x 1}
{y y 0}
4)
{y y 3}
12. 1) f(–1) = 3
f(0) = 0
f(1) = – 3
2) g(–2) = 0
g(0) = – 4
13. 1) g(x) = x2 – 2x
(1) g(2) = 0 (2) g(–3) = 15
2) f(s) = 1s
1
(1) f(4) = 5
1(2) f(0) = 1
14. 1) {x x R} {y y R}
2) {x x R} {y y R}
3) {x x R} {y y = 2
1}
4) {x x 2} {y y 0}
5) {x x –2} {y y 0}
6) {x x R} {y y 0}
7) {x x R} {y y –1}
8) {x x R} {y y 0}
9) {x x R} {y y 0}
10) {x x R} {y y 0}
37
38
1.2
1. 1) y1 = 5x + 3 y2 = 5x – 3
2) y1 = –x + 3 y2 = –x – 3
3) y1 = 5 – x y2 = 5 + x
4) y1 = x + 2 y2 = –x – 2
Y
X
y2
3
y1 –3
Y
X
3
–3
y2
y1
Y
X
y1 y2
5
Y
X
y2 y1
2
–2
0
–3 3
–5 0 5
–2 0
38
39
2. 1) ( ) 2) ( ) 3) ( )
3. 1) (3, 5) y
y = 1x7
2
y x = 3
y = 1)3(7
2
y = 7
61
x = 3 y = 7
61
(3, 7
61 ) y (3, 5)
2) (– 4, –5) y
y = –7 – 2x
y x = – 4
y = –7 – 2(– 4)
y = 1
x – 4 y = 1
(– 4, 1) (– 4, –5)
(– 4, 1)
0
(– 4, –5)
Y
X– 4
–5
1
(3, 5)
0
Y
X
2
3
(3, 7
61 )
39
40
3) y = –1
(x, –1) x
(4, –5) y = –1
4. 1) x
f(x)
150 5
f(x) = 150 + 5x
x f(x)
0
1
2
3
4
150
155
160
165
170
0 1 2 3 4 5
170
x
f(x)
165
160
155
150
(4, –5)
Y
Xy = –1
–5
40
40
41
2) x
f(x)
6,000 5%
f(x) = 6,000 + 0.05x
x f(x)
0
1,000
2,000
3,000
4,000
6,000
6,050
6,100
6,150
6,200
3) 2.54
x
y
y = 2.54x
x y
0
1
2
3
4
0
2.54
5.08
7.62
10.16
x
1,000 2,000 3,000 4,000
f(x)
6,000
6,100
6,200
Y
0 1 2 3 4 5X
2
4
6
8
10
41
42
5. 1)
( )
34,000
200,000 28,000 150,000
A
b
c
x ( )
A = c + bx
34,000 200,000
34,000 = c + 200,000 x --------------- (1)
28,000 150,000
28,000 = c + 150,000 x --------------- (2)
(1) – (2) 6,000 = 50,000 x
x = 100
12
100
12 12
2) x = 100
12 (1)
34,000 = c + 200,000 100
12
34,000 = c + 24,000
c = 10,000
10,000
3)
s
f(s) s
f(s) = 10,000 + 100
12(s)
42
43
6. 1) 12,000 /
10%
x
f(x) x
f(x) = 12,000 + 12,000(100
10)x
= 12,000 + 1,200x
= 12,000(1 + 10
x)
2) x = 5
f(5) = 12,000(1 + 10
5)
= 12,000 2
3
= 18,000
1.3.1
1.
1) y = 2x2
2) y = –2x2
Y
X0
Y
X0
43
44
3) y = 2x2 + 1
4) y = 2x2 – 1
5) y = –2x2 + 1
6) y = –2x2 – 1
Y
X0
Y
X
0
1
–1
Y
X0
1
–1
Y
X0
44
45
7) y = (x – 1)2
8) y = (x + 1)2
9) y = (x – 1)2 – 1
10) y = (x + 1)2 + 1
1
–1
–1
1
Y
X0
Y
X0
Y
X
Y
X0–1
1
0
45
46
2. 1) y1 = x2
y2 = 2x2
y3 = 5x2
y4 = 11x2
2) y1 = x2
y2 = 2x2
1
y3 = 2x5
1
3) y1 = 2x2
y2 = –2x2
4) y1 = 0.5x2
y2 = –0.5x2
5) y1 = (x – 3)2
y2 = (x – 4)2
y3 = (x – 5)2
Y
X
Y
X
y1y2
y3
Y
X
y1
y2
Y
X
y1
y2
Y
X
y1 y2 y3
3 4 50
y1
y2
y3
y4
0
0
0
0
46
47
6) y1 = –(x + 1)2
y2 = –(x + 2)2
y3 = –(x + 3)2
7) y1 = x2
y2 = (x – 1)2
y3 = (x – 1)2 + 2
8) y1 = x2
y2 = (x + 1)2
y3 = (x + 1)2 – 1
9) y1 = x2
y2 = (x – 1)2
y3 = (x – 1)2 + 1
10) y1 = –x2
y2 = –(x – 1)2
y3 = –(x – 1)2 + 1
Y X
y3 y2 y1
–3 –2 –1
Y
X
y2
y3
y1
–1
Y
X
y1
y3
y2
1
1
y3
y2
y1
Y
X1
–1
0
0
0
y2y1
y3
2
1
Y
X0
47
48
11) y1 = x2
y2 = (x – 2)2
y3 = 5(x – 2)2
y4 = 5(x – 2)2 – 5
12) y1 = x2
y2 = (x + 3)2
y3 = 8(x + 3)2
y4 = 8(x + 3)2 + 3
13) y1 = x2
y2 = (x + 4)2
y3 = –(x + 4)2
y4 = –(x + 4)2 + 7
14) y = –(x + 4)2 – 7
15) y = (x + 4)2 + 7
Y
X
y1 y2
y3
y4
2
–5
y2
y4Y
X
y3
–3 0
y1
Y
X
y2
y3
y4
–4
–7 y1
YX–4 0
–7
Y
X–4
7
0
0
0
48
49
X
Y
0
