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( .) 2544
1 3
. .
.
1 3
1. 2.
3. 3
4.
1)
2) 6
3)
5. 6.
3
1.
2. 3. 4.
1.
2.
3. 4.
1
3
1
2
3
4
5
9
15
12
12
12
1 1
1
2
2
2
1.2 3
3
1.3 3
4
2 11
11
12
2.1 12
12
12
2.2 13
13
2.3 14
14
14
2.4 16
16
18
23
34
34
3.2 34
34
3.3 36
36
38
42
4 50
51
4.1 51
51
4.2 y = ax 2 a ≠ 0 52
52
52
4.3 y = ax 2 + k a ≠ 0 53
53
54
4.4 y = a(x – h) 2 + k a ≠ 0 55
55
55
4.5 y = ax 2 + bx + c a ≠ 0 56
56
57
79
83
84
84
5.2 87
88
97
(9 )
1.1 a a > 0 (2 )
1.2 (4 )
1.3 (3 )
2 2
a b = ab a > 0, b > 0 b
a =
b
a a > 0, b > 0
1.1 a a > 0 (2 )
a a > 0
1) a b = ab a > 0, b > 0
2) b
a a > 0, b > 0
1. “” 2. “ a
2 ”
a 2 a
a 2 a
3. 8 22
q p25 5pq 4 5q
4 p
5pq 4 5q
4 p
4. a b = ab a > 0, b > 0 b
a =
b
a a > 0,
3
1.2 (4 )
1. a a > 0
a a > 0 2. “” a a > 0
3. ba = ab a > 0, b > 0 b
a =
b
a
a > 0, b > 0 12
4. “”
1.3 (2 )
4
2. 4 3. “” 4 6 4. “” “” 1.2
5. “ ” 6. “”
“ ”
1.
5) 9
4 -
2.
5
“ a 2 ”
1. 2. 3. 4. 5. 6. 7. 8. 111
9. 10.
1.1
43 yx
4.
“ ”
1. 53 = 533 ⋅⋅ = 15 3 ⋅
35 = 355 ⋅⋅ = 15 5 ⋅ 3 5
53 < 35
2. 85 = 83 + 82 53 83 32 82 53 + 32 83 + 82
53 + 32 ≠ 85
3. 5 2 1 + 5 3
1 + 5 > 3
4. 3 2 32 4
32 – 3 < 1
5. 3 2 32 2 × 2 4 33 3 × 2 6
32 + 33 < 10
6. 2 ≈ 1.4, 3 ≈ 1.7 5 2.5
2 + 3 ≈ 3.1 5
2 + 3 > 5
7. 57 – 56 = 5 5 2
57 – 56 > 2
8. 35 – 34 = 3 1 35 – 24 35 – 34
35 – 24 > 1
7
9. 3 + 5 ≈ 1.7 + 5 6.7 5 + 3 ≈ 2.2 + 3 5.2
3 + 5 > 5 + 3
10. 2 + 3 + 5 ≈ 1.4 + 1.7 + 2.2 5.3
2 + 3 + 5 > 5
1.2
1.
2
27
1.
4.
2
“”
1. 3 2 315 15 × 2 30
315 < 30
2. 310 = 31010 ⋅⋅ 30 10 ⋅ 103 = 1033 ⋅⋅ 30 3 ⋅ 10 3
310 > 103
3. 3 × 5 = 15 4
3 × 5 < 5
25 × 34 ≠ 69
9
5. 2 × 7 = 14 3 × 5 = 15
2 × 7 < 3 × 5
6. 2
20 = 10 10 9 3
2
20 > 3
7. 26 5 5
26 1
14 =
2
3
3 <
3
5
10. 3× 01.0 = 3× 0.1 = 0.3 01.0
3 =
1.0
3) 16
4) 99
“ ”
1 = 1
123454321 = 1 1 1 1 1
11234565432 = 1 1 1 1 1 1
3211234567654 = 1 1 1 1 1 1 1
543211234567876 = 1 1 1 1 1 1 1 1
1 1 1234321 = 1111
1 4 4 1 2 3 4 3 2 1
“”
1.
