专题2.3幂函数与二次函数七大题型（原卷版）_11.4-2_2026-11-04

      
      
专题2.3 幂函数与二次函数【七大题型】

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 !"
Þ 1I23-24 Œß·àá·âãHäOßå
W
 WI    O
A´𝑦=
1
𝑥3
B´𝑦= 2𝑥
C´𝑦= 2𝑥2
D´𝑦= −𝑥−1
æ5 1-1I23-24 Œ™·çèéêëì·í]OßårÕî– WI    O
A´
 ()Œ!ïð<
B´𝛼= 1,3,
1
2¥D
𝑦= 𝑥𝛼Wñ
C´
 ()òxy[Tó)ô
D´𝑦= 2𝑥2õW12
DöW
æ5 1-2I23-24 Œ™·÷øùú·í]Oßå
W
û[(−∞,0)Wñ
 WI    O
A´𝑦=
1
𝑥
B´𝑦= 𝑥3 +1
C´𝑦= 𝑥−2
D´𝑦=
|𝑥|
æ5 1-3I23-24 Œ™·üéýþ·í]Oy‹ßå
①𝑦= 𝑥3€②𝑦= 4𝑥2€③𝑦= 𝑥5 +1€④𝑦=
(𝑥−1)2€⑤𝑦= 𝑥D]
 
ŸI    O
      
      
A´4
B´3
C´2
D´1
 2  #$% &'
Þ 2I2023·™ÿĀā·ŒĂO¶ƒ𝑎D𝑏∈RDă“𝑎> 𝑏”W“𝑎3 > 𝑏3” I    O.
A´ĄBą¦=’“
B´¦=ąĄB’“
C´Ą=’“
D´õąĄBĆą¦=’“
æ5 2-1I2024·ćĈ·ĂĉĊċO¶ƒ𝑎= log510D𝑏= log48D𝑐= 4𝑏−7
6Dă a;b;c &'̟
I    O
A´𝑎> 𝑏> 𝑐
B´𝑏> 𝑐> 𝑎
C´𝑐> 𝑏> 𝑎
D´𝑐> 𝑎> 𝑏
æ5 2-2I2024·QéČč·ĂĉĊċO¶ƒ
𝑓(𝑥) = (𝑚−1)𝑥𝑛 ()ï<(𝑚,8)´Ď𝑎= 𝑓(20.3)D
𝑏= 𝑓(0.32)D𝑐= 𝑓(log20.3)Dă𝑎D𝑏D𝑐 &'ÌWI    O
A´𝑏< 𝑐< 𝑎
B´𝑎< 𝑐< 𝑏
C´𝑎< 𝑏< 𝑐
D´𝑐< 𝑏< 𝑎
æ5 2-3I2023·àáďĐ·ĂĉĊċO¶ƒ𝑓(𝑥)Ÿđ
Dª0 ≤𝑥≤2¥D𝑓(𝑥) = 2𝑥−𝑥2Dª𝑥> 2¥D
𝑓(𝑥) = |𝑥−3|−1DăI    O
A´−𝑓(−26) > 𝑓(20.3) > 𝑓(30.3)
B´𝑓(20.3) > 𝑓(30.3) > −𝑓(−26)
C´−𝑓(−26) > 𝑓(30.3) > 𝑓(20.3)
D´𝑓(30.3) > 𝑓(20.3) > −𝑓(−26)
 3  
 ()*+, -./
Þ 3I2024·àèĒþ·ĂĉĊċOēÀ
𝑓(𝑥) = 𝑥𝛼ª𝛼= 2,3,
1
2,−1¥ +,DĔĕ
[!"ĖeŸ
đ
Dû[(0, + ∞)™JKėñDă𝛼= I    O
A´2
B´3
C´
1
2
D´-1
æ5 3-1I2023·óĘèĄ·ĂĉĊċO¶ƒ
𝑓(𝑥) = 𝑥
𝑚
𝑛(𝑚,𝑛∈Z)Dßåę9Ÿ“𝑓(𝑥)W R ™ Ě