16) y = (x + 4)2 – 7
17) y = 3(x – 3)2 + 3
18) y = –2(x + 2)2 + 1
3. 1) y = (x – 4)2 – 3
( )
–3
4
Y
X
3
3
Y
X–2
1
Y
X–4
–7
0
0
0
49
50
X
Y
0
X
Y
0
2) y = –(x – 4)2 + 3
( )
3) y = (x + 4)2 – 3
( )
4) y = –(x + 4)2 + 3
( )
3
4
–3
– 4X
Y
0
3
– 4
50
51
5) y = 2(x – 2)2
( )
6) y = (x + 3)2 – 4
( )
7) y = 3)1x(2
1 2
( )
Y
X0
Y
X–3 3
-4
0
–3
–1X
Y
0
2
51
52
8) y = –2(x + 3)2 + 2
( )
9) y = x2 – 2x + 3
= (x2 – 2x + 1) + 2
= (x – 1)2 + 2
( )
10) y = 2x2 – 4x + 5
= 2(x2 – 2x) + 5
= 2(x2 – 2x + 1) +5 – 2
= 2(x – 1)2 + 3
( )
2
–3X
Y
0
2
0 1
Y
X
Y
X
3
-1 10
52
53
4. 1) y = x2 – 2x – 3
y = (x2 – 2x + 1) – 3 – 1
y = (x – 1)2 – 4
h = 1 k = – 4
(1, – 4)
x2 y
y
2) y = x2 – 4x + 8
y = (x2 – 4x + 4) + 8 – 4
y = (x – 2)2 + 4
h = 2 k = 4
(2, 4)
x2 y
y
Y
X0
– 4(1, – 4)
1
Y
X0
(2, 4)4
2
53
54
3) y = 2x2 + 4x + 8
y = 2(x2 + 2x + 4)
= 2[(x2 + 2x + 1) + 3]
= 2[(x + 1)2 + 3]
= 2(x + 1)2 + 6
h = –1 k = 6
(–1, 6)
x2 y
y
4) y = 3x2 + 12x + 3
y = 3x2 + 12x + 3
= 3(x2 + 4x + 1)
= 3[(x2 + 4x + 4) + 1 – 4]
= 3[(x + 2)2 – 3]
= 3(x + 2)2 – 9
h = –2 y = –9
(–2, –9)
x2 y
y
Y
X
(–1, 6)
–1 0
6
Y
X
(–2, –9)
0
–2
–9
54
55
5) y = –x2 + 2x + 1
y = –(x2 – 2x – 1)
= –[(x2 – 2x + 1) – 1 – 1]
= –[(x – 1)2 – 2]
= –(x – 1)2 + 2
h = 1 k = 2
(1, 2) x2
y y
5. 1) y = –3x2 + 6x + 3
y = ax2 + bx + c , a 0 x =
a2
b
y = –3x2 + 6x + 3 a = –3 b = 6
x =)3(2
)6( = 1
y = –3(1)2 + 6(1) + 3
= –3 + 6 + 3
= 6
(1, 6)
x2
y
Y
X0
(1, 2)2
1
Y
X0 1
(1, 6)6
55
56
2) y = 2x2 – 4x
y = ax2 + bx + c , a 0 x
a2
b
y = 2x2 – 4x a = 2 b = – 4
x = )2(2
)4( = 1
y = 2(1)2 – 4(1)
= –2
(1, –2)
x2
y
3) y = 2x2 + 4x + 2
y = ax2 + bx + c, a 0 x
a2
b
y = 2x2 + 4x + 2 a = 2 b = 4
x = )2(2
)4( = –1
y = 2(–1)2 + 4(–1) + 2
= 2 – 4 + 2
= 0
(–1, 0)
x2
y
Y
X0
(1, –2)–2
1
56
57
4) y = 2x2 – 2x – 24
y = ax2 + bx + c, a 0 x =
a2
b
y = 2x2 – 2x – 24 a = 2 b = –2
x = )2(2
)2( =
4
2 =
2
1
y = 24)2
1(2)
2
1(2 2
= 2412
1
= –242
1
)2
124,
2
1(
x2
y
YX
0
)2
124,
2
1(
1
Y
X(–1, 0) 0
57
58
1.3.2 (1)
1. 1) x2
= 16
x2 – 16 = 0
y = x2 – 16 y
y X y = 0 , x = – 4, 4
x2 – 16 = 0 x
2 = 16 x = – 4, 4
2) 3x2
= 27
3x2 – 27 = 0
y = 3x2 – 27 y
y X y = 0 , x = –3, 3
3x2 – 27 = 0 3x
2 = 27 x = –3, 3
3) 2x2
= 8
2x2 – 8 = 0
y = 2x2 – 8 y
X
Y
0– 4 4
(0, –16)
X
Y
0–3 3
(0, –27)
58
59
y = 0 x = –2, 2
2x2 – 8 = 0 2x
2 = 8 x = –2, 2
4) x2 = 0
y = x2
y
y X (0, 0) y = 0 , x = 0
x2 = 0 x = 0
5) x2 = –8
x2 + 8 = 0
y = x2 + 8 y
X
Y
0–2 2
(0, –8)
X
Y
0 (0, 0)
X
Y
0
(0, 8)
–8
8
59
60
y 8
x y = 0
y x x2 + 8 = 0
x2 + 8 = 0 x
2 = –8
2. 1) x2 + 8x + 16 = 0
(x + 4)2 = 0
y = (x + 4)2
y
y X (– 4, 0)
x2 + 8x + 16 = 0 x = – 4
2) 8x2 = 16x – 3
8x2 – 16x + 3 = 0
8(x2 – 2x) + 3 = 0
8(x2 – 2x + 1) + 3 – 8 = 0
8(x – 1)2 – 5 = 0
y = 8(x – 1)2 – 5 y
y X
8x2 + 8x + 16
X
Y
(– 4, 0)
X
Y
1
–5
0
(1, –5)
60
61
3) 6x2 = 4x + 3
6x2 – 4x – 3 = 0
y = ax2 + bx + c, a 0 x
a2
b
y = 6x2 – 4x – 3 a = 6 b = – 4
x = )6(2
)4 (=
3
1
y = 3)3
1(4)
3
1(6 2
= 33
4
3
2
= 3
11
x2
y )3
11,
3
1(
y X
6x2 = 4x + 3
4) 2x2 – 4x + 1 = 0
2(x2 – 2x) + 1 = 0
2(x2 – 2x + 1) + 1 – 2 = 0