(15 )
2.1 (2 )
2.2 (4 )
2.3 (6 )
2.4 (3 )
2 2 ax
2 + bx + c a, b, c a ≠ 0
2.4 4
2.1 (2 )
2.1
1. 2.1
2. ( a )
2 = a a > 0
ba = ab a > 0 b > 0
b
a =
b
3. 5x 4
1 2
− 2
5x 4
1 2
= 4
1 (x – 52 )(x + 52 )
4. 8 – (x – 3) 2 3
13
2.2 (4 )
2.2
1. 2.2 2. ax
2 + bx + c a = 1
1) x
2 + 2px + c x
2 – 2px + c p
2) x
2 + 2px x
2 – 2px 1)
p 2
x 2 + 2px + c = (x
2 + 2px + p
2 – 2px + p
3) 2) p 2 – c = d
2 d
x 2 + 2px + c = (x + p)
2 – d
2 – d
2 (x – p)
2 – d
2 = (A + B)(A – B) A
14
+−
535 x
4. “”
2.3 (6 )
2.3 2.3
3. A
3 + B
2 A 2 + AB + B
2 A 2 – 2AB + B
2
A 2 + 2AB + B
2
4. 13 14 2.3 4 x
3 – 5x
3 + 27) – (5x
3 + 27
5x 2 – 15x 5x
x 3 – 5x
2 – 15x + 27 = (x
= (x + 3)[(x 2 – 3x + 9) – 5x]
= (x + 3)(x 2 – 8x + 9)
4 3 9x
4 – y
4 ) – (y
2 – (y + 3)]
16
2.4 (3 )
1. = ( × ) + x – a a x P(x) 2. 6 x
3 – x
2 – 8x + 12
12 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12 -12 x P(x) 0 1, -1, 2, -2, ... 0 x
3 – x
2 – 8x + 12
12 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12 -12
P(1) = 1 3 – 1
P(-1) = (-1) 3 – (-1)
P(2) = 2 3 – 2
P(-2) = (-2) 3 – (-2)
P(3) = 3 3 – 3
2 – 8(-3) + 12 = (-27) – 9 + 24 + 12 = 0
x – 2 x + 3 x 3 – x
2 – 8x + 12
x 3 – x
2 – 8x + 12 x
2 + x – 6
2 – 8x + 12 = (x – 2)(x + 3)(x – 2)
= (x – 2) 2 (x + 3)
x P(x) = x 3 – x
2 – 8x + 12
3. 2.4 9 x
4 + 3x
3 – 27x – 81
P(3) = 3 4 + 3(3)
3 – 27(3) – 81
3 – 27x – 81
3 + 6x
2 + 18x + 27
Q(-3) = (-3) 3 + 6(-3)
2 + 18(-3) + 27
2 + 18x + 27
2 + 3x + 9)
2 + 3x + 9)
18
5. “”
2.1
1. (x + 3 )(x – 3 ) 2. (x + 7 )(x – 7 )
3. ( 52 + x)( 52 – x) 4. ( 23 + x)( 23 – x)
5.
5
9. ( x7 + 62 )( x7 – 62 ) 10. (x – 1 + 6 )(x – 1 – 6 )
11. (x + 3 + 10 )(x + 3 – 10 ) 12. (x – 2 + 33 )(x – 2 – 33 )
13. ( 25 + x – 4)( 25 – x + 4) 14. ( 24 + x + 5)( 24 – x – 5)
15. (2x + 3 + 62 )(2x + 3 – 62 ) 16. (3x – 2 + 132 )(3x – 2 – 132 )
17. (5x – 1 + 34 )(5x – 1 – 34 ) 18. ( 26 + 4x + 3)( 26 – 4x – 3)
2.2
1. (x + 14)(x + 10) 2. (x + 33)(x – 17)
3. (x – 13)(x – 15) 4. (x + 5)(x – 31)
5. (x + 4 + 6 )(x + 4 – 6 ) 6. (x + 1 + 6 )(x + 1 – 6 )
7. (x – 3 + 7 )(x – 3 – 7 ) 8. (x – 1 + 11 )(x – 1 – 11 )
9. (x + 5 + 62 )(x + 5 – 62 ) 10.
− −
1. (x + 7)(3x – 2) 2. (11x + 1)(x – 13)
3. (x – 5)(15x – 2) 4. -2(x + 3 + 11 )(x + 3 – 11 )
5. -3(x – 4 + 21 )(x – 4 – 21 ) 6.
+ +
1. (x – 5 ) 2 2.
