Basic m4-2-chapter1

2
1.
2. r x
y
r
r =
2x
1
yRRy,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 = 2
y
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 = 2
x } (x R, y 0)
(16) r = {(x, y) R R y = 5x2
} (x R, y 5 )
(17) r = {(x, y) R R y = 2
x7 } ( 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 = 3x
y = 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 = 3x2
, y = 5x2
y = ax2
a > 1
y = x2
2) 0 < a < 1 y = 2
x
2
1
, y = 2
x
3
1
, y = 2
x
5
1
y = ax2
0 < a < 1
y = x2
Y
X0
Y
X0
Y
X0
11

12
3) y = ax2
, a < 0 y = –x2
y = ax2
a > 0 a < 0 2
2. y = x2
+ c, c > 0
y = x2
+ c, c > 0 y = x2
y = x2
y = x2
+ 1
y = x2
+ 2
y = x2
+ 3
3. y = x2
– c, c > 0
y = x2
– c, c > 0 y = x2
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 = x2
y = x2
y = (x + 1)2
y = (x + 2)2
y = (x + 3)2
Y
X0
–3 –2 –1
Y
X
0
3
2
1
X
Y
0
–3
–2
–1
Y
X0
12

13
5. y = (x – c)2
, c > 0
y = (x – c)2
, c > 0 y = x2
y = x2
y = (x – 1)2
y = (x – 2)2
y = (x – 3)2
y = ax2
, y = (x – a)2
y = x2
+ c
y = a(x – h)2
+ k
y = 2(x – 1)2
+ 2 y = x2
1) y = x2
2) y = (x – 1)2
y = x2
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
X
1
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
X
0
Y
X0
Y
X
0
a < 0
Y
X
0
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
–5
50
y = x – 5
y2 = 8
Y
X
3
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
X
1
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 = x2
+ 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
– 4
4
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 = x2
– 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
4
x
3
2
y = 3
4
x
3
2
Y y = 3
4
x
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
4
x
3
2
0
22

23
2) y = 4 – x2
y = 4 – x2
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
X
0
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
X
1
0–1
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
X
0
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
X
0
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 – 4 2
3
–1 –3 3
0 –2
1 –1 –3
2 0
2
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 = (x2
– 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
X0
1
y = x + 1
Y
X0
4
y1
2–2
y2
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 = h10
2
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
= a2
+ a2
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
a
b
c
1
2
3
A B
a
b
c
1
2
3
A B
a
b
c
d
A B
1
2
3
4
a
b
c
d
A B
1
2
3
4
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
2
3
4
10
15
20
A B
–7
–2
0
1
3
4
7
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)
a
b
c
1
2
A B
a
b
c
1
2
3
A B
a
b
c
1
2
3
A B
a
b
c
1
2
3
A B
a
b
c
1
2
3
A B
a
b
c
1
2
3
A B
a
b
c
1
2
3
A B
a
b
c
1
2
A B
a
b
c
1
2
A B
a
b
c
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
x2
2
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 = 1x
7
2
y x = 3
y = 1)3(
7
2
y =
7
6
1
x = 3 y =
7
6
1
(3,
7
6
1 ) 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
6
1 )
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 5
X
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
X
0
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 = 2
x
2
1
y3 = 2
x
5
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
y1
y2
y3
Y
X
y1
y2
Y
X
y1
y2
Y
X
y1 y2y3
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
y4
Y
X
y3
–3 0
y1
Y
X
y2
y3
y4
–4
–7 y1
Y
X–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
– 4 X
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
–1 X
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
–3 X
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
= 241
2
1
= –24 2
1
)
2
1
24,
2
1
(
x2
y
Y
X0
)
2
1
24,
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 x2
= 16 x = – 4, 4
2) 3x2
= 27
3x2
– 27 = 0
y = 3x2
– 27 y
y X y = 0 , x = –3, 3
3x2
– 27 = 0 3x2
= 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 2x2
= 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 x2
= –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
= 3
3
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 x2
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) = (x2
– 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 = (x2
– 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
X
0–2 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 = (x2
+ 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(x2
+ 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(x2
– 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
72
2
= 6
26
2
= 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
1
x( 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 = (x2
– 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 = (x2
– 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
x1 x2
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
X
3
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
8
4
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
45
40
35
30
25
20
15
10
5

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 + y2
= 6
x + y2
= 6
y2
= 6 – x
xy2
x y2
= 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 = – 2002
x + 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 x2
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
–1 0
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
03.0
1(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

Basic m4-2-chapter1

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