2(x – 1)2 – 1 = 0
y = 2(x – 1)2 – 1 x
2
y (1, –1)
X
Y
–5)
3
11,
3
1(
0
61
62
X
2x2 – 4x + 1 = 0
5) –8x2 – 24 = 0
y = –8x2 – 24
y
y = –8x2 – 24 X
–8x2 – 24 = 0
3. 1) –(x + 1)2 + 1 = 0
y = –(x + 1)2 + 1 y
y X
–(x + 1)2 + 1 = 0
X
Y
–1
1
X
Y
–24
X
Y
–1
1
(1, –1)
(–1, 1)
0
0
0
62
63
2) 7(x + 2)2 = 0
y = 7(x + 2)2 y
y X
7(x + 2)2 = 0
3) (x – 4)2 = – 4 (x – 4)
2 + 4 = 0
y = (x – 4)2 + 4 y
y X
(x – 4)2 = – 4
4) (x + 7)2 = 3 (x + 7)
2 – 3 = 0
y = (x + 7)2 – 3 y
y X
(x + 7)2 = 3
X
Y
(–2, 0)
X
Y
4
4
X
Y
–7–3
(–7, –3)
0
0
0
63
64
4. 1) (1) {x x R}
{y y 0}
(2) (– 4, 0)
(3) y = 0
2) (1) x x R
{y y – 4}
(2) (–3 , – 4)
(3) y = – 4
3) (1) {x x R}
{y y 2}
(2) (–3, 2)
(3) y = 2
4) (1) x x R
{y y –3}
(2) (–1, –3)
(3) y = –3
5) (1) x x R
{y y –1}
(2) (2, –1)
(3) y = –1
5. 1) y = x2 – 8x + 15 y a(x – h)
2 + k
(x2 – 8x + 15) = (x
2 – 8x + 16) + 15 – 16
= (x – 4)2 – 1
a = 1, h = 4 k = –1
a > 0 y (4, –1)
y = (x – 4)2 – 1
64
65
1) Df = {x x R}
Rf = {y y –1}
2) (4, –1)
3) y –1
4) X X
x2 – 8x + 15 = 0
(x – 3)(x – 5) = 0
x = 3, 5
X (3, 0) (5, 0)
2) y = x2 – 2x – 4 y a(x – h)
2 + k
x2 – 2x – 4 = (x
2 – 2x + 1) – 4 – 1
= (x – 1)2 – 5
a = 1 , h = 1 k = –5
a > 0 f (1, –5)
y = (x – 1)2 – 5
1) Df = {x x R}
Rf = {y y –5}
2) (1, –5)
Y
X0–2 1
–5(1, –5)
Y
X0
(4, –1)
65
66
3) y –5
4) X X x = a2
ac4bb 2
x2 – 2x – 4 = 0 a = 1 b = –2 c = –4
x =)1(2
)4 )(1(4)2()2( 2
=2
1642
= 51
X (1 – 5 , 0) (1 + 5 , 0)
3) y = x2 + 8x + 13 y a(x – h)
2 + k
x2 + 8x + 13 = (x
2 + 8x + 16) + 13 – 16
= (x + 4)2 – 3
a = 0, h = – 4 k = –3
a > 0 y (– 4, –3)
y = (x + 4)2 – 3
1) Df = {x x R}
Rf = {y y R, y –3}
2) (– 4, –3)
3) y –3
4) X X x = a2
ac4bb 2
x2 + 8x + 13 = 0 a = 1, b = 8, c = 13
x =)1(2
)13)(1(488 2
=2
52648
Y
X0
(– 4, – 3)
– 4
–3
66
67
=2
128
=2
328= 34
X (– 4 – 3 , 0) (– 4 + 3 , 0)
4) y = 2x2 + 4x + 4 y a(x – h)
2 + k
2x2 + 4x + 4 = 2(x
2 + 2x + 2)
= 2[(x2 + 2x + 1) + 2 – 1]
= 2[(x + 1)2 + 1]
= 2(x + 1)2 + 2
a = 2, h = –1 k = 2
a > 0 y (–1, 2)
y = 2(x + 1)2 + 2
1) Df = {x x R}
Rf = {y y R, y 2}
2) (–1, 2)
3) y 2
4) X
5) y = 3x2 – 12x + 6 y a(x – h)
2 + k
3x2 – 12x + 6 = 3(x
2 – 4x + 2)
= 3[(x2 – 4x + 4) + 2 – 4]
= 3[(x – 2)2 – 2]
= 3(x – 2)2 – 6
Y
X0
(–1, 2)2
–1
67
68
a = 3, h = 2 k = –6
a > 0 y (2, –6)
y = 3(x – 2)2 – 6
1) Df = {x x R}
Rf = {y y R, y –6}
2) (2, –6)
3) y y –6
4) X X x =a2
ac4bb 2
3x2 – 12x + 6 = 0 a = 3, b = –12, c = 6
x =)3(2
)6)(3(4)12()12( 2
=6
7214412
=6
722
=6
262
= 22
X (2 + 2 , 0) (2 – 2 , 0)
6) y = x(x – 1) – 1 y a(x – h)2 + k
x(x – 1) – 1 = x2 – x – 1
= (x2 – x +
4
1) – 1 –
4
1
= 4
5)
2
1x( 2
a = 1, h = 2
1 k =
4
5
Y
X0
(2, –6)
2
–6
68
69
a > 0 y )4
5,
2
1(
y = x(x – 1) – 1
1) Df = {x x R}
Rf = {y y R, y 4
5}
2) (4
5,
2
1)
3) y 4
5
4) X X x = a2
ac4bb 2
x2 – x – 1 = 0 a = 1, b = –1, c = –1
x =)1(2
)1)(1(4)1()1( 2
=2
411
=2
51
X (2
51, 0) (
2
51, 0)
7) y = x2 – 4x – 7 y a(x – h)
2 + k
x2 – 4x – 7 = (x
2 – 4x + 4) – 7 – 4
= (x – 2)2 – 11 = 0
a = 1, h = 2 k = –11
a > 0 y (2, –11)
Y
X0
(4
5,
2
1)
–2 2
–2