2
2
2
3
1. (x + 3)(x 2 – 3x + 9) 2. (y + 4)(y
2 – 4y + 16)
2 – 20z + 25)
2 ) 6. (x + 7)(x
2 – 77x + 151)
2 + 6z + 36)
11. (5y – 4)(25y 2 + 20y + 16) 12. (10 – 6x)(100 + 60x + 36x
2 )
2 ) 14. (4x – 2)(16x
2 – 383x + 379)
1. (x 2 + 25)(x + 5)(x – 5) 2. (9y
2 + 25)(3y + 5)(3y – 5)
3. (9x 2 + 16y
2 + x + 2)(x
2 – x + 2)
2 + 4x + 8)(x
2 – 4x + 8)
2 – 6y + 18) 8. (y + 1)(y – 1)(y
2 – y + 1)(y
2 + y + 1)
2 – xy + y
2 + y
2. (y + 2)(y – 1)(y 2 + y +1)
3. (z – 4)(z 2 + 5z + 16) 4. (y – 6)(y
2 + 15y + 36)
2 + 4y
7. x(x – 3)(x + 2)(x – 2) 8. (3x 2 + y + 3)(3x
2 – y – 3)
2 – y + 3)(3x
2 – y – 3)
2 ) 12. (x
2 – a + y – b)(2x
2 – a – y + b)
2.4
1) 121 2) -60
3) 85 4) 0
5) 14 6) 0
3. x 3 – 2x
2 – 2x + 121 x + 2
0 x + 2 x 3 – 2x
2 – 2x + 12
2 – 2x + 12
P(-2) P(x) x + 2
P(-2) = (-2) 3 – 2(-2)
x + 2 x 3 – 2x
2 – 2x + 12
4. x 4 – 23x
2 + 18x + 40 x – 4
0 x – 4 x 4 – 23x
2 + 18x + 40
2 + 18x + 40
P(4) P(x) x – 4
P(4) = 4 4 – 23(4)
x – 4 x 4 – 23x
2 + 18x + 40
5.
1) (x – 1)(x – 3)(x – 4) 2) (x + 2)(x 2 – 4x + 6)
3) (x – 5)(x + 2)(x + 3) 4) (x – 1) 2 (x + 6)
5) (x + 2)(x + 4)(x – 4) 6) (x – 4)(x 2 + 3x + 1)
7) (x – 2) 2 (x + 3)
2 8) (x – 4)(x + 5)(x – 3)
2
9) (x – 5)(x + 5)(x + 3)(x – 3) 10) (x + 2)(x + 1)(x – 3)(x + 4)(x – 4)
“ k ”
1. -6 2. 59
3. -27 4. -6
1. x 2 + 2x – 3 = (x – 1)(x + 3)
2. x 2 + 4x + 1 = (x + 2 – 3 )(x + 2 + 3 )
3. 81x 2 – 169 = (9x + 13)(9x – 13)
4. 27x 3 – 1 = (3x – 1)(9x
2 + 3x + 1)
2 + 4x + 8)(x
2 – 4x + 8)
2
2 (x – 1)
2 + x [ x(3x + 1) ]
3. 2x(x – 3) + 3(x – 3) [ (x – 3)(2x + 3) ]
4. 3(2x – 1) 2 + 4(2x – 1) [ (2x – 1)(6x + 1) ]
5. x 2 + 4x – 5 [ (x + 5)(x – 1) ]
6. 2x 2 – 5x + 3 [ (2x – 3)(x – 1) ]
7. a 2 – a – 2 [ (a – 2)(a + 1) ]
8. a 2 + 6a + 9 [ (a + 3)
2 ]
2 ]
2 ]
12. p 2 – 1 [ (p – 1)(p + 1) ]
13. 4x 2 – 3
2 [ (2x + 3)(2x – 3) ]
17. (2x – 1) 2 – 4 [ (2x – 3)(2x + 1) ]
18. (x – 3) 2 – y
2 [ (x – y – 3)(x + y – 3) ]
2.2
1. x
2 – 8x – 16 = x
2 + 2(4x)(3) + 3
2 + 2(a)(3) + 3
2 – 2(y)(5) + 5
2 – 2(x)(9) + 9
2 + 2(2x)(2) + 2