” ĄB’“ WI    O
A´𝑚= −3,𝑛= 1
B´𝑚= 1,𝑛= 2
C´𝑚= 2,𝑛= 3
D´𝑚= 1,𝑛= 3
æ5 3-2I23-24 ·™·™ÿāø·âãHäOě(ĜĝW
𝑦= 𝑥
𝑚
𝑛I𝑚,𝑛ĞŸîğ
û𝑚,𝑛Ġ,O 
()DăI    O
      
      
A´𝑚,𝑛Wđ
û
𝑚
𝑛< 1
B´𝑚WĚ
D𝑛Wđ
Dû
𝑚
𝑛< 1
C´𝑚WĚ
D𝑛Wđ
Dû
𝑚
𝑛> 1
D´𝑚,𝑛Wđ
Dû
𝑚
𝑛> 1
æ5 3-3I2023·÷øġĢ··ĂO¶ƒ
𝑓(𝑥) = 𝑥3 + (𝑎−2)𝑥2 +2𝑥+ 𝑏[[−2𝑐−1,𝑐+ 3]™Ÿđ
D
ăuN5𝑓(2𝑥+ 1) + 𝑓(𝑎+ 𝑏+ 𝑐) > 0 3ģĤĥI    O
A´(−2,4]
B´(−3,5]
C´ −5
2 ,2
D´(−2,2]
 4  012
 345
Þ 4I23-24 Œ™·Ħáħ!·íĨOĩxŒĪ¥ī‹ßåó+,] ·+, 12
𝑓(𝑥) =
      
´
①𝑓(𝑥) 7'%Ÿ−1€②𝑓(𝑥) Œ2ĬÌ
Ÿ−4€③𝑓(0) = 3€④𝑓(𝑥) = 𝑓(−𝑥+ 2)´
æ5 4-1I2023 ··ćĈ·HäO¶ƒ12
𝑓(𝑥) ½¼<BĭW 0 ± 5D()ÂÃʙDû
𝑓(𝑥)[—˜[−1,4]™ 7&%Ÿ 12Dă
𝑓(𝑥) 345Ÿ      
´
æ5 4-2I23-24 Œ™·ĮįİıIJ·í]O¶ƒ12
𝑓(𝑥) = 𝑎𝑥2 +𝑏𝑥+ 𝑐 (𝑎> 0,𝑎,𝑏,𝑐∈R)D
𝑓(1) = 1DLijĴ𝑥∈RD𝑓(𝑥−2) = 𝑓(−𝑥)Dû𝑓(𝑥) ≥𝑥89:´ă12
𝑓(𝑥) ĵğ345Ÿ      
´
æ5 4-3I23-24 Œ™·ĶQķĸ·ÂĹO¶ƒ12
𝑓(𝑥) = 𝑎𝑥2 +𝑏𝑥+ 𝑐 LMœW𝑥= 1Dûu
N5𝑓(𝑥) ≤2𝑥 3ģŸ[1,3]Dă𝑓(𝑥) 345W𝑓(𝑥) =      
´
 5  12
 ()6
Þ 5I2020·÷ø·>O¶ƒ12
𝑦= 𝑎𝑥2 +𝑏𝑥+ 𝑐 (ĺě(ĜĝDăuN5𝑎𝑥2 +𝑏𝑥+ 𝑐> 0
 3ģWI    O
      
      
A´(−2,1)
B´(−∞,−2) ∪(1, + ∞)
C´[−2,1]
D´(−∞,−2] ∪[1, + ∞)
æ5 5-1I23-24 Œ™·àèĻļ·âãHäOuN5𝑐𝑥2 +𝑎𝑥+ 𝑏> 0 3ģŸ 𝑥∣−1 < 𝑥< 1
2 Dă
𝑦= 𝑎𝑥2−𝑏𝑥−𝑐 ()&ĽŸI    O
A´
  