69
70
y = (x – 2)2 – 11
1) Df = {x x R}
Rf = {y y R, y –11}
2) (2, –11)
3) y –11
4) X X x = a2
ac4bb 2
x2 – 4x – 7 = 0 a = 1, b = – 4, c = –7
x =)(
))((4) () ( 2
12
7144
=2
28164
=2
1124
= 112
X (2 + 11 , 0) (2 – 11 , 0)
8) y = x2 – 2x + 5 = 0 y a(x – h)
2 + k
x2 – 2x + 5 = (x
2 – 2x + 1) + 5 – 1
= (x – 1)2 + 4
a = 1, h = 1 k = 4
a > 0 y (1, 4)
y = (x – 1)2 + 4
Y
X0
(2, –11)
2
–11
70
71
1) Df = {x x R}
Rf = {y y R, y 4}
2) (1, 4)
3) y 4
4) X
6.
x1 x2 X
x1, x2 y = 0
x2 – 2x – 8 = 0
(x + 2)(x – 4) = 0
x = –2, 4
x1 = –2 x2 = 4
y = x2 – 2x – 8 y a(x – h)
2 + k
= (x2 – 2x + 1) – 8 – 1
= (x – 1)2 – 9
a = 1, h = 1, k = –9
(1, –9)
y1 –9
Y
X0
(1, 4)4
1
x1x2
Y
X0
y1
71
72
7. 1) y = (x – 3)(x – 6)
(x – 3)(x – 6) = 0
x = 3, 6
2) y = (x – 6)(x + 4)
(x – 6)(x + 4) = 0
x = – 4, 6
3) y = x(5 – x)
x(5 – x) = 0
x = 0, 5
4) y = x2 + 2
Y
X3
6
Y
X–4 6
Y
X50
Y
X
0
0
0
2
72
73
5) y = x2 + 4x + 12
= (x2 + 4x + 4) + 8
= (x + 2)2 + 8
6) y = 2x2 – 12x + 6
2x2 – 12x + 6 = 0
x2 – 6x + 3 = 0
x = )(
))(()()( 2
12
31466
= 2
246 = 63
x = 63 63
7) y = –x2 – 2x – 1
y = –(x2 + 2x + 1)
= –(x + 1)2
y = 0
x = –1
8) y = 15 + 2x – x2
15 + 2x – x2 = 0
(x – 5)(x + 3) = 0
x = –3, 5
Y
X
84
0–2
Y
X0
–6
–12
Y
X(–1, 0)
Y
X
(1, 16)
–3 0 1 5
3
73
74
2
1
1.3.2 (2)
1. 1) x2 1
y1 = x2 y2 = 1
y1 y2
x2 1 x 1 x –1
2) 4x2 1
y1 = 4x2 y2 = 1
y1 y2
4x2 < 1
2
1 < x <
2
1
Y
X
y1
y2 = 1
–1 0 1
Y
X
y1
2
1
2
1
y2 = 1
1
–1 1
74
75
3) 5 – x2 1
5 – x2 – 1 > 0
4 – x2 > 0
y = 4 – x2
y
y > 0 –2 < x < 2
y1 = 5 – x2 y2 = 1 y1 y2
y1 > y2 –2 < x < 2
4) –(x – 1)(x + 5) 0
y = –(x – 1)(x + 5)
x2 y
X (–5, 0) (1, 0)
y 0 x –5 x 1
Y
X–2 0 2
4
Y
X–5 0 1
Y
X–2 0 2
1
5
y2
75
76
2. 1) x2 – x – 2 0
(x – 2)(x + 1) 0
x2
y X (–1, 0) (2, 0)
y
x2 – x – 2 0 x –1 x 2
2) x2 – 3x – 1 < 3
x2 – 3x – 4 < 0
(x – 4)(x + 1) < 0
y = (x – 4)(x + 1)
x2 y
X (4, 0) (–1, 0) y
y < 0 –1 < x < 4
3) x2 + 2x 3
x(x + 2) 3
y1 = x(x + 2) y2 = 3
x2
y1 X (0, 0) (–2, 0)
y1 y2
Y
X0–1 2
Y
X0–1 4
76
77
y1 = y2
x2 + 2x = 3
x2 + 2x – 3 = 0
(x + 3)(x – 1) = 0
x = 1, –3
y1 y2
y1 y2 (1, 3) (–3, 3)
y1 y2 x2 – 2x 3 –3 x 1
4) –x2 – 6x 7x
–x2 – 6x – 7x 0
–x2 – 13x 0
–x(x + 13) 0
x2
y1 X (0, 0) (–13, 0)
y 0 x –13 x 0
Y
X
y1
(1, 3)(–3, 3)
–3 1
y2 = 3
0
77
45403530252015105
78
1.3.3
1. 1) x 45
y
x + y = 45
y = 45 – x
2) xy
xy = x(45 – x)
= 45x – x2
3) 45x – x2 = 164
x2 – 45x + 164 = 0
(x – 4)(x – 41) = 0
45 164 4 41
4) x y 45
x y 1 44
x 44 y = 1
2. xy2 x + y
2 = 6
x + y2 = 6
y2 = 6 – x
xy2 x y
2 = 6 – x
xy2 = x(6 – x)
g = x(6 – x)
x(6 – x) x2 g
X (0, 0) (6, 0)
78
79
g g = x(6 – x)
g x = a2
b x =
)1(2
6 = 3
x = 3 y = x(6 – x)
= 3(6 – 3)
= 9
g (3, 9)
xy2 9
3. 1) x
y x
y = [(100 – 0.1(x)] x
= 100x – 0.1x2
2) y = 100x – 0.1x2
x2
y y
y = ax2 + bx + c x =
a2
b
y = 100x – 0.1x2
a = –0.1 b = 100
x = )1.0(2
)100( = 500
y = 100(500) – 0.1(500)2
= 25,000
9
(0, 0)3
(6, 0)
Y
X
79
80
y
x = 500
500
3) 500 25,000
4. 1) x 200
y
4000 + 200x 80 – x
y = (80 – x)(4,000 + 200x)
= 320,000 + 16,000x – 4,000x – 200x2
= –200x2 + 12,000x + 320,000