2 – 2(2x)(5) + 5
2 – 2(10x) + 1
26
2. x
2 + 6x + 9 = (x + 3)
2
2
2 ]
2 ]
2 ]
2 ]
2 ]
2 ]
2.3
a 3 + b
3 = (a + b)(a
2 – ab + b
3 )
1. 5 1 a 2 b ( b ∠ a)
3 b b a 4 b a – b a
5 a – b a a
28
2. 1 .................................... [ a
3 ]
5 .................................... [ a 2 (a – b) ]
3. 3 4 5 a 1 []
4. 1 2 [ a 3 + b
3 ]
5. 2 3
a
5
b
29
6. 4 5
7. [a 3 + b
3 ]
30
8. 4 6 7 4 7 a
3 + b
2 )
2 3 = b 2 (a + b)
4 5 = a(a – b)(a + b) 3 3 4 5 1
2 3 4 5 2 1
b 2 (a + b) + a(a – b)(a + b) = b
3 + a
3 = (a + b)[b
2 )
a
2.3
a 3 – b
3 = (a – b)(a
2 + ab + b
2 )
1. 5 1 a 2 b b < a
3 a a – b 4 a – b a b
5 b a – b b
2. 1 .................................... [ a
3 ]
3 .................................... [ a 2 (a – b) ]
4 .................................... [ ab(a – b) ]
5 .................................... [ b 2 (a – b) ]
a
a
a
1
32
3. 2 3 4 5 1
[]
4. 2 a
3 – b
3 []
5. 3 4 5
[ a 2 (a – b) + ab(a – b) + b
2 (a – b) ]
6. 5 [ (a – b)(a
2 + ab + b
2 ) ]
7. 4 6 [ a 3 – b
3 = (a – b)(a
2 + ab + b
3.1 (2 )
3.2 (6 )
3.3 (4 )
ax 2 + bx + c = 0 a, b, c
a ≠ 0 ax 2 + bx + c
2 2 ax
2 + bx + c = 0 a, b, c a ≠
0 x = 2a
4ac b b- 2 −±
3.1 (2 )
ax 2 + bx + c = 0 a, b, c
a ≠ 0
1. 2. 1, 2, 3 4 3
3. 5, 6 7
3.2 (6 )
ax 2 + bx + c = 0 a, b, c
a ≠ 0 x = 2a
4ac b b 2 −− ±
3.2 3.2
35
2. 5 ax
2 + bx + c = 0 a ≠ 1
x 2 1
a 3. 3.2 ax
2 + bx + c = 0 a, b, c a ≠ 0
x = 2a
4ac b b 2 −− ±
b 2 – 4ac > 0
ax 2 + bx + c = 0 b
2 – 4ac
3 b 2 – 4ac < 0
4. “” b
2 – 4ac = 0
k
5. “ ” 3.2 6. 3.2 “” ax
2 + bx + c
24x
2 + 7x – 5
1. 2.
3.3
1. 1 x x
2 – 22x + 105 (x – 15)(x – 7) x
2. 3.3 3.3 3. “ ” 1 2 “” 121 – 122
ABCD x E AB E EC AB F
37
AFHD BC = a BE = a
2
2 = BC
2 + BE
5
EF = a 2
AF = 2
2
5
= 2
a)51( +
AFHD AF : AD
AD
51 + 1.618
A E B
3.1
5
5) 3
15) 3
3) 6 -6 4)
5) 2 + 60 2 – 60 6)
7) 2
3
15) 16) 9
2
1. 103 + 103 − 2. 104- + 104- −
3. 4. 2
293- +
7. 3
72 +
9. 2
13. 3
4
302- +
1. -12 12
1.
40
9) 10) 2.
1) 1 11 2) -2 5
3) -2 + 3 -2 – 3 4)
5) 2
3) 4
331 +
5) 0 4
10 -2
1. a
b - 2.
