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【332257】1.2反比例函数的图象与性质

时间:2025-01-21 12:31:20 作者: 字数:2779字
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第一章 反比例函数

1.2 反比例函数的图象与性质


基础导练

1.已知点(11)在反比例函数y= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> (k为常数,k≠0)的图象上,则这个反比例函数的大致图象是( )

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a>

2.若反比例函数y= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> 的图象经过第二、四象限,则k的取值范围是( )

A.k <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> B.k <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> C.k= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> D.不存在

3.若点A(1y1)B(2y2)都在反比例函数y= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> (k0)的图象上,则y1,y2的大小关系为( )

A.y1y2 B.y1≤y2 C.y1y2 D.y1≥y2

4.如图,直线x=2与反比例函数y= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> y=- <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> 的图象分别交于AB两点,若点Py轴上任意一点,则△PAB的面积是 .

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a>

  1. 如图,反比例函数y= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> 的图象经过点P,则k= .

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a>

  1. 已知反比例函数y= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> ,当x>0时,y值随x值的增大而减小,则k取值范围是 (写出满足条件的一个值即可).


能力提升

7.如图是反比例函数y= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> 的图象的一支,根据图象回答下列问题:

(1)图象的另一支在哪个象限?常数m的取值范围是什么?

(2)在这个函数图象的某一支上任取点A(ab)B(a′b′),如果a>a′,那么bb′有怎样的大小关系?

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a>






8.如图,已知一次函数y=k x+b的图象与反比例函数y=- <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> 的图象交于AB两点,且点A的横坐标与点B的纵坐标都是-2.求:

(1)一次函数的解析式;

(2)△AOB的面积.

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a>







9.如图,一次函数y1=x1的图象与反比例函数y2= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> (k为常数,且k≠0)的图象都经过点A(m2).

(1)A点的坐标及反比例函数的表达式;

(2)结合图象直接比较:当x0时,y1y2的大小.

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a>











参考答案


1.C 2.B 3.C 4. 1.5 5.-6 6.大于2

7.解:(1)另一支在第三象限.由题意可知,m-5>0,解得m>5.

(2)由图象可知,在每一象限内,函数值随自变量的增大而减小,所以a>a′时,b<b′.

8.解:(1)x A=-2y B=-2代入y=- <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> 中,得到y A=4x B=4

所以A(-24)B(4-2).把这两个点分别代入y=k x+b,得

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> 解得 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a>

所以一次函数的解析式为:y=-x+2.

(2)一次函数的解析式y=-x+2y轴的交点C的坐标为(02).

所以S△AOC= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> OC|x A|= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> ×2×2=2S△BOC= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> OC|x B|= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> ×2×4=4.

所以AOB的面积=S△AOC+S△BOC=6.

9.解:(1)因为一次函数y1=x1的图象经过点A(m2)所以2=m1.解得m=1.

所以A的坐标为A(12).

因为反比例函数y2= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> 的图象经过点A(12)所以2= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> .解得k=2.

所以反比例函数的表达式为y2= <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/411/" title="反比例" class="c1" target="_blank">反比例</a> <a href="/tags/885/" title="函数" class="c1" target="_blank">函数</a> <a href="/tags/898/" title="图象" class="c1" target="_blank">图象</a> .

(2)由图象得:当0x1时,y1y2;当x=1时,y1=y2;当x1时,y1y2.