2) y = 375,000
375,000 = – 2002x + 12,000x + 320,000
200x2 – 12,000x + 55,000 = 0
200(x2 – 60x + 275) = 0
200(x – 55)(x – 5) = 0
x = 5, 55
375,000
2
1 4,000 + 200(5) = 5,000 5
2 4,000 + 200(55) = 15,000 55
25000(500, 25000)
5000
Y
X
80
81
3)
y = –200x2 + 12,000x + 320,000
x2 y
y y x = a2
b
y = –200x2 + 12,000x + 320,000
a = –200, b = 12,000
x = )200(2
000,12 =
400
000,12 = 30
4000 + 200(30) 10,000
4)
y = –200(x)2 + 12,000x + 320,000 x = 30
= –200(30)2 + 12,000(30) + 320,000
= 500,000
500,000
10,000 30 80
5. 1) x 5 75
75 + x
475 – 5x
y
y = (75 + x)(475 – 5x)
= –5x2 + 100x + 35,625
= –5(x2 – 20x – 7,125)
y = –5x2 + 100x + 35,625 x
2
y y
x = a2
b
y = –5x2 + 100x + 35,625 a = –5 b = 100
x = )5(2
100 = 10
y = –5(10)2 + 100(10) + 35,625 36,125
81
82
y
(10, 36125)
y 36,125 x = 10
(75 + 10) 85
2) 36,125
1.4
1. 1) y = 3x
(R)
{y y > 0}
(10, 36125)
Y
X
y = 3x
1
0
0
82
83
2) y = x
3
1
{y y > 0}
3) y = 2x + 1
{y y > 1}
4) y = 3x – 1
{y y > –1}
y = x
3
1
Y
X
y = 2x + 1
0
2
1
Y
X
y = 3x – 1
–10
83
+
-
84
5) y = 2x+1
{y y > 0}
2. f(x) = 10
x
)3.1(000,4 10 f(10)
f(10) = 10
10
)3.1(000,4
= 4,000(1.3)
= 5,200
20 f(20)
f(20) = 10
20
)3.1(000,4
= 4,000(1.3)2
= 6,760
3. f(x) = 850,000(1.08)n
5 f(5)
f(5) = 850,000(1.08)5
1,248,928.865
1,248,928.865 – 850,000 398,928.87
4. A(t) = 10(0.8)t 8
A(8)
A(8) = 10(0.8)8
10(0.168)
1.7
8 1.7
y = 2x + 1
84
85
5. v(t) = 0.78C(0.8)t-1
t
800,000 3
v(3)
C = 800,000 t = 3
V(3) = 0.78(800,000)(0.83-1
)
= 0.78(800,000)(0.82)
= 399,360 400,000
5 t = 5
V(5) = 0.78(800,000)(0.85-1
)
= 0.78(800,000)(0.84)
= 255,590.4 256,000
10 t = 10
V(10) = 0.78(800,000)(0.810-1
)
= 0.78(800,000)(0.89)
83,751 84,000
6. S = P(1 + i)n
S
P
i
n
P = 100,000, i = 4
03.0, n = 3 4 12
S = 43)4
03.01(000,100
= 12)0075.1(000,100
100,000(1.09380)
109,380
100,000 3 3%
3 109,380 – 100,000 9,380
85
86
7. S = P(1 + i)n
P = 50,000
i =12
12.0 = 0.01
n = 6
S = 50,000(1 + 0.01)6
= 50,000(1.01)6
50,000(1.06152)
53,076
53,076
8. S = P(1 + i)n
P = 10,000
i = 0.01
n = 12
S = 10,000(1 + 0.01)12
= 10,000(1.01)12
10,000(1.1268)
11,268
11,268
86
87
1.5
1. 1) y = x + c
c = – 3 y = x – 3 y
{y y –3}
c = 1 y = x + 1 y
{y y 1}
Y
X0
–3
Y
X0
1
87
88
2) y = x – c
c = –3 y = x + 3 y
{y y 0}
c = 1 y = x – 1 y
{y y 0}
3) y = x + c – 2
c = –3 y = x – 3 – 2 y
{y y –2}
Y
X0–3
3
Y
X0 1
Y
X0
–2
88
89
c = 1 y = x + 1 – 2 y
{y y –2}
2. 1) 3 + x = 0 y = 3 + x y
y = 0, x = –3
3 + x = 0 x = –3
2) x – 5 = 0 y = x – 5 y
y = 0 , x = 5
x – 5 = 0 x = 5
Y
X–1
–2
Y
X0
Y
X0 5
–3
89
90
3) 5 – x = 0
y = 5 – x y
y = 0 , x = 5
5 – x = 0 x = 5
4) x – 1 = 0 y = x – 1 y
y = 0 x = –1, 1
x – 1 = 0 x = 1 x = –1
5) x + 7 = 7 x + 7 – 7 = 0
y = x + 7 – 7 y
y = 0 , x = 0 –14
x + 7 – 7 = 0 x + 7 = 7 x = 0 x = –14
Y
X0
Y
X–1
Y
X0
–7
–14
5
90
91
3. 1) y = x + 7 y
x –7 , x + 7 0
2) y = 5 – x y
x 5 , 5 – x 0
3) y = 2 – x y
x < 2 , 2 – x > 0
–7
Y
X
5
Y
X
2
Y
X0
0
0
91
92
1.6
1. f Y
0 50 100 250 500 1,000 1,500 2,000
2
4
6
8
10
12
14
16
18
20
Y ( )
X
( )
92
93
2.