3.3
41
AC = 13 3. BC = 8 4. 25 27
5. 8 15 6 6. 4 7. 5 11
3.2
“ ” 82
r 1 r 2 ax 2 + bx + c = 0 a, b, c
a ≠ 0
r 1 + r
k
1) kx 2 – 2x + 7 = 0
5
2
2) 4x 2 – 11x + (k – 3) = 0
2
3
1) kx 2 – 2x + 7 = 0
r 1 + r 2 = 5
2
5
21
r r ⋅ = 2
9
44
p
1. 3px 2 – 5x – p = 0 5
1
3
3
4
3 -
2
3. 5m 2 – (p + 1)m – 2 = 0
5
4. 6x 2 + x – 3p = 0
2
5. px 2 – 6x – 1 = 0
5
3.2
ax
2 + bx + c
ax 2 + bx + c = 0 a, b, c
a ≠ 0
x = 2a
b 2 – 4ac > 0
ax 2 + bx + c
a, b, c b 2 – 4ac > 0 a ≠ 0 r
1 =
2a
4ac b b- 2 −+
r 2 = 2a
4ac b b- 2 −−
ax 2 + bx + c ax
2 + bx + c = a(x – r 1)(x – r 2)
ax
2 + bx + c a, b, c b
2 – 4ac > 0
a ≠ 0
ax 2 + bx + c =
ax 2 + bx + c =
−
−+
ax
2 + bx + c = a(x – r 1)(x – r 2) r 1 r 2 ax
2 + bx + c = 0
2 – 3x + 1
1) 2x
2 – 3x + 1
x = 2(2)
2 =
2
1
x = 2a
4ac b b- 2 −±
x = 2(30)
3
x = 2a
4ac b b- 2 −±
x = 2(3)
3
2 =
3
+
2 + 11x – 6 [(x + 6)(2x – 1)]
2. 5x 2 – 8x – 4 [(5x + 2)(x – 2)]
3. 3m 2 – 4m – 1 3
2 7 2 7 x x
3 3
3 14 3 14 5 x x
5 5
8 3 −+
9 33 9 33 2 x x
4 4
6 33 6 33 3 x x
3 3
1 2 1 2 25 x x
5 5
3.3
1. 5
1 + +
, 2 2
2. 3 120 [ 17.06 ]
3. 7 2 [ 1.26 ]
4. 74 10 205 [ 1.85 ]
1 0
4.1 (2 )
4.2 y = ax 2 a ≠ 0 (2 )
4.3 y = ax 2 + k a ≠ 0 (2 )
4.4 y = a(x – h) 2 + k a ≠ 0 (3 )
4.5 y = ax 2 + bx + c a ≠ 0 (3 )
y = ax
2 + bx + c x, y a, b, c a ≠ 0
y = ax
2 + bx + c
4.1 (2 )
1. 94 95 2 2
y = ax
2 + bx + c x, y a, b, c
a ≠ 0
2. “” y = ax 2 + bx + c
a ≠ 0 a = 0 3. “” 1 5) 6) y = ax
2 + bx + c a, b c
2y = 3x – x 2 – 5
y =
52
4.2 y = ax 2 a 0 (2 )
1. y = ax 2 a ≠ 0
2. y = ax 2 a ≠ 0
3. y y = ax 2 a ≠ 0
4. y = ax 2 a > 0 a < 0
1. y = ax
2 a ≠ 0
a ≠ 0 2 a > 0 a < 0
2. “ y = x 2 ”
y = ax 2 a = 1
y y x > 0 x 1 y x < 0 x 1 y y = x
2
y = ax 2 a > 0
3. “ y = ax 2 a > 0”
y = ax 2 a > 0 a
y = ax 2 a > 0 a y = ax
2
a > 0 4. “ y = -x
2 ” “ y = ax
2 a < 0”
y = ax
2 a > 0 y = ax
2 a < 0
53
5. “” y = ax
2 y = -ax
2 a > 0 x
2
X y = 3x
2 y = -3x
2
y = 2x 2
y = -2x 2
6. 4.2 1 x y x 0 y x 1 -1 y y
X Y x y x x y 6
4.3 y = ax 2 + k a 0 (2 )
1. y = ax 2 + k a ≠ 0
2. y = ax 2 + k a ≠ 0
2 + k
a ≠ 0 y = ax 2 a ≠ 0 y = ax
2
y = ax 2 + k k = 0
y = ax 2 + k y = ax
2 Y
a y = ax
2 + k k
k > 0 k < 0
2. “ y = ax 2 + k, a > 0” “ y = ax
2 + k, a < 0”
Y y = 2x
2 Y y = 2x
2 + 2 y
y = ax
2 + k a ≠ 0
3. 1 2 1 4 x y x Y x 4. 4.3 1 X Y
2 y = ax
2 + k a ≠ 0
55
4.4 y = a(x – h) 2 + k a 0 (3 )
1. y = a(x – h) 2 + k a ≠ 0
2. y = a(x – h) 2 + k
a ≠ 0 3. y y = a(x – h)
2 + k a ≠ 0
1. 4.3 y = a(x – h)
2 + k a ≠ 0 y = ax
2 + k a ≠ 0
y = a(x – 0) 2 + k h = 0 h ≠ 0
2. “ y = a(x – h)
2 ”
h ≠ 0 y = 2x
2 y = 2(x – 0) 2 y = 2(x – 1)
2
y = 2(x + 1) 2 y = 2x
2
X h y = a(x – h) 2 a ≠ 0
3. 1 2 y = a(x – h)
2 a ≠ 0 h ≠ 0
1 x 4. y = a(x – h)
2 + k a ≠ 0 h ≠ 0 k ≠ 0
y = ax 2 + k y = a(x – h)
2
y = a(x – h) 2 + k
y = 2(x – 1) 2 , y = 2(x – 1)
2 + 2 y = 2(x – 1)
2 – 2
“ y = a(x – h) 2 + k”
y = a(x – h)
2 + k
(-1 2 )
3
5. 4.4 3
4.5 y = ax 2 + bx + c a 0 (3 )
1. y = ax 2 + bx + c a ≠ 0
2. y = ax 2 + bx + c
a ≠ 0 3. y y = ax
2 + bx + c a ≠ 0
4.5
1. 3 y = ax
2 + bx + c
a ≠ 0 y = a(x – h) 2 + k 1 2
1 y = 3x
2 – 6x + 1
= -2(x 2 + 6x) – 17
= -2(x 2 + 6x + 3
2 ) – (-2)(3
2 ) – 17
2. y = ax
2 + bx + c a ≠ 0
y = a(x – h) 2 + k a ≠ 0
-2
-2
57
3. 4.5 2 4) X 4.5
4. “” “” 5. “” “” y = a(x – h)
2 + k a ≠ 0
“”
“”
1.
