Basic m4-2-chapter1

93
1 ( 28 ) 1. 2. 3. 4.

Transcript of Basic m4-2-chapter1

Page 1: Basic m4-2-chapter1

1

( 28 )

1.

2.

3.

4.

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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

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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

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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

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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

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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)

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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

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(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

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(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

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(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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

Page 39: Basic m4-2-chapter1

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

Page 40: Basic m4-2-chapter1

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

Page 41: Basic m4-2-chapter1

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

Page 42: Basic m4-2-chapter1

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

Page 43: Basic m4-2-chapter1

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

Page 44: Basic m4-2-chapter1

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

Page 45: Basic m4-2-chapter1

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

Page 46: Basic m4-2-chapter1

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

Page 47: Basic m4-2-chapter1

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

Page 48: Basic m4-2-chapter1

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

Page 49: Basic m4-2-chapter1

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

Page 50: Basic m4-2-chapter1

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

Page 51: Basic m4-2-chapter1

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

Page 52: Basic m4-2-chapter1

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

Page 53: Basic m4-2-chapter1

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

Page 54: Basic m4-2-chapter1

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

Page 55: Basic m4-2-chapter1

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

Page 56: Basic m4-2-chapter1

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

Page 57: Basic m4-2-chapter1

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

Page 58: Basic m4-2-chapter1

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

Page 59: Basic m4-2-chapter1

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

Page 60: Basic m4-2-chapter1

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

Page 61: Basic m4-2-chapter1

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

Page 62: Basic m4-2-chapter1

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

Page 63: Basic m4-2-chapter1

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

Page 64: Basic m4-2-chapter1

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

Page 65: Basic m4-2-chapter1

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

Page 66: Basic m4-2-chapter1

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

Page 67: Basic m4-2-chapter1

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

Page 68: Basic m4-2-chapter1

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

Page 69: Basic m4-2-chapter1

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

Page 70: Basic m4-2-chapter1

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

Page 71: Basic m4-2-chapter1

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

Page 72: Basic m4-2-chapter1

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

Page 73: Basic m4-2-chapter1

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

Page 74: Basic m4-2-chapter1

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

Page 75: Basic m4-2-chapter1

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

Page 76: Basic m4-2-chapter1

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

Page 77: Basic m4-2-chapter1

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

Page 78: Basic m4-2-chapter1

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

Page 79: Basic m4-2-chapter1

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

Page 80: Basic m4-2-chapter1

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

Page 81: Basic m4-2-chapter1

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

Page 82: Basic m4-2-chapter1

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

Page 83: Basic m4-2-chapter1

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

+

-

Page 84: Basic m4-2-chapter1

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

Page 85: Basic m4-2-chapter1

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

Page 86: Basic m4-2-chapter1

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

Page 87: Basic m4-2-chapter1

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

Page 88: Basic m4-2-chapter1

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

Page 89: Basic m4-2-chapter1

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

Page 90: Basic m4-2-chapter1

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

Page 91: Basic m4-2-chapter1

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

Page 92: Basic m4-2-chapter1

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

Page 93: Basic m4-2-chapter1

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