( )
1,000
1,000 1,000
( 1,000 1,000 )
15.00
10.00
f ( Y
)
5
10
15
1,000 X
( )
Y ( )
25
35
45
55
5,0002,000 3,000 4,000
93
2
( 12 )
0 90
0 90
95
1.
2.
0 89
sin cos tan
302
1
2
3
3
3
452
2
2
2 1
602
32
13
3.
x
1) 2) 3)
4) 5) 6)
x18
39
32
51x
94221
x 75
12
x
44x
14025
16x
96
1.
1) ABC C 90 A 30
BC
AB
BC
AB
2)
A
A
C
30
B
AC
3
30
B
A C
5
B C
A
30
2
97
A C
B
3 5 10
3012 15
98
1.
1) 1) sin A
2) cos B
3) tan A
4) tan B
2)
1) sin A sin B
2) cos A cos B
3) tan A tan B
3)
1) sin A sin B
2) cos A cos B
3) tan A tan B
A C
B
ca
b
A C
B
2520
15
8 15
17B
C
A
99
2. x
1) 2)
3) 4)
5) 6)
3. ABC
1) sin 2) cos
3) tan 4) AC
18x
30
12
60x
x
60
AC
B
5021
42x
50
32
42
25
10
xx 7
3x
100
4. a, b c
1) 2)
3) 4)
5. ABC AB 18 AD A
AD BC AD
6. 1)
C
A B
C
D
18
100
30
30
C
A B8
b a
60
c
b6
B A45 45
b
B
a
C A
45
30
A
BaC
c7
60
101
2)
3)
60
x
12
30
x
30
102
1.
1) 1) sin A = c
a
2) cos B = c
a
3) tan A = b
a
4) tan B = a
b
2)
sin A =25
20=
5
4, sin B =
25
15=
5
3
cos A =25
15=
5
3, cos B =
25
20=
5
4
tan A =15
20=
3
4, tan B =
20
15=
4
3
3)
sin A =17
8, sin B =
17
15
cos A =17
15, cos B =
17
8
tan A =15
8, tan B =
8
15
A C
B
ca
b
A C
B
2520
15
8 15
17B
C
A
103
2. x
1) sin 30 = 18
x
x = 18 sin 30 = 18 0.50 = 9
2) cos 60 = 12
x
x = 12 cos 60 = 12 0.5 = 6
3) tan 60 = 42
x
x = 42 tan 60 = 42 1.732 = 72.744
4) tan x =50
32= 0.64
tan 33 = 0.649
x 33
5) cos x =7
3= 0.42
cos 65 = 0.423
x 65
6) sin x =25
10= 0.40
sin 24 = 0.407
x 24
18x
30
12
60
x
x
60
42
X
50
32
73
X
25X
10
104
3. ABC
1) sin
sin = 50
21 0.42
2) cos
sin 25 = 0.423
25
cos 25 = 0.906
3) tan
tan = tan 25
tan 25 = 0.466
4) AC
cos =50
AC
AC = 50 cos
= 50 cos 25
= 50 0.906
= 45.3
4. a, b c
1) sin 30 = 8
a, a = 8 sin 30
= 2
18 = 4
sin 60 = 8
b, b = 8 sin 60
= 8 0.866 = 6.92830
C
A B8
b a
60
AC
B
5021
105
2) tan 60 =7
a
a = 7 1.732
= 12.124
B = 90 – 60 = 30
sin B = sin 30 = c
7
c = 7 sin 30
= 7 0.5 = 3.5
3) sin A =c
6
c =Asin
6=
45
6
sin
=0.707
6= 8.487
ABC
( A = B = 45 )
b = 6
4) sin 45 =30b
b = 30 sin 45 = 30 0.707
= 21.21
A = 90 – 45 = 45
A = B = 45
ABC
a = b = 21.21
5. ABC AB 18 AD A
AD BC AD
A B
C
D
18
60
a
bC
B
A
45
30
cB A
b6
C
45 45
106
ABC
AB = AC = BC = 18
BAC = CBA = ACB = 60
AB = AC
ABC
BD = CD = 2
18 9
ABD D
sin B = 18AD
AD = 18 sin B
= 18 sin 60
sin 60 = 0.866
AD = 18 0.866
= 15.588
6. 1) x
tan 30 = 100
x
x = 100 tan 30
= 100 0.577 57.77
57.77
2) x
tan 60 = 12x
x = 12 tan 60
= 12 1.732
= 20.784
20.8
18
9
A
D
B
100 30
x
12
60
107
30
x
30
3) x
tan 30 = 30x
x = 30 tan 30
= 30 0.577
= 17.31
17.3
108
2.1
1. 1) sin A = 13
5cos A =
13
12
sin B = 13
12cos B =
13
5
tan A = 12
5tan B =
5
12
2) sin A = 5
3cos A =
5
4
sin B = 5
4cos B =
5
3
tan A = 4
3tan B =
3
4
3) sin A = 41
4cos A =
41
5
sin B = 41
5cos B =
41
4
tan A = 5
4tan B =
4
5
2. 1) sin A = AB
BC
2) sin B = AB
AC
3) cos A = AB
AC
4) cos B = AB
BC
5) tan B = BC
AC
6) tan A = AC
BC
A
B
C
13
12
5
C
35
4
C
4 5
41
B A
B C
A
A
B
109
3.