4) a = 2
5) a = 1 , b = 6 c = 9
6) a = -1, b = -1 c = 4
1 -
2.
1) y = ax 2 + bx + c
a = 1, b = 0 c = 0
2) y = ax 2 + bx + c
a ≠ 0
58
3) y = ax 2 + bx + c
a = 1, b = 2 c = -1
4) y = ax 2 + bx + c
a = 1, b = 2 c = 1
5) y = ax 2 + bx + c
a = -1, b = -2 c = -6
6) y = ax 2 + bx + c
a ≠ 0
1. 2. 16
4. 3 -3
5. x = 0 Y 6. 7. 0
8. 9. 0 x 0
10. y
“ y = ax 2 , a > 0”
1. x = 0 Y
2. (0, 0) y 0
1. 2. -9
3. -9
4. 4 -4
5. x = 0 Y 6. 7. 0
8. 9. 0 x 0
10. y
“ y = ax 2 , a < 0”
1. x = 0 Y
2. (0, 0) y 0
3. a a a
“”
y = 2
x 4
1 9
y = 2
x 2
3 27
y = 2
x 3
2 - -6 -
y = 2
x 3
4 - -12 -
-14
y = 3x 2
y = 21
x 3
1) x = 0 Y
2) (0, 0) 3) 0
3.
y = -4x 2
y = 21
- x 4
-1 - 1
1) x = 0 Y
2) (0, 0) 3) 0
4.
y = 25
x 2
10 5
1) x = 0 Y
2) (0, 0) 3) 0 5.
1) a a > 0
2) x = 0 Y
3) (0, 0) 6.
1) a a < 0
2) x = 0 Y
3) (0, 0)
Y
“ y = ax 2 + k, a > 0”
1. 2. x = 0 Y
3. y = 2x 2 + 2 (0, 2) y 2
y = 2x 2 (0, 0) y 0
y = 2x 2 – 2 (0, -2) y -2
4. y = 2x 2 + 2 X
y = 2x 2 – 2 X
5. y = 2x 2 + 2 y = 2x
2
Y X 2 y = 2x 2 – 2
y = 2x 2 Y X 2
“ y = ax 2 + k, a < 0”
1. 2. x = 0 Y
3. y = -2x 2 + 2 (0, 2) y 2
y = -2x 2 (0, 0) y 0
y = -2x 2 + 2 (0, -2) y -2
4. y = -2x 2 + 2 X
y = -2x 2 – 2 X
5. y = -2x 2 + 2 y = -2x
2
Y X 2 y = -2x 2 – 2
y = -2x 2 Y X 2
4.3
1) y = 5x 2 + 4
65
3. Y 4. y = 5x
2 + 4
x 0 1 2
y = 5x 2 + 4 4 9 24
2) y = -3x 2 – 2
1. 2. (0, -2)
3. Y 4. y = -3x
2 – 2
x 0 1 2
X
2 3
+x
1. 2. (0, 2)
3. Y 4.
y = - 21
-2
-4
-6
-8
0
Y
- 1 4
x
1. 2. (0, -1)