B´
  
C´
  
D´
  
æ5 5-2I23-24 1ß·áľĿŀ·íĨOĔuN5𝑎𝑥2−𝑥−𝑐> 0 3ģŸ{𝑥|−1 < 𝑥<
1
2}Dă
𝑦= 𝑐𝑥2
−𝑥−𝑎 ()•ŁŸI    O
A´
B´
C´
D´
æ5 5-3I2024 Œ·ćĈ·HäOuN5𝑎𝑥2−𝑏𝑥+𝑐>0 3ģŸ{𝑥|−2< 𝑥<1},ă
𝑦= 𝑎𝑥2 +𝑏𝑥+ 𝑐
 (ĺ&ĽŸI    O
      
      
A´
B´
C´
D´
 6  12
 7%6
Þ 6I23-24 1ß·UVĦé·íĨOßłÅ
𝑓(𝑥) = 𝑥2 +3𝑥+ 4 Ńvî– WI    O
A´𝑓(𝑥) > 089:
B´𝑓(𝑥)7&%W 5
C´𝑓(𝑥)* y œń¾<
D´𝑓(𝑥)Ņ‹7'%
æ5 6-1I2024 ··ćĈ·HäOĎ12
𝑓(𝑥) = (𝑎−2)𝑥2 +3𝑎𝑥+ 2[R™‹7&%D7&%Ÿ
𝑚(𝑎)Dª𝑚(𝑎)Æ7'%¥D𝑎= I    O
A´0
B´1
C´
1
2
D´ 2
æ5 6-2I23-24 Œ™·ZņŇňʼn·âãHäO𝑓(𝑥) = 2017𝑥2−2018𝑥+ 2019 × 2020,𝑥∈[𝑡,𝑡+ 2].ăª
𝑡æ¥D𝑓(𝑥)max−𝑓(𝑥)min 7'%ŸI    O
A´2020
B´2019
C´2018
D´2017
æ5 6-3I21-22 Œ™·ĶQŊŋ·íĨO¶ƒ
𝑓(𝑥) = 𝑎𝑥2 +2𝑥 !"ĖŸ—˜[mDn]D]
𝑎,𝑚,𝑛∈𝑅DĔ fIxO %ĖŸ[-4D4]Dă𝑛−𝑚 Æ%ØÙWI    O
A´[4D4 2]
B´[2 2D8 2]
C´[4D8 2]
D´[4 2D8]
 7  12
 89:6
Þ 7I2024·Ōúō÷·1ĂO¶ƒª𝑥> 0¥DuN5
𝑥2−𝑚𝑥+ 16 > 089:DăŎ
𝑚 Æ%ØÙW
I    O
A´(−8,8)
B´(−∞,8]
C´(−∞,8)
D´(8, + ∞)
æ5 7-1I2023·Ōúō÷·1ĂOĔLijĴ 𝑥∈(0, + ∞),𝑥2−𝑚𝑥+ 1 > 089:Dă m Æ%ØÙW
I    O
A´(−2,2)
B´(2, + ∞)
C´(−∞,2)
D´(−∞,2]
æ5 7-2I2023·Ōú&ŏ·ĂĉĊċOŐ“∃𝑥> 0,𝑎𝑥2 +𝑥+ 1 < 0”ŸőŐDăŐ9: ĄBu¦=
      
      
’“WI    O
A´𝑎≥−
1
4
B´𝑎≥0
C´𝑎≥1
D´𝑎< 1
æ5 7-3I2024·QéŒQ·ĂĉĊċOńÕ𝑥Æœ%¥DuN5𝑥2−2𝑘𝑥+ 4 > 089:Dă𝑘 Æ%ØÙ
WI    O
A´(−∞,−2)
B´(−∞,−4)
C´(−4,4)
D´(−2,2)
Œ;J
1´I2024·ŔøŔŋ·ĂĉĊċOĔ
𝑓(𝑥) = (𝑚2−𝑚−1)𝑥2𝑚−3[(0, + ∞)™JKėñDăŎ
𝑚 %Ÿ
I    O
A´2
B´1
C´−1
D´−2
2´I2023·àèĒþ·ĂĉĊċOě(D¶ƒ
𝑦= 𝑥𝑎,𝑦= 𝑥𝑏,𝑦= 𝑥𝑐[(0, + ∞)™ ()BĭWßŕDŖ
ŗ™ŘDřŚ™ŘDăI    O
A´𝑐< 𝑏< 𝑎
B´𝑎< 𝑐< 𝑏
C´c<a<b
D´𝑎< 𝑏< 𝑐
3´I2023·áľÿś·ŒĂO¶ƒ12
𝑓(𝑥)DLijĴ 𝑥∈𝑅D‹𝑓(2𝑥) < 2𝑓(𝑥)Dă𝑓(𝑥) ()•ęW
I    O
A´
B´
C´
D´
4´I2024·ĶQ·ĂĉĊċOĔuN5𝑘𝑥2 + (𝑘−6)𝑥+ 2 > 0 3ŸćŜŎ
DăŎ
𝑘 Æ%ØÙWI    O
      