1) sin A =c
a2) cos B =
c
a
3) cos A =c
b4) sin B =
c
b
4. 1) sin 32 = 0.530 2) sin 4 = 0.070
3) sin 81 = 0.988 4) sin 17 = 0.292
5) cos 29 = 0.875 6) tan 18 = 0.325
7) tan 81 = 6.314 8) sin 51.5 0.7825
9) cos 67.5 0.383 10) tan 42.5 0.9165
5. 1) sin = 0.53
sin 32 = 0.53
= 32
2) sin = 0.899
sin 64 = 0.899
= 64
3) sin = 5
2 0.4
sin 24 = 0.407
= 24 ( )
4) sin = 8
5 0.625
sin 39 = 0.629
= 39 ( )
5) sin = 23
15 0.652
sin 41 = 0.656
= 41 ( )
C
b a
A Bc
110
6. 1) cos = 0.5
cos 60 = 0.500
= 60
2) cos = 0.99
cos 8 = 0.990
= 8
3) cos = 0.75
cos 41 = 0.755
= 41 ( )
4) cos = 5
3 0.6
cos 53 = 0.602
= 53 ( )
5) cos = 13
9 0.692
cos 46 = 0.695
= 46 ( )
7. 1) tan = 0.404
tan 22 = 0.404
= 22
2) tan = 4.011
tan 76 = 4.011
= 76
3) tan = 4
9 2.250
tan 66 = 2.246
= 66 ( )
4) tan = 7
4 0.571
tan 30 = 0.577
= 30 ( )
5) tan = 13
28 2.154
tan 65 = 2.145
= 65 ( )
111
8.
1) cos = 0.616 cos 52 = 0.616 = 52
2) tan = 0.488 tan 26 = 0.488 = 26
3) sin = 0.982 sin 79 = 0.982 = 79
4) tan = 2.356 tan 67 = 2.356 = 67
5) cos = 0.707 cos 45 = 0.707 = 45
9. 1
AB2 + BC
2= AC
2
BC2
= AC2 – AB
2
= 152 – 9
2
= 144
BC = 12
BC BC = 12
sin A =AC
BC =
15
12 0.8
tan A =AB
BC =
9
12 1.33
cos C =AC
BC =
15
12 0.8
tan C =BC
AB =
12
9 0.75
2
cos A =15
9 0.6
cos 53 = 0.602
A = 53 ( )
C = 180 – 90 – 53 = 37 ( )
sin A = sin 53 = 0.799 0.8
tan A = tan 53 = 1.327 1.33
A B
C
15
9
112
cos C = cos 37 = 0.799 0.8
tan C = tan 37 = 0.754 0.75
10. 1) sin 45 =14
x
x = 14 sin 45
sin 45 = 0.707
x = 14 0.707
= 9.898
2) tan 30 =35
x
x = 35 tan 30
=3
335
= 5
cos 30 =y
35
y =30cos
35
=
2
3
35
= 10
14x
45
30 x
y
35
113
3) cos 60 =8
x
x = 8 cos 60
cos 60 = 0.500
x = 8 0.500
= 4
sin 60 =8
y
y = 8 sin 60
sin 60 = 0.866
y = 8 0.866
= 6.928
tan 45 =z
y
z =45tan
y
45tan
928.6
tan 45 = 1.0
z = 6.928
4) sin 60 =8
x
x = 8 sin 60
sin 60 = 0.866
x = 8 0.866
= 6.928
sin 30 =8
y
y = 8 sin 30
sin 30 = 0.50
y = 8 0.5
= 4
11. ABC C cos A = 2
1
cos A =2
1
sin B = cos A
45
y
z
A
B C
21
sin B = 2
1 0.5
y 8
60x
x
30 60
y
8
114
sin 30 = 0.5 B = 30
cos 60 = 0.5 A = 60
1) sin A = sin 60
sin 60 = 0.866
2) tan A = tan 60
tan 60 = 1.732
3) sin B = cos A = 2
1 0.5
4) cos B = sin A = 0.866
5) tan B = tan 30
tan 30 = 0.577
12. 1)
ABD
BD2 = AB
2 + AD
2
BD2 = 8
2 + 6
2
BD2 = 64 + 36
BD2 = 100
BD = 10
BD
BD = 10
2) 1sin =BD
AB=
10
8
5
4
3) 1cos =BD
AD=
10
6
5
3
4) 2sin =BD
AD=
10
6
5
3
5) 2cos =BD
AB=
10
8
5
4
A B
D C
8
6
24
31
115
ABCD
BC = 6 DC = 8
6) 3sin =BD
BC=
10
6
5
3
7) 3cos =BD
CD=
10
8
5
4
8) 4sin =BD
CD=
10
8
5
4
9) 1sin = 5
4 0.8
sin 53 = 0.799
1 53
10) 2sin =5
30.6
sin 37 = 0.602
2 37
1 53
1 + 2 = 90
53 + 2 90
2 37
11) 3sin =5
30.6
sin 37 = 0.602
3 37
12) 4sin =5
40.8
sin 53 = 0.799
4 53
116
13.