3. Y 4.
y = 21
3
2 – 5
c 4 y = 1x
4
1 -
2 +
Y
X
2
1. 2. y = 2(x + 1)
2 x = -1
y = 2x 2 x = 0
y = 2(x – 1) 2 x = 1
3. y = 2(x + 1) 2 (0, -1)
y = 2x 2 (0, 0)
y = 2(x – 1) 2 (0, 1)
4. y = 2(x – 1) 2 Y
5. y = 2(x + 1) 2 Y
6. y = 2(x – 1) 2 y = 2x
2
X 1 y = 2(x + 1)
2 y = 2x
2
X 1 7. y = -2x
2 (0, 0)
y = -2(x – 1) 2 (0, 1)
y = -2(x + 1) 2 (0, -1)
8. y = -2(x – 1) 2 y = -2x
2
X 1 y = -2(x + 1)
2 y = -2x
2
4.4
1. 2. (-1, 0)
x -1 0 1
y = (x + 1) 2
1. 2. (1, 0)
3. x = 1 4. y = -3(x – 1)
2
1. 2. (-2, 0)
3. x = -2 4. y = -4(x + 2)
2
Y
1. 2. (3, 0)
3. x = 3 4. y = (x – 3)
2
2
2
“ y = a(x – h) 2 + k”
1. 2. x = 1 3. y = 2(x – 1)
2 + 2 (1, 2)
y = 2(x – 1) 2 (1, 0)
y = 2(x – 1) 2 – 2 (1, -2)
Y
2
6
4
72
4. y = 2(x – 1) 2 + 2 X
y = 2(x – 1) 2 – 2 X
5. y = a(x – h) 2 + k y = a(x – h)
2
Y X k k > 0 X k k < 0
6. y = -2(x – 3) 2 (3, 0)
y = -2(x – 3) 2 + 2 (3, 2)
y = -2(x – 3) 2 – 2 (3, -2)
7. y = -2(x – 3) 2 + 2
y = -2(x – 3) 2 Y X 2
y = -2(x – 3) 2 – 2
y = -2(x – 3) 2 Y X 2
4.4
3 (x – 1)
2 – 2
1. 2. (1, -2)
3. x = 1 4.
y = 1
2) y = -(x + 1) 2 – 3
1. 2. (-1, -3)
3. x = -1 4. y = -(x + 1)
2 – 3
x -1 0 1
X
2
3) y = -(x + 1) 2 + 3
1. 2. (-1, 3)
3. x = -1 4. y = -(x + 1)
2 + 3
x -1 0 1
4) y = 1
5 (x + 2)
2 + 2
1. 2. (-2, 2)
3. x = -2 4.
y = 1
2 – 1
2
2
3.
1) y = 2 2) y = -1
3) y = (x – 2) 2 – 5
y = (x – 2) 2 x = 2 5
y = (x – 2) 2
y = (x – 2) 2 – 5 x = 2 5
4) Y = 0 y = (x + 4)
2 + 1
y = (x – 4) 2 + 1 y = 1 8
y = (x – 4) 2 + 1
y = (x + 4) 2 + 1 y = 1 8
4.5
y = (x + 3) 2 – 1
y = (x + 3) 2 – 1
2
6
4
76
1. 2. (-3, -1)
3. x = -3 4. y = x
2 + 6x + 8
x -3 -2 -1
2) y = -x 2 – 4x – 2
y = -(x + 2) 2 + 2
y = -(x + 2) 2 + 2
1. 2. (-2, 2)
3. x = -2 4. y = -x
2 – 4x – 2
x -2 -1 0
2
4
6
8
-2
77
2.
y = 2x 2 + 5x – 2 y =
2
5 -
− 0 ,
4
415-
y = -x 2 + 6x – 4 y = -(x – 3)
2 + 5
1) 2) (3, 5)
3) x = 3 4) X ( 3 20 , 0+ ) ( 3 20 , 0− )
Y
“ ”
1. 8 64 2. 63 3. 3.1 12.9
1. 5 2. 5 50 3. 10.27
“”
1. 25 × 25 2
2. 15.5 × 15.5 2
3. p
4 ×
p
4.5
X
y = 0 X 1. y = 2x
2 – 4x y
2x 2 – 4x = 0 [ y = 0]
2. 2x 2 – 4x = 0 [0 2]
3. y = 0 X []
4. y = 2x 2 – 4x y = 0 X
1) [(0, 0) (2, 0)]
2) x [0 2]
2 4 6-2-4-6
81
3) x 2) 2x 2 – 4x = 0
[]
4) 2x 2 – 4x = 0
y = 2x 2 + 4x X []
5. ax
2 + bx + c = 0 a, b c a ≠ 0 x
y = ax 2 + bx + c a, b c a ≠ 0 y = 0
X
6. X
y = x
2 – 2x – 3 X
x 2 – 2x – 3 = 0 2 -1 3
y = -x 2 – 4x – 4 X
-x 2 – 4x – 4 = 0 -2
y = x 2 – 4x + 7 X x
2 – 4x + 7 = 0
-6
Y
82
7.