      
A´2 ≤𝑘≤18
B´−18 < 𝑘< −2
C´2 < 𝑘< 18
D´0 < 𝑘< 2
5´I23-24 Œ™·ĶQ·JŝċĹOĎ
𝑓(𝑥) = 𝑥2 +2(4−𝑎)𝑥+ 2[—˜(−∞,3]™WŞ
DăŎ
 a 
Æ%ØÙWI    O
A´𝑎≥−7
B´𝑎≥7
C´𝑎≥3
D´𝑎≤−7
6´I2023·óĘşŋ·ŒĂO¶ƒ<(2,
1
8)[
𝑓(𝑥) = 𝑥𝛼 ()™DĎ𝑎= 𝑓(log23)D𝑏= 𝑓(ln3)D𝑐= 𝑓(3−1
2
)Dă aDbDc &'̟IŠŠO
A´bšašc
B´ašbšc
C´bšcša
D´ašcšb
7´I2023·Ħè·ĂĉĊċO¶ƒ
𝑓(𝑥) ()ï 1
2 ,
2
4 D𝑃(𝑥1,𝑦1)D𝑄(𝑥2,𝑦2)I𝑥1 < 𝑥2OW
()™
 ijĴuĪ½<DăßårÕ]î– WI    O
A´𝑥1𝑓(𝑥1) > 𝑥2𝑓(𝑥2)
B´𝑥1𝑓(𝑥2) < 𝑥2𝑓(𝑥1)
C´
𝑓(𝑥1)
𝑥2 >
𝑓(𝑥2)
𝑥1
D´
𝑓(𝑥1)
𝑥1 <
𝑓(𝑥2)
𝑥2
8´I2023·QéèĿ·1ĂO¶ƒ
𝑓(𝑥) = 𝑥3 +𝑎𝑥2 +𝑏𝑥+ 𝑐 ·¼<BĭŸ 1D𝑥1,𝑥2(0 < 𝑥1 < 𝑥2)DĔ

𝑓(𝑥+ 1)Ÿđ
Dă𝑓(2) Æ%ØٟI    O
A´[0,1]
B´(0,1)
C´(0,2)
D´[0,2]
1;Ţ
9´I2024·ćĈ·ĂĉĊċOßå
]õWđ
DöW!"Ė™ Ş
 WI    O
A´𝑓(𝑥) = −3𝑥5
B´𝑓(𝑥) = 2𝑥
C´𝑓(𝑥) =
1
𝑥
D´𝑓(𝑥) = −2𝑥
1
3
10´I2023·QRŏçţ·ĂĉĊċOĔLÅijĴŎ
 xDuN5(𝑎−1)𝑥2−2(𝑎−1)𝑥−4 < 089:DăŎ
 a
•ęWI     O
A´−2
B´0
C´−4
D´1
11´I2024·QRRŋ·ĂĉĊċO¶ƒ
𝑓(𝑥) = 𝑥+ 1DĎ𝑔1(𝑥) = 𝑓(𝑥)D𝑔𝑛(𝑥) = 𝑓(𝑔𝑛−1(𝑥))
(𝑛> 1,𝑛∈N∗)´ûÅ𝑥 
𝑦= 𝑥2 +
𝑛
𝑖=1𝑔𝑖(𝑥)(𝑛∈N∗)´ăI    O
A´𝑔𝑛(𝑥) = 𝑥+ 𝑛»𝑔𝑛(𝑥) = 𝑛𝑥+ 1
B´𝑦= 𝑛2+2𝑛
4
+ 𝑥+ 𝑛
2
2
C´ª𝑛≤2¥DŤ[Å𝑥 
𝑦[—˜(−∞,−1]™ 7'%Ÿ 6D𝑛= 0
      