1) tan A = 2 a = 10
tan A = b
a
2 = b
10
b = 2
10
= 2
52
= 25
2) tan A = 4
1 b = 3
tan A = b
a
4
1=
3
a
a = 4
3
3) a = 5 b = 12
sin A = c
a
c c2 = a
2 + b
2
c2
= 52 + 12
2
c2
= 25 + 144
c2
= 169
c = 13
c
b
a
B
C A
117
c
c = 13
sin A = 13
5
4) cos B = 3
2 a = 5
cos B = c
a
3
2=
c
5
c = 2
35 7.5
5) a = 4 c = 6
tan B = a
b
b c2 = a
2 + b
2
62
= 42 + b
2
b2
= 62 – 4
2
b2
= 36 – 16
b2
= 20
b = 52
b b = 52
tan B = 4
52
2
5
14. 1)
BG = 12 GH = 8
B =
A C
D
G
B
H
E
118
BGH
sin = BG
GH
= 12
8
= 0.667
sin 42 = 0.669
42
BDE
BE x
cos = BD
BE
cos 42 = 15
x
cos 42 = 0.743
0.743 = 15
x
x = 0.743 15 11.145
BE 11.145
2) AC = 9 DE = 6 AB = 15
B
G H8
12
x
B
D E
15
C
E
B
D
A
6
9
15
119
BC
AB2
= AC2 + BC
2
152
= 92 + BC
2
BC2
= 152 – 9
2
BC2
= 225 – 81
BC2
= 144
BC = 12
BC BC = 12
EC = BC – BE = 12 – 11.145 = 0.855
3) A = 50 BH = 4
ABC BGH
A = G = 50
tan 50 =GH
4
tan 50 = 1.192
1.192 =GH
4
GH =192.1
4
GH = 3.356
B
G H
4
50
120
4) AB = 210
AC = 56
ABC A =
cos = 210
56 = 0.949
cos 18 = 0.951
18
ABC = 180
CBA 180 – 90 – 18 72
15. DEF DG DE 18
DEF
60
DGE E = 60
sin E = 18
DG
sin 60 = 18
DG
DG = 18 sin 60
= 18 0.866
= 15.588
18 E
G
F
D
18
A C
B
210
56
121
16. JKL LM JL =16
JKL J = K
J = K =
180 = 90 + +
= 45
JLM
sin J =16
LM
LM = 16 sin 45
sin 45 = 0.707
LM = 16 0.707
= 11.312
17. ABC ACBC A = 27 BC = 10
ABC AB
sin A =AB
BC
sin 27 =AB
10
AB =27sin
10
sin 27 = 0.454
L
M K
16
J
27
A
BC10
122
AB =454.0
10
22.03
AC
tan A =AC
BC
tan 27 =AC
10
AC =27tan
10
tan 27 = 0.510
AC =510.0
10
AC 19.61
ABC = AB + BC + AC
22.03 + 10 + 19.61
51.64
18. 14, 14 18
AD CAB ABC
AD CAB ABC AD
BC
A
CB D
18CB
14 14
A
123
BD = DC = 9
B = C =
cos B =AB
BD
cos =14
9
cos = 0.643
cos 50 = 0.643
= 50
180 = BACACBCBA
180 = 50 + 50 + BAC
BAC = 180 – 100
BAC = 80
50CBA , 50ACB 80BAC
124
2.2
1. 40
20
A
B
C
tan B =BC
AC
AC = 40 tan 20
tan 20 = 0.364
AC = 40 0.364
AC = 14.56
14.56
40 .20
A
CB 20
40
125
2. 6.5
4
AB
AC 6.5
BC 4
BCA BCA =
cos =AC
BC=
5.6
4
= 0.615
cos 52 = 0.616
52
sin =AC
AB
AB = 6.5 sin 52
sin 52 = 0.788
AB = 6.5 0.788
= 5.122
5.122 5
C4
A
B
6.5
126
3. ABCD 6 55
ABC
sin A =AC
BC
BC = AC sin 55
sin 55 = 0.819
BC = 6 0.819
= 4.914
cos A =AC
AB
AB = AC cos 55
sin 55 = 0.574
AB = 6 0.574
= 3.444
4.914
3.444
55A B
D C
6
127
4.
AB
AC 16
cos A =AB
AC
AB =39cos
16
cos 39 = 0.777
AB =777.0
16
= 20.592
20.6 ( )
5.
AB 2
BC
sin A =AB
BC
BC = AB sin A
= 2 sin 7
sin 7 = 0.122
BC = 2 0.122
= 0.244
= 0.244 1000
= 244
244
C39
16 A
B
A
B
C7
2 .
128
6.
AB 9.5
BC
sin A =AB
BC
BC = 9.5 sin 58
sin 58 = 0.848
BC = 9.5 0.848
= 8.056
8.056 + 1.2 9.2 ( )
7. 45 12
1.5
AB
BC ( )
45
12
A
CB
A C
B
58
9.5
1.5 .
12 .
45
129
ABC
tan C =BC
AB
AB = BC tan 45
AB = 12 1 12
12 + 1.5 13.5
8.
AC 38
BC + CD
tan 13 = AC
BC
BC = AC tan 13
tan 13 = 0.231
BC = 38 0.231
= 8.778
tan 16 = AC
CD
CD = AC tan 16
tan 16 = 0.287
CD = 38 0.287
= 10.906
8.778 + 10.906 = 19.684 19.7
9.
AA // CD
CAA = DCA = 40 ( )
CD // BB
BCD = BBC = 55 ( )
D
A
B
C1316
40
55
C
A
B .
.
x
5
D4055
A
B
130
AB
BD 5
BCD BDC
tan 55 = CD
BD
CD = 55tan
5
tan 55 = 1.428
CD = 428.1
5
= 3.5
ACD CDA
tan 40 = CD
AD
AD = 3.5 tan 40
tan 40 = 0.839
AD = 3.5 0.839
= 2.9365 2.9
5 + 2.9 7.9 ( )
10.
OB
OA
15.07
1
2
3
4
567
8
9
10
1112
AO
9 .
B
131
12 3 15 1
15.07 . 7
715
90 42
cos =OB
OA
OA = OB cos 42
cos 42 = 0.743
OA = 9 0.743
= 6.687
6.7
12
3
132
4 ( 4 - 6) 2
°√–∑√«ß»÷°…“∏‘°“√