1) -x 2 + x + 6 = 0
[-2 3]
[ 3]
[]
8.
1) 2x 2 – 4 = 0
[ 2 - 2 ]
[5]
[]
[]
y = 2x 2 – 4
(12 )
5.1 (6 )
5.2 (6 )
5.1 (6 )
1. 2.
5.1
1. 2. “” 3. “” 4 1) E
ABCD E A, B, C D
[]
2) AX = BX BY = CY [ ]
A B
D C
85
3) EX ⊥ AB EY ⊥ BC [ ]
4) EXBY [ ]
5) EX = YB EY = XB [ ]
EX EY OX OY
4. “ ” “ ” 2 5. “ ”
O AB
1) BOA ∧
]
2) ACB ADB [ ]
A B O
86
3) ACB O [1 : 2]
4) O [1 : 2]
5) OACB O ACB O []
BOA ∧
= 90 o
OACB O ACB O “ ”
6. 3 “ ”
π
7. 5.1 π 8. 5.1 9. 2 “ ”
D
1. 2.
5.2
1. 2. 2 4 5.2 3. 5.2 9 10 4. “” 2
3
2
3
1
88
7. “” “ ” 8. “” 5.2 9. “”
“”
1. 2. 3. ...
“”
1. 2. OX OY 3. AB BC
4. BX BY
5.
“ ”
1. 2. 196 3. 24 4. 5. 672 6. 868
“ ”
1. 8 la 2
1
5.1
5.1
1. 204.1 ( π ≈ 3.14)
2. 42,743
≈
3. 75.36
37.68 ( π ≈ 3.14)
4. 942 ( π ≈ 3.14)
5. 378 6. 3.5 6
≈
3
2
2 l ≈ 14
≈ 7
≈ 7
1.
≈
≈
3) 233.15 ( π ≈ 3.14)
4) 550
≈
22 π
2. 72π 226.08 ( π ≈ 3.14)
72 2 π 319.68 ( π ≈ 3.14
2 ≈ 1.414)
1) 39,689.6 ( π ≈ 3.14)
2) 1.25 × 10 8 ( π ≈ 3.14)
3) 000,10
41
5. 943.4 ( π ≈ 3.14)
6. 30 ( π ≈ 3.14)
1,413
π 2 l = 1413
2πr 2
2πr 2
r ≈ 15
30
5.2
1. 8 2. 2 : 1
3. l
2πr = l r
r = π2
l
πr 2 = π
2
π2
l
4
l
4
l ×
4
l
= 16
2 l
4. 10 5. 749.66
6. 666.85
93
8. 39,800 ( π ≈ 3.14)
6 × 2
a 4
πr 2
= π(1.5) 2
3 6
≈ 166.27 – 7.07
≈ 159.2 159.2 × 250
≈ 39,800 9. 3 2,112
8
7 × 2,112 = 1,848
2
28 = 14
7
22 ×14 2 ≈ 616
616
848,1 = 3
3 10. 2
3
4 π(7 3 )
x 2
3
4 π(2 3 – x
49 × 3
4 π(2 3 – x
“”
1. 2πr 3
6. 6πr 2
7. 4πr 2
3
4 π
2. 2
3
2 π
3. 2 : 1
5. 3
“”
1. 0.00072 720 2. 30 3. 9.36 4. 71.6 5. 6. 0.68 × 10
3
7. 8. 1.5
“”
6.344 × 10 7
97.6% 13
100
100
6.97 × 13 × 10
≈ 6.344 × 10 7
“ ”
47,0255.9 180 × (37 × 10
3 ) × (77 × 10
≈ 51,282 × 10 7
0.917 0.917 × 51,282 × 10
7
=
96
1 × 10 3
1 × 47025.59 × 10 10
1. 1 : 7
2. 1,469.52 3. 1 : 7
1. 0.5 3 2. ( ) 1 3.
1 2 ( )
1. 1 10 []
2. ( )
[ ] 3. 2 [ r π r
2
]
4.
πr 2
5. 3 4 4
[2π r 2]
6. 5
[ 2 × 2π r 2
= 4π r 2]
5.2
178
AFC ∼ AGE
a 3
3
= h 3
1 (a
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