      
D´ª𝑛> 2¥DŤ[Å𝑥 
𝑦[—˜(−∞,−1]™ 7'%Ÿ 6D𝑛= 4
·;{|
12´I2024·áľťņ·ŒĂO¶ƒ
𝑓(𝑥) = 𝑥𝛼(0 < 𝛼< 1)[—˜(−1,0)™JKėŞDă𝛼 ŒÆ%
Ÿ      
´
13´I2024·óĘČŦ·ĂĉĊċO¶ƒ
𝑦= 𝑎(𝑥−4) + 2(𝑎> 0Dû𝑎≠1) (ĺ8ï!< PDû P [

𝑓(𝑥) (ĺ™Dă𝑓(𝑥) =      
´
14´I2024·Ħè·ĂĉĊċO¶ƒ
𝑓(𝑥) = |𝑥2−6𝑥+ 7|[[1,𝑚](𝑚> 1)™ 7&%Ÿ𝐴D[[𝑚,2𝑚−1]™ 
7&%Ÿ𝐵DĔ𝐴≥2𝐵DăŎ
𝑚 Æ%ØÙW      
´
ó;3ŧ
15´I2024·÷ø·1ĂO¶ƒ𝑓(𝑥)W12
Dû𝑓(1) = 4,𝑓(0) = 1,𝑓(3) = 4´
(1)0𝑓(𝑥) 345€
(2)Ĕ𝑥∈[−1,5]D0
𝑓(𝑥) 7'%±7&%´
16´I2023·÷ø·ŒĂO¶ƒ12
𝑓(𝑥)Ĥĥ𝑓(0) = −1Dº<Ÿ(1,−2).
(1)0
𝑓(𝑥) 345€
(2)Ĕ
𝑓(𝑥)[—˜[𝑎−1,4]™JKėñD0Ŏ
𝑎 Æ%ØÙ.
17´I23-24 Œß·™ÿ·í]O¶ƒ
𝑓(𝑥) = 𝑥𝑚2−2𝑚−3(𝑚∈Z)Ÿđ
Dû[—˜(0, + ∞)™WŨũ
Ş
.
(1)0
𝑦= 𝑓(𝑥) Ūū5€
(2)LijĴŎ
𝑥∈1
2 ,1 DuN5𝑓(𝑥) ≤𝑡+ 4𝑥89:D0Ŏ
 t Æ%ØÙ.
      
      
18´I23-24Œ™·Ōú·âãHäO¶ƒ
𝑓(𝑥) = (𝑚2 + 2𝑚−2)𝑥𝑚2−7I𝑚∈𝑍O !"ĖŸ𝑅Dû[[0, + ∞)
™JKėñ´
(1)0 m %€
(2)∀𝑥∈[1,2]DuN5𝑎𝑓(𝑥)−3𝑥+ 2 > 089:D0Ŏ
 a Æ%ØÙ´
19´I23-24 Œ™·QR·âãHäOĎ
𝑓(𝑥) = 𝑎𝑥2 +(1−𝑎)𝑥+ 𝑎−2.
I1OĔÅ𝑥 uN5𝑓(𝑥) ≥−2‹Ŏ
3D0Ŏ
𝑎 Æ%Øـ
I2OĔuN5𝑓(𝑥) ≥−2LÅŎ
𝑎∈[−1,1]¥89:D0Ŏ
𝑥 Æ%Øـ
I3O3Å𝑥 uN5
𝑓(𝑥) < 𝑎−1,(𝑎∈𝑅).