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指数函数及性质

一、创设情境,导入新知
问题
问题1 问题2

对应关系

定义域
x

y ? 1.073
?1 P?? ? ?2?

x ? ? * ,x ? 20

t ? 5730

t?0

思考: 这两个解析式有什么共同特征?

共同特征:
两个解析式都具有 y ? a
x

的形式.

二、新课讲解
指数函数的定义:

形如 y=a (a?0,且a ?1)的函数 叫做指数函数,其中x是自变量 .函数的 定义域是R . 思考:为何规定a?0,且a?1?

x

随堂练习:下列函数中,哪些是指数函数?

(1)y ? 4

x
4

( 2) y ( 4) y

? ?4 ?4

x

(3) y ? x
(5) y

x ?1

? 4 +1 (6) y ? 2
x

2x

y ? a x的结构特点: 指数函数解析式
(1)底数:大于0且不等于1的常数 (2)指数:自变量x (3)幂系数:1

研究指数函数性质
根据前面所学的函数知识,我们想了解 函数,需研究函数的什么内容? 观察图像
定义域 值域

研究性质

单调性
奇偶性

……

x
y ? 2x
?1? y ?? ? ?2?
x

… … …

-3
1 8

-2
1 4

-1
1 2

0 1 1
y

1 2
1 2

2 4
1 4

3 8
1 8

… … …

8

4

2
?1? y?? ? ?2?
x

y ? 2x

用描点法画出它们的图象

1

0

1

x

y
1 y ? ( )x a

y ? ax

1

0

1

x

结论:指数函数的底数互为倒数时,图像关于y轴对称。

y

?图像的位置
?1? y?? ? ?2?
x

?1? y?? ? ? 3?

x

y ? 3x

y ? 2x

?图像经过的定点 ?图像的变化趋势

1

0

1

x

y

y

y

?1? y?? ? ? 2 ?? y

x

a

?1? y?? ? x ? 3?

x

y ? 3x

y ? 2x

(a ? 1)

y ? ax
(0 ? a ? 1)

研究性质

1
1 0

1

x

0

1

x

0

x

完成下面表格,两函数的共性和不同分别是什么
函 数

y ? a (a ? 1)
x

y ? a (0 ? a ? 1)
x




R

定义域 值 域 单调性 过定点 函数值变 化情况

R

R

(0,+∞)
在R上是增函数 (0,1) x > 0时,y > 1 x < 0时,0< y <1

(0,+∞)

(0,+∞)
在R上是减函数 (0,1)

(0,1)

x > 0时,0< y <1 x < 0时,y > 1

左右无限上冲天, 永与横轴不沾边.

大 1 增,小 1 减, 图象恒过(0,1)点.

学以致用
例7 比较下列各题中两个值的大小: 同底比较大小 不同底比较大小

(1)1.72.5 , 1.73 (2)0.8-0.1 , 0.8-0.2

(3)1.70.3 , 0.93.1


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